Solutions to a System of Equations for $C^m$ Functions

TL;DR

This paper develops algorithms to determine when a system of linear equations with semialgebraic function coefficients has a $C^m$ solution, by identifying a finite set of differential operators that characterize solvability.

Contribution

It introduces algorithms for computing differential operators that characterize the existence of $C^m$ solutions to systems with semialgebraic coefficients.

Findings

Algorithms for computing differential operators are provided.

Characterization of solvability via annihilation by differential operators.

Applicable to systems with semialgebraic function coefficients.

Abstract

Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=\left( f_{1},\cdots ,f_{N}\right) \in C^{\infty }\left( \mathbb{R}^{n},\mathbb{R}^{N}\right) $, we consider the following system of equations \begin{equation} \sum_{j=1}^{M}A_{ij}\left( x\right) F_{j}\left( x\right) =f_{i}\left( x\right) \text{ }\left( i=1,\cdots ,N\right) \text{.} \end{equation} In this paper, we give algorithms for computing a finite list of linear partial differential operators such that $AF= f$ admits a $C^m(\mathbb{R}^n, \mathbb{R}^M)$ solution $F=(F_1,\cdots, F_M)$ if and only if $f=(f_1,\cdots, f_N)$ is annihilated by the linear partial differential operators.

Equations

$$\sum _{j=1}^M{A}_{ij}\left(x\right)F_j\left(x\right)=f_i\left(x\right)\left(i=1,\cdots ,N\right)\text{,}$$

$$x^2{F}_1+y^2{F}_2+xy{z}^2{F}_3=f\left(x,y,z\right)$$

$$\left[\begin{array}[]{l}f\left(x,y,z\right)=\frac{\partial f}{\partial x}\left(x,y,z\right)=\frac{\partial f}{\partial y}\left(x,y,z\right)=0\\ \text{and}\\ \frac{\partial^{2}f}{\partial x\partial y}\left(x,y,z\right)=\frac{\partial^{3}f}{\partial x\partial y\partial z}\left(x,y,z\right)=0\end{array}\right.\begin{array}[]{l}\text{for }x=y=0,z\in\mathbb{R}\\ \\ \text{at }x=y=z=0\text{.}\end{array}$$

$$L_{\nu }f\left(x\right)=\sum _{i=1}^N\sum _{|\alpha |\le \bar{m}}{a}_{\nu i\alpha }\left(x\right){\partial }^{\alpha }{f}_i\left(x\right)$$

$${\mathbb{I}}_{x=y=0}{\partial }^0,{\mathbb{I}}_{x=y=0}{\partial }_x,{\mathbb{I}}_{x=y=0}{\partial }_y,{\mathbb{I}}_{x=y=z=0}{\partial }_{xy}^2,{\mathbb{I}}_{x=y=z=0}{\partial }_{xyz}^3,$$

$$f=\left(f_1,\cdots ,f_N\right)\text{ such that }{L}_{\nu }\left(Pf\right)=0\text{ on }{\mathbb{R}}^n\text{ for each }\nu =1,\cdots ,K,$$

$$J_x^{\left(m\right)}F\in H_x\text{ for all }x\in {\mathbb{R}}^n\text{.}$$

$${\hat{H}}_x=\left\{P=\left(P_1,\cdots ,P_M\right)\in {\mathcal{P}}^{\left(m\right)}\left({\mathbb{R}}^n,{\mathbb{R}}^M\right):\sum _{j=1}^M{A}_{ij}\left(x\right)P_j\left(x\right)=f_i\left(x\right),\left(i=1,\cdots ,N\right)\right\}.$$

$$\sum _{j=1}^M{A}_{ij}\left(x\right)F_j\left(x\right)=f_i\left(x\right)$$

$${\partial }^{\alpha }\left\{{\pi }_x^{\overline{\stackrel{ˉ}{m}}\to \bar{m}}{J}_y^{\left(\overline{\stackrel{ˉ}{m}}\right)}F-J_x^{\left(\bar{m}\right)}F\right\}\left(x\right)$$

$$C_0{\Vert F\Vert }_{C^{\overline{\stackrel{ˉ}{m}}}\left({\mathbb{R}}^n,{\mathbb{R}}^D\right)}\cdot {\left|x-y\right|}^{\overline{\stackrel{ˉ}{m}}-|\alpha |}\text{,}$$

$$\left|{\partial }^{\alpha }\left\{{\pi }_x^{\overline{\stackrel{ˉ}{m}}\to \bar{m}}{J}_y^{\left(\overline{\stackrel{ˉ}{m}}\right)}F-J_x^{\left(\bar{m}\right)}F\right\}\left(x\right)\right|$$

$$h\left(x\right)=\left\{\begin{array}[]{c}\frac{1}{x^{2}}\\ 17\end{array}\right.\begin{array}[]{c}\text{if }x\not=0\\ \text{if }x=0\end{array}\text{.}$$

$$f\left(x\right)=\left\{\begin{array}[]{c}0\\ 17\end{array}\right.\begin{array}[]{c}\text{if }x\not=0\\ \text{if }x=0\end{array},$$

$$\underset{\begin{array}{c}{y}_1,\cdots ,y_N\to x\\ \left(y_1,\cdots ,y_N\right)\in G\end{array}}{\lim }f(y_1,\cdots ,y_N)=L$$

$$\text{For every }x\in E_1,\text{ there exists }y\in {\mathbb{R}}^{n_2}\text{ such that }(x,y)\in E.$$

$$\Phi (x_1,\cdots ,x_{n_1}):\left(\exists y_1\right)\cdots \left(\exists y_{n_2}\right)\left((x_1,\cdots ,x_{n_1},y_1,\cdots ,y_{n_2})\in E\right).$$

$$\Phi (x_1,\cdots ,x_n):\left(\forall \epsilon >0\right)\left(\exists y_1\right)\cdots \left(\exists y_n\right)\left(\left[\left(y_1,\cdots ,y_n\right)\in E\right]\wedge \left[{\left(x_1-y_1\right)}^2+\cdots +{\left(x_n-y_n\right)}^2<{\epsilon }^2\right]\right)\text{.}$$

$$\tilde{E}=\left\{\begin{array}[]{c}\left(x_{1},\cdots,x_{n},\underline{x}_{1},\cdots,\underline{x}_{n}\right)\in E\times\underline{E}:\left(\underline{x}_{1},\cdots,\underline{x}_{n}\right)\text{ is at least as close as}\\ \text{any point of }\underline{E}\text{ to }\left(x_{1},\cdots,x_{n}\right)\end{array}\right\},$$

$$\begin{gathered} \Phi \left(x_1,\cdots ,x_n,{\underline{x}}_1,\cdots ,{\underline{x}}_n\right):\left[\left(x_1,\cdots ,x_n\right)\in E\right]\wedge \left[\left({\underline{x}}_1,\cdots ,{\underline{x}}_n\right)\in \underline{E}\right]\wedge \\ \left[\left(\forall y_1\right)\cdots \left(\forall y_n\right)\left\{\left[\left(y_1,\cdots ,y_n\right)\in \underline{E}\right]\to {\left(x_1-{\underline{x}}_1\right)}^2+\cdots +{\left(x_n-{\underline{x}}_n\right)}^2\le {\left(x_1-y_1\right)}^2+\cdots +{\left(x_n-y_n\right)}^2\right\}\right]\text{.} \end{gathered}$$

$$\mathrm{dist}(y,S_2(x))\ge C(x)\cdot {\left[\mathrm{dist}\left(y,S_1(x)\cap S_2(x)\right)\right]}^K$$

$$\left|A\left(x,y\right)\right|\le A_1\left(x\right)\text{ for all }y\in E^{+}\left(x\right).$$

$$\left|A\left(x,y\right)\right|\le A_1\left(x\right)\cdot {\left[\mathrm{dist}\left(y,{\underline{E}}^{+}\left(x\right)\right)\right]}^{-K}\text{ for all }y\in E^{+}\left(x\right)\setminus {\underline{E}}^{+}\left(x\right).$$

$$\frac{1}{A(x,y)}$$

$$A_1(x)=\{\begin{array}{l}\text{smallest }{B}^{+}\ge 0\text{ for which }\text{eqref}wa2\text{ holds, if }{\underline{E}}^{+}(x)=\varnothing \\ \text{smallest }{B}^{+}\ge 0\text{ for which }\text{eqref}wa3\text{ holds, if }{\underline{E}}^{+}(x)\ne \varnothing \end{array}\}.$$

$$|A(x,y)|$$

$$\left(P_0,P_1,\cdots ,P_k\right)\mapsto Q\left(x_0,P_0,x_1,P_1,\cdots ,x_{k,}{P}_k,z_1,\cdots ,z_L\right)$$

$$Q\left(x_0,P_0,x_1,P_1,\cdots ,x_{k,}{P}_k,z_1,\cdots ,z_L\right)\le A\left(x_0,\cdots ,x_k\right)\cdot \left[\sum _{|\alpha |\le m}|{\partial }^{\alpha }{P}_0(x_0){|}^2+\sum _{i=1}^k\sum _{|\alpha |\le \bar{m}}{\left|{\partial }^{\alpha }{P}_i\left(x_i\right)\right|}^2\right]$$

$$E^{+}(x_0)=\{(x_1,\cdots ,x_k)\in {\mathbb{R}}^n\times \cdots \times {\mathbb{R}}^n:(x_0,x_1,\cdots ,x_k)\in E^{+}\}$$

$$E^{++}(x_0)=\{(x_1,\cdots ,x_k,z_1,\cdots ,z_L)\in {\mathbb{R}}^n\times \cdots \times {\mathbb{R}}^n:(x_0,x_1,\cdots ,x_k,z_1,\cdots ,z_L)\in E^{++}\}.$$

Full text

Solutions to A System of Equations for $C^{m}$ Functions

Charles Fefferman, Garving K. Luli

The first author is supported in part by NSF Grant DMS-1608782, AFOSR Grant FA9550-12-1-0425, and Grant No 2014055 from the United States-Israel Binational Science Foundation. The second author is supported by NSF Grant DMS-1554733.

1. Introduction

Here and in [11], we study systems of linear equations

[TABLE]

for unknown functions $F_{1},\cdots,F_{N}\in C^{m}\left(\mathbb{R}^{n}\right)$ for fixed $m$.111$C^{m}\left(\mathbb{R}^{n}\right)$ denotes the vector space of $m$-times continuously differentiable functions $\mathbb{R}^{n}$, with no growth conditions assumed at infinity. Similarly, $C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of all such $\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$. These notations remain in force during the introduction, but will be changed later.

Because $m$ is fixed, we’re not allowed to lose derivatives.

The most interesting systems (1.0.1) are underdetermined. An example due to Epstein and Hochster [5] is the single equation

[TABLE]

for unknown continuous functions $F_{1},F_{2},F_{3}$ on $\mathbb{R}^{3}$.

For a system of the form (1.0.1), we pose three problems.

Problem 1**.**

Suppose the $A_{ij}$ and $f_{i}$ are given functions. For fixed $m$, how can we decide whether (1.0.1) admits a solution $F=\left(F_{1},\cdots,F_{M}\right)\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$?

Problem 2**.**

Suppose the $A_{ij}$ are given polynomials. For fixed $m$, the vectors $f=\left(f_{1},\cdots,f_{N}\right)$ of polynomials $f_{1},\cdots,f_{N}$ for which (1.0.1) admits a $C^{m}$ solution $F$ form a module $\mathcal{M}$ over the ring $\mathcal{R}$ of polynomials on $\mathbb{R}^{n}$. Exhibit generators for $\mathcal{M}$.

Problem 3**.**

Suppose the $A_{ij}$ and $f_{i}$ are polynomials and suppose (1.0.1) admits a $C^{m}$ solution $F$. Can we take our $C^{m}$ solution $F$ to be semialgebraic?

For $m=0$, these problems were posed by Brenner [3], and Epstein-Hochster [5], and solved by Fefferman-Kollár [8] and Kollár [12].

In particular, for $m=0$, the answer to Problem 3 is affirmative; $F\in C^{0}$ can be taken to be semialgebraic. An example in Kollár-Nowak [13] shows that it isn’t always possible to take $F_{1},\cdots,F_{N}$ to be rational functions. See Brenner-Steinbuch [4], Kollár [12], Kollár-Nowak [13], and Kucharz-Kurdyka [14] for several related questions and results.

For $m\geq 1$, Problem 1 was solved in Fefferman-Luli [9], with no restriction on the functions $A_{ij}$, $f_{i}$.

In this paper and [11], we solve Problem 2 for $m\geq 1.$

We needn’t assume that the given matrix elements $A_{ij}$ are polynomials; we may take them to be (possibly discontinuous) semialgebraic functions.

So far, Problem 3 for $m\geq 1$ is still unsolved.

Observe that Problem 2 can’t be solved using only analysis, because it concerns generators for a module over a polynomial ring. On the other hand, it can’t be solved using only algebra, because it concerns $C^{m}$ functions. To make a clean splitting into an analysis problem and an algebra problem, we pose the analogue of Problem 2 for vectors $f=\left(f_{1},\cdots,f_{N}\right)$ of $C^{\infty}$ functions.

Problem 4**.**

Fix a nonnegative integer $m$ and a matrix $\left(A_{ij}\right)$ of semialgebraic functions on $\mathbb{R}^{n}$. Characterize all the $f=\left(f_{1},\cdots,f_{N}\right)\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ for which (1.0.1) admits a $C^{m}$-solution.

To illustrate our result on Problem 4, consider the Epstein-Hochster equation (1.0.2), for unknown continuous $F_{1},F_{2},F_{3}$. For $f\in C^{\infty}\left(\mathbb{R}^{3}\right)$, one checks that a continuous solution exists if and only if $f$ satisfies

[TABLE]

Note that a third derivative of $f$ enters into (1.0.3), even though we are merely looking for solutions $F=\left(F_{1},F_{2},F_{3}\right)\in C^{0}$.

For general systems (1.0.1), our result on Problem 4 is as follows.

Theorem 1**.**

Fix $m\geq 0$, and let $\left(A_{ij}\left(x\right)\right)_{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$. Then there exist linear partial differential operators $L_{1},L_{2},\cdots,L_{K}$, for which the following hold.

• •

Each $L_{\nu}$ acts on vectors $f=\left(f_{1},\cdots,f_{N}\right)\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$, and has the form

[TABLE]

where the coefficients $a_{\nu i\alpha}$ are semialgebraic. (Perhaps $\bar{m}>m$.)

• •

Let $f=\left(f_{1},\cdots,f_{N}\right)\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$. Then the system (1.0.1) admits a solution $F=\left(F_{1},\cdots,F_{M}\right)\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ if and only if $L_{\nu}f=0$ on $\mathbb{R}^{n}$ for each $\nu=1,\cdots,K$.

For the Epstein-Hochster equation (1.0.2), with $m=0,$ the operators $L_{\nu}$ are

[TABLE]

where $\mathbb{I}$ denotes the indicator function. (Compare with (1.0.3).)

Our proof of Theorem 1 is constructive. In principle, we can compute the operators $L_{\nu}$ from the data $m$, $\left(A_{ij}\left(x\right)\right)_{1\leq i\leq N,1\leq j\leq M}$.

In [11], we apply Theorem 1 to solve Problem 2. The idea is as follows.

Given linear partial differential operators $L_{1},\cdots,L_{K}$ with semialgebraic coefficients (not necessarily given by Theorem 1), we introduce the $\mathcal{R}$-module of all polynomial vectors

[TABLE]

and for every polynomial $P$. (Recall $\mathcal{R}$ is the ring of polynomials on $\mathbb{R}^{n}$.) Let us call this $\mathcal{R}$-module $\mathcal{M}\left(L_{1},\cdots,L_{K}\right)$.

If, as in Theorem 1, the polynomial vectors $f$ annihilated by the $L_{\nu}$ already form an $\mathcal{R}$-module, then that $\mathcal{R}$-module coincides with $\mathcal{M}\left(L_{1},\cdots,L_{K}\right)$.

In particular, the module $\mathcal{M}$ in Problem 2 satisfies $\mathcal{M=\mathcal{M}}\left(L_{1},\cdots,L_{K}\right)$ for the $L_{1},\cdots,L_{K}$ given by Theorem 1.

Consequently, Problem 2 is reduced to the following problem of computational algebra.

Problem 5**.**

Given linear partial differential operators $L_{1},\cdots,L_{K}$ with semialgebraic coefficients, exhibit generators for the $\mathcal{R}$-module $\mathcal{M}\left(L_{1},\cdots,L_{K}\right)$.

We solve Problem 5 in [11], completing the solution of Problem 2.

Thus, by posing Problem 4, we have split Problem 2 into an analysis problem and an algebra problem.

Let us sketch the proof of Theorem 1. We oversimplify to bring out the main ideas. The correct discussion appears in Sections 2, $\cdots$, 8 below.

To prepare the way, we introduce notation. For $x\in\mathbb{R}^{n}$ and $F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, we write $J_{x}^{\left(m\right)}F$ (the “$m$-jet” of $F$ at $x$) to denote the $m^{th}$ order Taylor polynomial of $F$ at $x$.

Thus, $J_{x}^{\left(m\right)}F$ belongs to $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, the vector space of all $\mathbb{R}^{D}$-valued polynomials of degree at most $m$ on $\mathbb{R}^{n}$.

If $D=1$, we write $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n}\right)$ in place of $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}\right)$.

For $F,G\in C^{m}\left(\mathbb{R}^{n}\right)$ and $x\in\mathbb{R}^{n}$, we have $J_{x}^{\left(m\right)}\left(FG\right)=J_{x}^{\left(m\right)}F\odot_{x}J_{x}^{\left(m\right)}G$, where $P\odot_{x}Q:=J_{x}^{(m)}\left(PQ\right)$ for $P,Q\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n}\right)$. The multiplication $\odot_{x}$ makes $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n}\right)$ into a ring $\mathcal{R}_{x}^{(m)}$, the “ring of $m$-jets at $x$”. Similarly, the multiplication $Q\odot_{x}\left(P_{1},\cdots,P_{M}\right):=\left(Q\odot_{x}P_{1},\cdots,Q\odot_{x}P_{M}\right)$ makes $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ into an $\mathcal{R}_{x}^{\left(m\right)}$-module.

We can now explain our solution [9] to Problem 1; later, we will apply what we have learned to Problem 4. Thus, let us fix $m\geq 0$, and let $A_{ij}$ and $f_{i}$ be given functions. We investigate whether (1.0.1) has a solution $F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$.

The idea is to construct families $\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$ of affine subspaces $H_{x}\subset\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, such that any $C^{m}$ solution of (1.0.1) necessarily satisfies

[TABLE]

To start with, we can simply take

• (1.0.5)

$\widehat{\mathscr{H}}=\left(\hat{H}_{x}\right)_{x\in\mathbb{R}^{n}},$ where

[TABLE]

We allow the empty set as an affine subspace of $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. This can already happen for $\widehat{\mathscr{H}}$ given by ((1.0.5)), if equations (1.0.1) are inconsistent for some $x$. Obviously, (1.0.4) cannot hold if some of the $H_{x}$ are empty.

The nonempty $H_{x}$ arising in our families $\mathscr{H}$ will have a special form; they are translates of $\mathcal{R}_{x}^{\left(m\right)}$-submodules of $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$.

Thus,

• (1.0.7)

$\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$, where for each $x$, either $H_{x}$ is empty, or $H_{x}=P_{x}+I\left(x\right)$, where $P_{x}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ and $I\left(x\right)\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ is an $\mathcal{R}_{x}^{(m)}$-submodule.

For instance, $\widehat{\mathscr{H}}$ given by ((1.0.5)) has this form.

We call any $\mathscr{H}$ of the form ((1.0.7)) a “bundle”, and we call $F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ a “section” of the bundle $\mathscr{H}$ if (1.0.4) holds. Also, if $\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$ and $\mathscr{H}^{\prime}=\left(H_{x}^{\prime}\right)_{x\in\mathbb{R}^{n}}$ are bundles, then we say that $\mathscr{H}^{\prime}$ is a “subbundle” of $\mathscr{H}$ if $H_{x}^{\prime}\subset H_{x}$ if all $x\in\mathbb{R}^{n}$. We write $\mathscr{H}$ $\supset\mathscr{H}^{\prime}$ to indicate that $\mathscr{H}^{\prime}$ is a subbundle of $\mathscr{H}$.

We call $H_{x_{0}}$ the “fiber” of $\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$ at $x_{0}$.

Immediately from the definition ((1.0.5)), we see that a $C^{m}$ solution of the system (1.0.1) is precisely a section of the bundle $\widehat{\mathscr{H}}$. Therefore, Problem 1 is a special case of the following.

Problem 6**.**

Given a bundle $\mathscr{H}$, decide whether $\mathscr{H}$ has a section.

We solve this problem using the notion of “Glaeser refinement”. The idea is as follows. Let $\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$ be a bundle, and let $x_{0}\in\mathbb{R}^{n}$. By definition, any section $F$ of $\mathscr{H}$ must satisfy $J_{x_{0}}^{\left(m\right)}F\in H_{x_{0}}$. However, $H_{x_{0}}$ may contain polynomials $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ that can never arise as the $m$-jet at $x_{0}$ of any section.

In that case, we may replace $\mathscr{H}$ by a subbundle $\tilde{\mathscr{H}}$ without losing any sections. Let us see how such an $\tilde{\mathscr{H}}$ can be defined.

Fix $x_{0}\in\mathbb{R}^{n}$ and $P_{0}\in H_{x_{0}}$. Suppose $F$ is a section of $\mathscr{H}$, with $J_{x_{0}}^{\left(m\right)}F=P_{0}$. Fix a large integer constant $k$ (determined by $m,n,M),$ and let $x_{1},\cdots,x_{k}\in\mathbb{R}^{n}$ lie close to $x_{0}$.

Setting $P_{i}=J_{x_{i}}^{\left(m\right)}F$ for $i=1,\cdots,k$, we have $P_{1}\in H_{x_{1}},P_{2}\in H_{x_{2}},\cdots,P_{k}\in H_{x_{k}}$, and

• (1.0.9)

$\sum_{0\leq i<j\leq k}\sum_{\left|\alpha\right|\leq m}\left(\frac{\left|\partial^{\alpha}\left(P_{i}-P_{j}\right)\left(x_{j}\right)\right|}{\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}}\right)^{2}\rightarrow 0$ as $x_{1},\cdots,x_{k}\rightarrow x_{0}$, by Taylor’s theorem.

Note that $P_{0},\cdots,P_{k}$ enter into ((1.0.9)), but $P_{0}$ plays a different rôle from $P_{1},\cdots,P_{k}$.

The above remarks lead us to define the Glaeser refinement of the bundle $\mathscr{H}=\left(H_{x}\right)_{x\in\mathbb{R}^{n}}$ by setting $\mathscr{G}\left(\mathscr{H}\right)=\left(\mathscr{\tilde{H}}_{x}\right)_{x\in\mathbb{R}^{n}}$, where for each $x_{0}\in\mathbb{R}^{n}$, $\tilde{H}_{x_{0}}$ consists of those $P_{0}\in H_{x_{0}}$ such that

• (1.0.11)

$\min\left\{\sum_{0\leq i<j\leq k}\sum_{\left|\alpha\right|\leq m}\left(\frac{\left|\partial^{\alpha}\left(P_{i}-P_{j}\right)\left(x_{0}\right)\right|}{\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}}\right)^{2}:P_{1}\in H_{x_{1}},\cdots,P_{k}\in H_{x_{k}}\right\}$ tends to zero

as $x_{1},\cdots,x_{k}\rightarrow x_{0}$.

The Glaeser refinement has three basic properties.

• •

$\mathscr{G}\left(\mathscr{H}\right)$ is a subbundle of $\mathscr{H}$.

• •

$\mathscr{G}\left(\mathscr{H}\right)$ and $\mathscr{H}$ have the same sections, as we saw above.

• •

$\mathscr{G}\left(\mathscr{H}\right)$ can in principle be computed from $\mathscr{H}$, thanks to the explicit nature of ((1.0.11)).

Note that $\mathscr{G}\left(\mathscr{H}\right)$ may have empty fibers, even if $\mathscr{H}$ has none. In that case, we know that $\mathscr{H}$ has no sections.

Starting from a given bundle $\mathscr{H}$, we can now perform “iterated Glaeser refinement” to pass to ever smaller subbundles $\mathscr{H}^{\left(1\right)},\mathscr{H}^{\left(2\right)}$, etc., without losing sections. We set $\mathscr{H}^{\left(0\right)}=\mathscr{H}$, and for $l\geq 0$, we set $\mathscr{H}^{\left(l+1\right)}=\mathscr{G}\left(\mathscr{H}^{\left(l\right)}\right)$. Thus, by an obvious induction on $l$, $\mathscr{H}=\mathscr{H}^{\left(0\right)}\supset\mathscr{H}^{\left(1\right)}\supset\mathscr{H}^{\left(2\right)}\supset\cdots,$ yet $\mathscr{H}$ and $\mathscr{H}^{\left(l\right)}$ have the same sections.

In principle, each $\mathscr{H}^{\left(l\right)}$ can be computed from $\mathscr{H}$.

The main result of [9] gives the

Solution to Problem 6: For a large enough integer constant * $l_{\ast}$ determined by $m,n,M,$ the following holds. *

Let $\mathscr{H}$ be a bundle, and let $\mathscr{H}^{\left(0\right)},\mathscr{H}^{\left(1\right)},\mathscr{H}^{\left(2\right)},\cdots$ be its iterated Glaeser refinements. Then $\mathscr{H}$ has a section if and only if $\mathscr{H}^{\left(l_{\ast}\right)}$ has no empty fibers.

In particular, this solves Problem 1 for systems of equations (1.0.1). This concludes our discussion of Problem 1.

We now want to apply the above to Problem 4. To do so, we have to understand how the iterated Glaeser refinements arising from the bundle $\widehat{\mathscr{H}}$ in ((1.0.5)) depend on the right-hand side $f=\left(f_{1},\cdots,f_{N}\right)$ in (1.0.1), assuming $f\in C^{\infty}.$

This gives rise to the study of bundles of the form

• (1.0.13)

$\mathscr{H}_{f}=\left(T\left(x\right)J_{x}^{\left(\bar{m}\right)}f+I\left(x\right)\right)_{x\in\mathbb{R}^{n}}$, where $I\left(x\right)\subset\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ is an $\mathcal{R}_{x}^{(m)}$-submodule depending semialgebraically on $x$, and $T\left(x\right):\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)\rightarrow\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ is a linear map, also depending semialgebraically on $x$.

We want to understand how the Glaeser refinement of the bundle $\mathscr{H}_{f}$ in ((1.0.13)) depends on $f\in C^{\infty}$. In particular, we want to know when that Glaeser refinement has no empty fibers. Under suitable assumptions on $T\left(x\right)$ in ((1.0.13)), we prove the following:

• (1.0.15)

The fibers of $\mathscr{G}\left(\mathscr{H}_{f}\right)$ are all non-empty if and only if $f$ is annihilated by finitely many linear partial differential operators $L_{1},\cdots,L_{K}$  with semialgebraic coefficients.

• (1.0.17)

If the fibers of $\mathscr{G}\left(\mathscr{H}_{f}\right)$ are all non-empty, then $\mathscr{G}\left(\mathscr{H}_{f}\right)$ again has the form ((1.0.13)), possibly with a smaller $I\left(x\right)$, a larger $\bar{m}$, and a different $T\left(x\right)$.

This allows us to keep track of the $f$-dependence of the iterated Glaeser refinements of the bundle $\widehat{\mathscr{H}}$ in ((1.0.5)), thus proving Theorem 1.

Let us say a few words about the proof of ((1.0.15)) and ((1.0.17)).

Because a quadratic form in ((1.0.11)) lies at the heart of the matter, we have to understand quadratic forms acting on the jets of a function $f\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ at points $x_{1},\cdots,x_{k}\in\mathbb{R}^{n}$. More precisely, suppose we are given a positive semidefinite quadratic form

• (1.0.19)

$\left(P_{0},P_{1},\cdots,P_{k}\right)\mapsto Q\left(x_{0},P_{0},x_{1},P_{1},\cdots,x_{k},P_{k}\right)$ depending semialgebraically on points $x_{1},\cdots,x_{k}\in\mathbb{R}^{n}$. Here, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, while $P_{1},\cdots,P_{k}\in\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$.

We fix $x_{0},P_{0},$ and let $x_{1},\cdots,x_{k}$ vary. We have to characterize the functions $f\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ such that $Q\left(x_{0},P_{0},x_{1},J_{x_{1}}^{\left(\bar{m}\right)}f,\cdots,x_{k},J_{x_{k}}^{\left(\bar{m}\right)}f\right)\rightarrow 0$ as $x_{1},\cdots,x_{k}\rightarrow x_{0}$.

Section 3.2 contains our results on this problem, namely Propositions 2 and 3. These propositions are proven by induction on the dimension of a relevant semialgebraic set. To make the induction work, we have to allow our quadratic form ((1.0.19)) to depend on additional points $z_{1},\cdots,z_{L}$. We refer the reader to Section 4 for full details. Section 4 contains the main work in our proof of Theorem 1.

We will first establish the following variant of Theorem 1.

Theorem 2**.**

Let $E\subset\mathbb{R}^{n}$ be compact, semialgebraic. Let $\left(A_{ij}\left(x\right)\right)_{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions defined on $E$. Let $m\geq 0$ be given. Then there exist linear partial differential operators $L_{\nu}$ $\left(1\leq\nu\leq\nu_{\max}\right)$, for which the following hold.

• •

Each $L_{\nu}$ has semialgebraic coefficients and carries functions in $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ to scalar-valued functions on $\mathbb{R}^{n}$.

• •

Let $f=\left(f_{1},\cdots,f_{N}\right)\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$. Then there exist $F_{1},\cdots,F_{M}\in C^{m}\left(\mathbb{R}^{n}\right)$ such that

[TABLE]

all $x\in E$, $i=1,\cdots,N$

if and only if $L_{\nu}f=0$ on $E$ for all $\nu=1,\cdots,\nu_{\max}$.

In Section 8, we show how to pass from the compact case to the noncompact case, and thus establish Theorem 1.

This concludes our explanation of the proof of Theorem 1. We again warn the reader that the explanation is oversimplified, and that the true story is to be found in Sections 2, $\cdots$, 8 below.

We should also warn the reader that although our results solve Problems 1 and 2 in principle, the calculations involved are prohibitive in practice, except in the simplest cases.

The results of this paper were announced in [10].

We are grateful to Matthias Aschenbrenner, Saugata Basu, Edward Bierstone, Zeev Dvir, János Kollár, Pierre Milman, Wieslaw Pawłucki, Ary Shaviv and the participants in the 9th-11th Whitney workshops for valuable discussions, and to the Technion – Israel Institute of Technology, College of William and Mary, and Trinity College Dublin, for hosting the above workshops.

2. Preliminaries

2.1. Notation

$\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the vector space of $\mathbb{R}^{D}$-valued polynomials of degree at most $m$ on $\mathbb{R}^{n}$. If $D=1$, we may write $\mathcal{P}^{(m)}\left(\mathbb{R}^{n}\right)$ in place of $\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}\right)$.

We depart from the notation used in the Introduction. From now on, $C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of all $\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$ whose derivatives up to order $m$ are continuous and bounded on $\mathbb{R}^{n}$; $C^{m}_{loc}(\mathbb{R}^{n},\mathbb{R}^{D})$ denotes the space of $\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$ with continuous derivatives up to order $m$; $C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of infinitely differentiable $\mathbb{R}^{D}$-valued functions of compact support on $\mathbb{R}^{n}$; $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of infinitely differentiable $\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$. If $D=1$, we write $C^{m}(\mathbb{R}^{n}),C^{\infty}(\mathbb{R}^{n}),C_{0}^{\infty}(\mathbb{R}^{n})$ in place of $C^{m}(\mathbb{R}^{n},\mathbb{R}^{D}),C^{\infty}(\mathbb{R}^{n},\mathbb{R}^{D}),C^{\infty}_{0}(\mathbb{R}^{n},\mathbb{R}^{D})$, respectively.

If $F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ and $x\in\mathbb{R}^{n}$, then $J_{x}^{\left(m\right)}F$ (the “$m$-jet” of $F$ at $x$) denotes the $m^{\text{th}}$ order Taylor polynomial of $F$ at $x$.

We write $\pi_{x}^{m^{\prime}\rightarrow m}:\mathcal{P}^{(m^{\prime})}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\rightarrow\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ for the natural projection from $m^{\prime}$-jets at $x$ to $m$-jets at $x$ $\left(m^{\prime}\geq m\right)$.

2.2. A simple consequence of Taylor’s Theorem

Let $F\in C^{\bar{\bar{m}}}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, let $\bar{\bar{m}}\geq\bar{m}$, and let $x,y\in\mathbb{R}^{n}$. Then for $\left|\alpha\right|\leq\bar{m}$, we have

[TABLE]

The quantity (2.2.1) has absolute value (i.e. norm in $\mathbb{R}^{D}$) at most

[TABLE]

by Taylor’s theorem, with $C_{0}$ depending only on $\bar{\bar{m}},n,D$. Therefore, the following holds:

Proposition 1**.**

[TABLE]

where $C$ depends only on $\bar{\bar{m}},n,D$.

2.3. Semialgebraic sets and functions

Let $A:E\rightarrow\mathbb{R}$, where $E\subset\mathbb{R}^{N}$ is semialgebraic. Recall that $A$ is a semialgebraic function if its graph $\left\{\left(x,y\right)\in E\times\mathbb{R}:y=A\left(x\right)\right\}$ is semialgebraic. In particular, semialgebraic functions needn’t be continuous.

Note that, by definition, a semialgebraic function is finite everywhere on $E$. Thus, the following functions are not semialgebraic on $\mathbb{R}$:

• •

f\left(x\right)=\left\{\begin{array}[]{c}\frac{1}{x^{2}}\\ \infty\end{array}\right.\begin{array}[]{c}\text{if }x\not=0\\ \text{if }x=0\end{array}

• •

f\left(x\right)=\left\{\begin{array}[]{c}\frac{1}{x^{2}}\\ \text{undefined}\end{array}\right.\begin{array}[]{l}\text{if }x\not=0\\ \text{if }x=0\text{.}\end{array}

However, the following function is semialgebraic:

[TABLE]

The dimension of a semialgebraic set $E\subset\mathbb{R}^{n}$ is the maximum of the dimensions of all the imbedded (not necessarily compact) submanifolds of $\mathbb{R}^{n}$ that are contained in $E$.

For instance, in $\mathbb{R}^{3}$, the union of the $x$-$y$ plane and the $z$ axis has dimension $2$.

A map $\phi:E\rightarrow\mathbb{R}^{N}$ is semialgebraic if $\left\{\left(x,y\right)\in E\times\mathbb{R}^{N}:y=\phi\left(x\right)\right\}$ is a semialgebraic set.

Again, semialgebraic maps $\phi:E\rightarrow\mathbb{R}^{N}$ are defined everywhere on $E$.

2.4. Limits

Let $E$ be a metric space, let $f:E\rightarrow\mathbb{R}$ be a function, and let $x\in E$ be given. As every student knows, $\lim_{y\rightarrow x}f\left(y\right)=L$ means that given $\varepsilon>0$ there exists $\delta>0$ such that $\left|f\left(y\right)-L\right|<\varepsilon$ for all $y\in E$ with $\text{dist}\left(y,x\right)<\delta$.

We point out here that if $x$ is an isolated point of $E$, then $\lim_{y\rightarrow x}f\left(x\right)=L$ means simply that $f\left(x\right)=L$.

Note that the function

[TABLE]

defined on $\mathbb{R}$, does not satisfy $\lim_{y\rightarrow 0}f\left(y\right)=0$.

Similarly, if $G\subset E\times\cdots\times E$, then the condition

[TABLE]

is defined in the usual way via $\varepsilon$’s and $\delta$’s.

In particular, (2.4.1) holds vacuously if $G$ fails to contain points arbitrarily close to $(x,x,\cdots,x)$.

2.5. Computations with Semialgebraic Sets

In this section, we present some known technology for computations involving semialgebraic sets. See the reference book [2].

We begin by describing our model of computation. Our algorithms are to be run on an idealized computer with standard von Neumann architecture [17], able to store and perform basic arithmetic operations on integers and infinite precision real numbers, without roundoff errors or overflow conditions. We suppose that our computer can access an ORACLE that solves polynomial equations in one unknown. More precisely, the ORACLE answers queries; a query consists of a non-constant polynomial $P$ (in one variable) with real coefficients, and the ORACLE responds to a query $P$ by producing a list of all the real roots of $P$.

Let us compare our model of computation with that of [2].

All arithmetic in [2] is performed within a subring $\Lambda$ of a real closed field $K$ (e.g. the integers sitting inside the reals). However, some algorithms in [2] produce as output a finite list of elements of $K$ not necessarily belonging to $\Lambda$. A field element $x_{0}$ arising in such an output is specified by exhibiting a polynomial $P$ (in one variable) with coefficients in $\Lambda$ such that $P(x_{0})=0$, together with other data to distinguish $x_{0}$ from the other roots of $P$.

In our model of computation, we take $\Lambda$ and $K$ to consist of all real numbers, and we query the ORACLE whenever [2] specifies a real number by means of a polynomial $P$ as above.

Next, we describe how we will represent a semialgebraic set $E$. We will specify a Boolean combination of sets of the form

• (2.5.1)

$\left\{\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{n}:P(x_{1},\cdots,x_{n})>0\right\}$,

• (2.5.3)

$\left\{\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{n}:P(x_{1},\cdots,x_{n})<0\right\}$, or

• (2.5.5)

$\left\{\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{n}:P(x_{1},\cdots,x_{n})=0\right\}$

for polynomials $P$ with real coefficients.

Of course it is possible to represent the same set $E$ by many different Boolean combination of the above form, but that doesn’t bother us.

We will specify a semialgebraic function by specifying its graph.

Let us mention a few basic algorithms from [2].

• •

Given a semialgebraic set $E\subset\mathbb{R}^{n}$, we compute its dimension. (See Algorithm 14.31 in [2].)

• •

Given a zero-dimensional (and consequently finite) semialgebraic set $E\subset\mathbb{R}^{n}$, we compute a list of all the elements of $E$. (See Algorithm 16.20 in [2].)

• •

Given a semialgebraic set, we produce a list of its connected components, exhibiting each component as a Boolean combination of sets of the form ((2.5.1)), ((2.5.3)), and ((2.5.5)). (See Algorithm 16.20 in [2].)

• •

Given semialgebraic sets $E_{1}\subset\mathbb{R}^{n_{1}}$, $E\subset E_{1}\times\mathbb{R}^{n_{2}}$, we check whether it is the case that

[TABLE]

If (2.5.7) holds, we compute a (possibly discontinuous) semialgebraic function $F:E_{1}\rightarrow\mathbb{R}^{n_{2}}$ such that $(x,F(x))\in E$ for all $x\in E_{1}$. (See Algorithm 11.3 as well as Section 5.1 in [2].)

Next, we discuss “elimination of quantifiers”, a powerful tool to show that certain sets are semialgebraic, and to compute those sets.

The sets in question consist of all $(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$ that satisfy a certain condition $\Phi(x_{1},\cdots,x_{n})$. Here, $\Phi(x_{1},\cdots,x_{n})$ is a statement in a formal language, the “first order predicate calculus for the theory of real closed fields”.

Rather than giving careful definitions, we illustrate with a few examples, and refer the reader to [1, 2].

• •

If $E\subset\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$ is a given semialgebraic set, and if $\pi:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\rightarrow\mathbb{R}^{n_{1}}$ denotes the natural projection, then we can compute the semialgebraic set $\pi E$, because $\pi E$ consists of all $(x_{1},\cdots,x_{n_{1}})\in\mathbb{R}^{n_{1}}$ satisfying the condition

[TABLE]

• •

Suppose $E\subset\mathbb{R}^{n}$ is semialgebraic. Then we can compute $E^{\text{closure}}$, the closure of $E$, because $E^{\text{closure}}$ consists of all $(x_{1},\cdots,x_{n})$ satisfying the condition

[TABLE]

In particular, $E^{\text{closure}}$ is semialgebraic.

• •

Let $E,\underline{E}\subset\mathbb{R}^{n}$ be given semialgebraic sets. Then we can compute the semialgebraic set

[TABLE]

because $\tilde{E}$ consists of all $\left(x_{1},\cdots,x_{n},\underline{x}_{1},\cdots,\underline{x}_{n}\right)\in\mathbb{R}^{2n}$ satisfying the condition

[TABLE]

With a single exception, all the semialgebraic sets and functions arising in our arguments in the following sections can be computed by obvious applications of the above standard algorithms, together with elimination of quantifiers. When that exception arises (in the next section), we explain how to deal with it.

2.6. Growth of Semialgebraic functions

We will use a special case of a result of Łojasiewicz and Wachta [15].

Theorem 3**.**

Let $S_{1},S_{2}\subset\mathbb{R}^{n}\times\mathbb{R}^{p}$ be compact semialgebraic sets. For $x\in\mathbb{R}^{n}$, let $S_{i}(x)=\{y\in\mathbb{R}^{p}:(x,y)\in S_{i}\}$ $(i=1,2)$. Then there exists a positive integer $K$ for which the following holds.

Given $x\in\mathbb{R}^{n}$ such that $S_{1}(x)\cap S_{2}(x)\not=\emptyset$, there exists a positive number $C(x)$ such that

[TABLE]

for all $y\in S_{1}(x)$.

Remarks: The result of [15] applies to subanalytic sets. We need only the semialgebraic case. Our notation differs from that of [15].

We will apply Theorem 3 to prove the following result.

Lemma 1** (Growth Lemma).**

Let $E\subset\mathbb{R}^{n_{1}}$ and $E^{+}\subset E\times\mathbb{R}^{n_{2}}$ be compact and semialgebraic, with $\dim E^{+}\geq 1$. Let $A$ be a semialgebraic function on $E^{+}$. Then there exist an integer $K\geq 1$, a semialgebraic function $A_{1}$ on $E$, and a compact semialgebraic set $\underline{E}^{+}\subset E^{+}$, with the following properties.

**(GL1): **

$\dim\underline{E}^{+}<\dim E^{+}$.

For $x\in E$, set $E^{+}\left(x\right)=\left\{y\in\mathbb{R}^{n_{2}}:\left(x,y\right)\in E^{+}\right\}$ and $\underline{E}^{+}\left(x\right)=\left\{y\in\mathbb{R}^{n_{2}}:\left(x,y\right)\in\underline{E}^{+}\right\}$. Then, for each $x\in E$, the following hold.

**(GL2): **

If $\underline{E}^{+}\left(x\right)$ is empty, then

[TABLE]

**(GL3): **

If $\underline{E}^{+}\left(x\right)$ is non-empty, then

[TABLE]

Proof.

By replacing $A(x,y)$ by $1+|A(x,y)|$, we may suppose that $A(x,y)\geq 1$ for all $(x,y)\in E^{+}$.

We define semialgebraic sets

• •

$S_{0}=\left\{(x,y,\frac{1}{A(x,y)})\in\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\mathbb{R}:(x,y)\in E^{+}\right\}$.

• •

$S_{1}=$ closure of $S_{0}$ in $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\mathbb{R}$.

• •

$S_{2}=E^{+}\times\{0\}\subset\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\mathbb{R}$.

In particular, $S_{1},S_{2}$ are compact semialgebraic subsets of $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\mathbb{R}$, so Theorem 3 applies.

Observe that $S_{1}\cap S_{2}$ has the form $\underline{E}^{+}\times\{0\}$ for a compact semialgebraic set $\underline{E}^{+}\subset E^{+}$. Moreover, $S_{1}\cap S_{2}\subset S_{0}^{\text{closure}}\setminus S_{0}$, hence $\dim\underline{E}^{+}=\dim S_{1}\cap S_{2}\leq\dim(S_{0}^{\text{closure}}\setminus S_{0})<\dim S_{0}=\dim E^{+}$.

Thus, $\dim\underline{E}^{+}<\dim E^{+}$.

Now let $x\in E$ be given. We write $\underline{E}^{+}(x)=\left\{y\in\mathbb{R}^{n_{2}}:(x,y)\in\underline{E}^{+}\right\}$.

Case 1: Suppose $\underline{E}^{+}(x)=\emptyset$. Then $S_{1}(x)\cap S_{2}(x)=\emptyset$.

Therefore, $S_{0}(x)$ avoids a neighborhood of $S_{2}(x)$ in $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\mathbb{R}$, which implies that $\frac{1}{A(x,y)}$ avoids a neighborhood of zero as $y$ varies over $E^{+}(x)$.

Therefore, for some constant $B^{x}$, we have

• (2.6.1)

$|A(x,y)|=A(x,y)\leq B^{x}$ for all $y\in E^{+}(x)$.

Estimate ((2.6.1)) holds if $\underline{E}^{+}(x)=\emptyset$.

Case 2: Suppose $\underline{E}^{+}(x)\not=\emptyset$. Then $S_{1}(x)\cap S_{2}(x)\not=\emptyset$.

Let $K$ and $C(x)$ be as in Theorem 3. Then, for any $y\in E^{+}(x)$, we have $(y,\frac{1}{A(x,y)})\in S_{1}(x)$ and $(y,0)\in S_{2}(x)$, therefore

[TABLE]

Thus, for some constant $B^{x}$, we have

• (2.6.3)

$|A(x,y)|=A(x,y)\leq\frac{B^{x}}{\left[\operatorname{dist}\left(y,\underline{E}^{+}(x)\right)\right]^{K}}$ for all $y\in E^{+}(x)\setminus\underline{E}^{+}(x)$.

Estimate ((2.6.3)) holds if $\underline{E}^{+}(x)\not=\emptyset$.

For $x\in E$, we now set

[TABLE]

Then $A_{1}(x)$ is a non-negative semialgebraic function, and we have

[TABLE]

The proof of Lemma 1 is complete. ∎

We thank M. Aschenbrenner and W. Pawłucki for pointing out that Lemma 1 follows easily from known results, thus subtracting approximately 20 pages from this paper. Pawłucki supplied the above proof of Lemma 1 based on [15].

We indicate how $\underline{E}^{+},K,A_{1}$ in Lemma 1 can be computed. The delicate point is the computation of $K$.

Proceeding as in the proof of Lemma 1, we first replace $A$ by $1+|A|$, then compute $\underline{E}^{+}$.

Given any positive integer $K$, we can then decide whether the following hold for each $x\in E$.

• (2.6.5)

$\sup\{A(x,y):y\in E^{+}(x)\}$ finite if $\underline{E}^{+}(x)$ is empty.

• (2.6.7)

$\sup\{A(x,y)\cdot[\operatorname{dist}(y,\underline{E}^{+}(x))]^{K}:y\in E^{+}(x)\}$ finite if $\underline{E}^{+}(x)$ is nonempty.

We successively test $K=1,2,3,\cdots$ until we find a $K$ for which ((2.6.5)) and ((2.6.7)) hold. We will eventually find such a $K$, thanks to the proof of Lemma 1.

Once we have found $\underline{E}^{+}$ and $K$, we can compute the function $A_{1}$ defined in the proof of Lemma 1. Thus, we compute $\underline{E}^{+},K,A_{1}$ as promised.

3. Semialgebraic Quadratic Forms and $C^{\infty}$ Functions

3.1. Setup

We are given the following objects and assumptions:

• (3.1.1)

$E\subset\mathbb{R}^{n}$, $E^{+}\subset E\times\underset{\text{k copies}}{\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}}$ , $E^{++}\subset E^{+}\times\underset{L\text{ copies}}{\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}}$.

• (3.1.3)

$E,E^{+},E^{++}$ are semialgebraic.

• (3.1.5)

$E$ and $E^{+}$ are compact. We do not assume $E^{++}$ compact.

• (3.1.7)

$\bar{m}\geq m\geq 0$ integers. $D,I$ are positive integers.

• (3.1.9)

$Q\left(x_{0},P_{0},x_{1},P_{1},\cdots,x_{k,}P_{k},z_{1},\cdots,z_{L}\right)$, a semialgebraic function of $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $P_{1},\cdots,P_{k}\in\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$.

• (3.1.11)

For fixed $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$, the map

[TABLE]

is a positive semidefinite quadratic form.

• (3.1.13)

$A\left(x_{0},\cdots,x_{k}\right)$ is a nonnegative semialgebraic function on $E^{+}$.

• (3.1.15)

[TABLE]

for all $\left(x_{0},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $P_{1},\cdots,P_{k}\in\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$.

For $x_{0}\in E$, we define

[TABLE]

and

[TABLE]

In ((3.1.1)), ((3.1.9)), ((3.1.11)), and ((3.1.15)), we allow $L=0$. That is, ((3.1.9)), ((3.1.11)), and ((3.1.15)) need not involve $z$’s.

3.2. Statement of main propositions on semialgebraic quadratic forms

Under the assumptions of Section 3.1, we have the following results:

Proposition 2**.**

There exist an integer $\bar{\bar{m}}\geq\bar{m}$ and a family of vector spaces

[TABLE]

*depending semialgebraically on $\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}$, such that the following holds.

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Then*

[TABLE]

if and only if

[TABLE]

for each $\left(y_{1},\cdots,y_{k}\right)\in E^{+}(x_{0})$.

Moreover, the above family of vector spaces $H^{\text{bdd}}$ can be computed from the data provided in Section 3.1.

Proposition 3**.**

There exist $m^{+}\geq\bar{m}$ and a family of vector subspaces

[TABLE]

*depending semialgebraically on $x_{0}\in E$, such that the following holds.

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Assume that (3.2.1) holds. Then*

[TABLE]

if and only if

[TABLE]

Moreover, the above family of vector spaces $H^{\lim}$ can be computed from the data provided in Section 3.1.

Note: In Proposition 2, the $\sup$ (3.2.1) is taken to be [math] if $E^{++}\left(x_{0}\right)$ is empty.

4. Proof of Propositions 2 and 3

Proof.

We prove Propositions 2 and 3 by induction on $\dim E^{+}$.

All the semialgebraic sets and functions arising in our proof of Propositions 2 and 3 will be computable by the methods of Sections 2.5 and 2.6.

In the base case, $\dim E^{+}=0$, i.e., $E^{+}$ is finite.

In that case, hypothesis ((3.1.15)) from Section 3.1 asserts that for some constant $A$, we have

[TABLE]

for all $\left(x_{0},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$ and all $P_{0},P_{1},\cdots,P_{k}$.

Consequently (3.2.1) in Proposition 2 is satisfied, for any $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. We take $\bar{\bar{m}}=\bar{m}$ and

[TABLE]

Then (3.2.1) and (3.2.2) both hold for any $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$, proving Proposition 2 in the base case $\dim E^{+}=0$.

To prove Proposition 3 in the base case, we note that since $E^{+}$ is finite, condition (3.2.3) in Proposition 3 is equivalent to the following:

[TABLE]

We define $H^{\lim}\left(x_{0}\right)$ to consist of all $\left(P_{0},P\right)\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\oplus\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ such that

[TABLE]

Then, by taking $m^{+}=\bar{m}$, we see that (3.2.3) is equivalent to (3.2.4) as in Proposition 3. Note that $H^{\lim}\left(x_{0}\right)$ is a vector subspace of $\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\oplus\mathcal{P}^{\left(\bar{m}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$, since

[TABLE]

is a semidefinite quadratic form for each $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$.

The semialgebraic dependence of $H^{\lim}\left(x_{0}\right)$ on $x_{0}\in E$ is trivial, since $E$ is finite (because $E^{+}$ is finite).

This completes the proof of Propositions 2 and 3 in the base case ($\dim E^{+}=0$).

For the induction step, we fix a positive integer $\Delta$, and assume that Propositions 2 and 3 hold whenever $\dim E^{+}<\Delta$. We will prove those propositions in the case $\dim E^{+}=\Delta$.

Let us assume that all is as in Section 3.1, and that $\dim E^{+}=\Delta$.

We apply Lemma 1 to the semialgebraic function $A\left(x_{0},x_{1},\cdots,x_{k}\right)$ in ((3.1.13)), ((3.1.15)) of Section 3.1.

Thus, there exist an integer $K\geq 1$, a compact semialgebraic subset $\underline{E}^{+}\subset E^{+}$, and a semialgebraic function $A_{1}$ defined on $E$, having the following properties:

• (4.0.1)

$\dim\underline{E}^{+}<\Delta$.

Let $x_{0}\in E$. Recall that

• (4.0.3)

$E^{+}\left(x_{0}\right)=\left\{\left(x_{1},\cdots,x_{k}\right)\in\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}:\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}\right\}$.

Define

• (4.0.5)

$\underline{E}^{+}\left(x_{0}\right)=\left\{\left(x_{1},\cdots,x_{k}\right)\in\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}:\left(x_{0},x_{1},\cdots,x_{k}\right)\in\underline{E}^{+}\right\}$.

Then, then for each $x_{0}\in E$, the following hold.

• (4.0.7)

If $\underline{E}^{+}\left(x_{0}\right)$ is empty, then

[TABLE]

• (4.0.9)

If $\underline{E}^{+}\left(x_{0}\right)$ is nonempty, then

[TABLE]

Let

• (4.0.11)

$\underline{E}=\left\{x_{0}\in E:\underline{E}^{+}\left(x_{0}\right)\not=\emptyset\right\}$.

Note that $\underline{E}$ and $\underline{E}^{+}$ are compact semialgebraic sets, and that

• (4.0.13)

$\underline{E}^{+}\subset\underline{E}\times\underset{k\text{ copies}}{\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}}$.

• (4.0.15)

Let $\underline{E}^{++}=\left\{\left(x_{0},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}:\left(x_{0},\cdots,x_{k}\right)\in\underline{E}^{+}\right\}$.

For $x_{0}\in\underline{E}$, define

• (4.0.17)

$\underline{E}^{++}(x_{0})=\{(y_{1},\cdots,y_{k},z_{1},\cdots,z_{L})\in\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}:(x_{0},y_{1},\cdots,y_{k},z_{1},\cdots,z_{L})\in\underline{E}^{++}\}.$

Thus, $\underline{E}^{++}$ is a semialgebraic subset of $\underline{E}^{+}\times\underset{L\text{ copies}}{\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}}$.

Thanks to $\eqref{P1}$, our induction hypothesis applies to $Q(x_{0},P_{0},x_{1},P_{1},\cdots,x_{k},P_{k},z_{1},\cdots,z_{k})$ restricted to $(x_{0},\cdots,x_{k},z_{1},\cdots,z_{L})\in\underline{E}^{++}$. Applying Proposition 2, we therefore learn the following.

There exist $\underline{m}^{\prime}\geq\bar{m}$, and a computable family of vector spaces

[TABLE]

depending semialgebraically on $\left(x_{0},\cdots,x_{k}\right)\in\underline{E}^{+}$, such that the following holds.

Let $x_{0}\in\underline{E}$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathcal{\mathbb{R}}^{n},\mathbb{R}^{I}\right)$.

Then

[TABLE]

if and only if

[TABLE]

for each $\left(y_{1},\cdots,y_{k}\right)\in\underline{E}^{+}(x_{0})$.

Later on, we will apply Proposition 3 in the same setting; see ((4.0.113)) below.

Let us sketch the arguments that follow, concentrating on the proof of Proposition 2.

Given $x_{0},P_{0},F,$ we must decide whether the quantity

[TABLE]

stays bounded as $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)$ varies over $E^{++}\left(x_{0}\right)$.

If we restrict attention to the set of such $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)$ where $\left(x_{1},\cdots,x_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$, then already the equivalence of (4.0.20) to (4.0.21) settles the issue. Hence, we may restrict attention to the set where $\left(x_{1},\cdots,x_{k}\right)\not\in\underline{E}^{+}\left(x_{0}\right)$. Also, ((3.1.15)) and ((4.0.7)) easily imply that the quantity (4.0.22) stays bounded whenever $\underline{E}^{+}\left(x_{0}\right)$ is empty. Hence, we may assume that $\underline{E}^{+}\left(x_{0}\right)$ is non-empty.

Thus, our problem is to decide whether the quantity (4.0.22) stays bounded as $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)$ varies over the set where

• (4.0.23)

$\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}\left(x_{0}\right)$, $\left(x_{1},\cdots,x_{k}\right)\not\in\underline{E}^{+}\left(x_{0}\right),$ assuming $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$.

In deciding this question, we are fighting against the factor $\left[\operatorname{dist}\left(\left(x_{1},\cdots,x_{k}\right),\underline{E}^{+}\left(x_{0}\right)\right)\right]^{-K}$ in estimate ((4.0.9)).

Our strategy is to pick $\bar{\bar{m}}$ much larger than $\bar{m}$, and approximate $J_{x_{i}}^{\left(\bar{m}\right)}F$ by $\pi_{x_{i}}^{\bar{\bar{m}}\rightarrow\bar{m}}J_{\underline{x}_{i}}^{\left(\bar{\bar{m}}\right)}F$ in (4.0.22), where $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is a point of $\underline{E}^{+}\left(x_{0}\right)$ lying as close as possible to $\left(x_{1},\cdots,x_{k}\right)$. According to Proposition 1, the derivatives of the error $J_{x_{i}}^{\left(\bar{m}\right)}F-\pi_{x_{i}}^{\overline{\overline{m}}\rightarrow\bar{m}}J_{\underline{x}_{i}}^{\left(\overline{\overline{m}}\right)}F$ at $x_{i}$ are

[TABLE]

If we pick $\bar{\bar{m}}$ large enough, then the small factor $\left[\operatorname{dist}\left(\cdots\right)\right]^{\bar{\bar{m}}-\bar{m}}$ overcomes the large factor $\left[\operatorname{dist}\left(\cdots\right)\right]^{-K}$ in ((4.0.9)).

Therefore, the induction step in the proof of Proposition 2 comes down to deciding whether the quantity

[TABLE]

remains bounded as $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ varies over the set where

[TABLE]

We are therefore tempted to define

[TABLE]

for $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ as in (4.0.26), and for $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $P_{1},\cdots,P_{k}\in\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$.

Our problem is then to decide whether the quantity

[TABLE]

remains bounded as $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ varies over the set $\hat{E}^{++}\left(x_{0}\right)$ defined by (4.0.26). We hope to decide this question by applying Proposition 2 to the quadratic form $\tilde{Q}$ and the sets

[TABLE]

in place of $Q,E^{++},E^{+},E$.

Here, our present $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ plays the rôle of $\left(x_{1},\cdots,x_{k}\right)$ in the statement of Proposition 2, while our present $\left(z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ plays the rôle of $\left(z_{1},\cdots,z_{L}\right)$. The key point is that $\dim\hat{E}^{+}=\dim\underline{E}^{+}<\Delta$, by ((4.0.1)), so we can hope to apply our induction hypothesis (Proposition 2 holds when $\dim E^{+}<\Delta$). If we could apply Proposition 2 to $\tilde{Q},\hat{E}^{++},\hat{E}^{+},\hat{E}$, then we could decide whether the quantity (4.0.28) satisfies the required boundedness condition, and thus complete our inductive proof of Proposition 2.

Unfortunately, the above doesn’t quite work, because the quadratic form $\tilde{Q}$ needn’t satisfy the analogue of hypothesis ((3.1.15)) of Proposition 2.

We rescue the argument by modifying $\tilde{Q}$. Given $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ we define a projection $\Pi_{\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)}$ from

[TABLE]

to a subspace on which $\tilde{Q}$ behaves well. We then define

[TABLE]

where $\left(\hat{P}_{0},\hat{P}_{1},\cdots,\hat{P}_{k}\right)=\prod_{\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)}\left(P_{0},P_{1},\cdots,P_{k}\right)$. (Compare with (4).)

Our induction hypothesis – Proposition 2 in lower dimension – applies to $\hat{Q},\hat{E}^{++},\hat{E}^{+},\hat{E},$ allowing us to argue as we hoped to do for $\tilde{Q},\hat{E}^{++},\hat{E}^{+},\hat{E}$. Thus, we can complete our induction on $\dim E^{+}$. This completes our preview of the proof of Proposition 2.

Proposition 3 is proven similarly. In some cases, we approximate $J_{x_{i}}^{\left(\bar{m}\right)}F$ in (3.2.1) by $\pi_{x_{i}}^{\bar{\bar{m}}\rightarrow\bar{m}}J_{x_{0}}^{\left(\overline{\overline{m}}\right)}F$, rather than by $\pi_{x_{i}}^{\bar{\bar{m}}\rightarrow\bar{m}}J_{\underline{x}_{i}}^{\left(\bar{\bar{m}}\right)}F$.

Having done with previews, let us return to the proof of Proposition 2.

We will now estimate $Q(x_{0},P_{0},x_{1},J_{x_{1}}^{(\bar{m})}F,\cdots,x_{k},J_{x_{k}}^{(\bar{m})}F,z_{1},\cdots,z_{L})$ under several different assumptions on $(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L})\in E^{++}$.

Case 1:

Suppose $\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}\setminus\underline{E}^{+}$, and suppose also that $\underline{E}^{+}\left(x_{0}\right)$ is nonempty. Let $F\in C^{\infty}_{0}(\mathbb{R}^{n},\mathbb{R}^{I})$.

Let $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$ be as close as possible to $\left(x_{1},\cdots,x_{k}\right)$. (Recall, $\underline{E}^{+}\left(x_{0}\right)$ is compact.)

Then, for $\bar{\bar{m}}>\bar{m}$ to be picked in a moment, Taylor’s theorem (see Proposition 1) gives

[TABLE]

for $\left|\alpha\right|\leq\bar{m}$, $j=1,\cdots k$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$.

Throughout this section, we write $c,C,C^{\prime},$ etc. to denote constants determined by $\bar{\bar{m}}$, $\bar{m},$ $n,k,L,I,K,D,$ and an upper bound for diameter of $E^{+}$. These symbols may denote different constants in different occurrences.

Since

[TABLE]

is a positive semidefinite quadratic form, we have the estimates

[TABLE]

and similarly

[TABLE]

From ((4.0.9)), (4.0.30), and hypothesis ((3.1.15)), we see that

[TABLE]

where the last inequality follows from the fact that $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is a closest point to $\left(x_{1},\cdots,x_{k}\right)$ in $\underline{E}^{+}\left(x_{0}\right)$.

We now pick $\bar{\bar{m}}$ so large that $2\left(\bar{\bar{m}}-\bar{m}\right)-K\geq 200$.

From (4), (4), (4.0.33), we then learn that

[TABLE]

and

[TABLE]

• (4.0.36)

Estimates (4.0.34) and (4) hold in Case 1, i.e., when $(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L})\in E^{++}$,

$\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}\setminus\underline{E}^{+}$, $\underline{E}^{+}\left(x_{0}\right)$ is nonempty, $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$ is as close as possible to $\left(x_{1},\cdots,x_{k}\right)$, and $F\in C_{0}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{I})$.

This completes the discussion of Case 1.

Case 2: Now suppose $\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}$ and that $\underline{E}^{+}\left(x_{0}\right)$ is empty.

Then ((4.0.7)) and hypothesis ((3.1.15)) show that

[TABLE]

whenever $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ and $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$.

Moreover, because

[TABLE]

is a nonnegative quadratic form, we have

[TABLE]

and

[TABLE]

By ((4.0.7)), hypothesis ((3.1.15)), and Taylor’s theorem (see Proposition 1), we have

[TABLE]

since we picked $\bar{\bar{m}}$ so that (in particular) $\bar{\bar{m}}-\bar{m}\geq 100$.

Putting (4.0.41) into (4), (4), we see that

[TABLE]

and

[TABLE]

Estimates (4.0.38), (4), (4) hold in Case 2, i.e., when

• (4.0.44)

$\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}$, $\underline{E}^{+}\left(x_{0}\right)=\emptyset$, $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$, and $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$.

This completes our discussion of Case 2.

One of the two cases Case 1 and Case 2 above must occur whenever $\left(x_{0},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}$ and $\left(x_{0},\cdots,x_{k}\right)\in E^{+}\setminus\underline{E}^{+}$.

However, it is also useful to study the following subcase of Case 1.

Case 1’: Suppose $(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L})\in E^{++}$, $\left(x_{0},x_{1},\cdots,x_{k}\right)\in E^{+}\setminus\underline{E}^{+}$, $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$, and $\left(x_{0},\cdots,x_{0}\right)\not\in\underline{E}^{+}\left(x_{0}\right)$. Let $F\in C_{0}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{I})$.

Because $\underline{E}^{+}$, $\underline{E}^{+}\left(x_{0}\right)$ are compact, we have

[TABLE]

Suppose

[TABLE]

hence, by ((4.0.9)),

[TABLE]

Hypothesis ((3.1.15)) therefore yields

[TABLE]

for any $P_{0},P_{1},\cdots,P_{k}$; compare with (4.0.38). Hence, we may proceed as in the proof of (4), (4), to establish the following result.

• (4.0.46)

In Case 1’, if $\left|\left(x_{1},\cdots,x_{k}\right)-\left(x_{0},\cdots,x_{0}\right)\right|<\frac{1}{2}\operatorname{dist}\left(\left(x_{0},\cdots,x_{0}\right),\underline{E}^{+}\left(x_{0}\right)\right)$, then we have

[TABLE]

and

[TABLE]

This completes our analysis of Case 1’.

For fixed $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$ and

[TABLE]

we ask whether

[TABLE]

Note: In (4.0.48), $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ are held fixed, i.e., the sup is over $z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$ satisfying the constraints. The sup is taken to be [math] if the the set of points satisfying the constraints is empty.

Observe that the set of all $(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},P_{0},P_{1},\cdots,P_{k})$ satisfying (4.0.48) is semialgebraic, since $Q$, $E^{++}$, $E^{+}$, $\underline{E}^{+}$ are semialgebraic. Moreover, since

[TABLE]

is a nonnegative quadratic form for fixed $x_{0},x_{1},\cdots,x_{k}$, $z_{1},\cdots,z_{L}$, it follows that the set of all $\left(P_{0},P_{1},\cdots,P_{k}\right)$ satisfying (4.0.48) is a vector subspace of

[TABLE]

for fixed $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$. We denote this vector subspace by

[TABLE]

Thus,

• (4.0.49)

$\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\subset\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\oplus\underset{k\text{ copies}}{\underbrace{\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)\oplus\cdots\oplus\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)}}$ is a vector subspace depending semialgebraically on $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$,

and

• (4.0.51)

for every given $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$ and

[TABLE]

condition (4.0.48) holds if and only if

[TABLE]

Given $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$ and $\left(P_{0},P_{1},\cdots,P_{k}\right)\in\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$, we define

[TABLE]

to be the square root of the sup in (4.0.48).

Note that

[TABLE]

for any $\left(P_{0},P_{1},\cdots,P_{k}\right)$, $\left(P_{0}^{\prime},\cdots,P_{k}^{\prime}\right)\in\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ and any $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$, because

[TABLE]

is a positive semidefinite quadratic form in $\left(P_{0},P_{1},\cdots,P_{k}\right)$ for fixed $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)$.

Also, for $\lambda\in\mathbb{R}$ we have

[TABLE]

Thus, for fixed $(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k})\in\underline{E}^{+}$, the function $(P_{0},P_{1},\cdots,P_{k})\mapsto\operatorname{Norm}(P_{0},P_{1},\cdots,P_{k};x_{0},\underline{x}_{1},\cdots,\underline{x}_{k})$ is a seminorm on the finite-dimensional vector space $\hat{H}^{\text{bdd}}(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k})$.

It follows that, for each $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$, we have

[TABLE]

for all $\left(P_{0},P_{1},\cdots,P_{k}\right)\in\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$; here, $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is some finite constant.

Since $\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ depends semialgebraically on $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$, and since $\operatorname{Norm}\left(P_{0},\cdots,P_{k};x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is semialgebraic in $\left(P_{0},\cdots,P_{k};x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$, it follows that the least possible nonnegative $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ in (4.0.53) is a semialgebraic function of $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}$.

From now on, we define $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ to be the least possible nonnegative number for which (4.0.53) holds for all $\left(P_{0},P_{1},\cdots,P_{k}\right)\in\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$.

Thus, $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is a semialgebraic function; and from (4.0.53), we obtain the estimate

[TABLE]

whenever

[TABLE]

[TABLE]

denote the orthogonal projection from

[TABLE]

onto

[TABLE]

with respect to the quadratic form $\sum_{|\alpha|\leq m}|\partial^{\alpha}P_{0}(x_{0})|^{2}+\sum_{i=1}^{k}\sum_{\left|\alpha\right|\leq\bar{\bar{m}}}\left|\partial^{\alpha}P_{i}\left(\underline{x}_{i}\right)\right|^{2}$.

If

[TABLE]

then the following hold.

[TABLE]

[TABLE]

[TABLE]

Also, $\Pi_{\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)}\left(P_{0},P_{1},\cdots,P_{k}\right)$ is given by a semialgebraic map

[TABLE]

We prepare to invoke the induction hypothesis (Propositions 2 and 3 hold for $\dim E^{+}<\Delta$).

Let

[TABLE]

[TABLE]

[TABLE]

• (4.0.62)

For $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$ and

[TABLE]

define

[TABLE]

and then set

[TABLE]

Here, $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ plays the role of $\left(x_{1},\cdots,x_{k}\right)$ in Section 3.1, and $\left(z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ plays the role of $\left(z_{1},\cdots,z_{L}\right)$ in Section 3.1.

For $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\hat{E}^{+}=\underline{E}^{+}$, we define

[TABLE]

with $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ as in (4.0.54).

We check that $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ satisfy hypotheses ((3.1.1)), $\cdots$,((3.1.15)) in Section 3.1, and that $\dim\hat{E}^{+}<\Delta$. This will allow us to apply Propositions 2 and 3 to $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Hypothesis ((3.1.1)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ simply asserts that

[TABLE]

From (4.0.60) and (4.0.61), we see that $\hat{E}\subset\mathbb{R}^{n}$ and $\hat{E}^{+}\subset\hat{E}\times\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}$. Also (4.0.59) and (4.0.60) show that $\hat{E}^{++}\subset\hat{E}^{+}\times\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}$. Thus, Hypothesis ((3.1.1)) holds for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Hypothesis ((3.1.3)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$ are semialgebraic, which follows from ((4.0.5)), (4.0.59), (4.0.60), (4.0.61), since $E^{++}$, $\underline{E}^{++}$ are semialgebraic.

Hypothesis ((3.1.5)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that $\hat{E}$ and $\hat{E}^{+}$ are compact, which follows from (4.0.60), (4.0.61) and the compactness of $\underline{E}^{+}$.

Hypothesis ((3.1.7)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that $\bar{\bar{m}}\geq m\geq 0$ and $D,I\geq 1$, which we know from our selection of $\bar{\bar{m}}$, and from Hypothesis ((3.1.7)) for $E$, $E^{+}$, $E^{++}$, $Q$, $A$.

Hypothesis ((3.1.9)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that

[TABLE]

is a semialgebraic function of $x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$, $P_{0},P_{1},\cdots,P_{k}$. This follows from ((4.0.62)), since

[TABLE]

depends semialgebraically on $x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},P_{0},P_{1},\cdots,P_{k}$, the projection $\pi_{x_{i}}^{\bar{\bar{m}}\rightarrow\bar{m}}$ depends semialgebraically on $x_{i}$ and

[TABLE]

is semialgebraic in $\left(x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L},\hat{P}_{0},\hat{P}_{1},\cdots,\hat{P}_{k}\right)$.

Thus, Hypothesis ((3.1.9)) holds for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Hypothesis ((3.1.11)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that for fixed $x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$, the map

[TABLE]

is a positive semidefinite quadratic form. This follows from ((4.0.62)) and Hypothesis ((3.1.11)) for $E$, $E^{+}$, $E^{++}$, $Q$, $A$, since the maps

[TABLE]

for fixed $x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$.

Thus, Hypothesis ((3.1.11)) holds for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Hypothesis ((3.1.13)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that

[TABLE]

is a nonnegative semialgebraic function.

This is immediate from (4.0.64), since $M\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ is a semialgebraic function.

Hypothesis ((3.1.15)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$ asserts that

[TABLE]

for $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$.

By definitions ((4.0.62)), (4.0.64), this means that

[TABLE]

for $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$, where

[TABLE]

Let us verify that (4.0.65) holds.

Let $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$, and let

[TABLE]

By definition (4.0.59), and by Hypothesis ((3.1.1)) for $E$, $E^{+}$, $E^{++}$, $Q$, $A$, we see that (4.0.55a)$,\cdots$, (4.0.55c) hold. Moreover,

[TABLE]

belongs to $\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$, by (4.0.57).

Hence, (4.0.55a)$,\cdots$, (4.0.55d) all hold with $\left(\hat{P}_{0},\hat{P}_{1},\cdots,\hat{P}_{k}\right)$ in place of $\left(P_{0},\cdots,P_{k}\right);$ from which we obtain, via (4.0.54), that

[TABLE]

Together with (4.0.56), this implies (4.0.65), completing the proof of Hypothesis ((3.1.15)) for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Thus, Hypotheses ((3.1.1)), $\cdots$, ((3.1.15)) all hold for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$. Moreover, $\dim\hat{E}^{+}=\dim\underline{E}^{+}<\Delta$; see (4.0.60) and ((4.0.1)).

So we may apply Propositions 2 and 3 to $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}$.

Thus, we learn the following.

Proposition 2 for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}:$ There exist $\hat{m}\geq\bar{\bar{m}}$, and a computable family of vector spaces

[TABLE]

depending semialgebraically on $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\hat{E}^{+}=\underline{E}^{+}$, such that the following holds.

For $x_{0}\in\hat{E}$, define

[TABLE]

and

[TABLE]

Let $x_{0}\in\hat{E}$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ be given. Then

[TABLE]

Proposition 3 for $\hat{E}$, $\hat{E}^{+}$, $\hat{E}^{++}$, $\hat{Q}$, $\hat{A}:$ There exist $\hat{m}^{+}\geq\bar{\bar{m}}$, and a computable family of vector spaces

[TABLE]

depending semialgebraically on $x_{0}\in\hat{E}$, such that the following holds.

Let $x_{0}\in\hat{E}$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ be given. Assume that condition (4.0.68c) holds. Then

[TABLE]

Now let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ be given and suppose that

[TABLE]

(In the above $\sup$, $x_{0}$ has been fixed, while $x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}$ vary.)

Recall ((4.0.1)), $\cdots$, (4.0.19) and (4.0.20), (4.0.21). We have the following, from the equivalence of (4.0.20) and (4.0.21):

• (4.0.71)

If $x_{0}\in\underline{E}$, then $\left(P_{0},J_{x_{1}}^{\left(\underline{m}^{\prime}\right)}F,\cdots,J_{x_{k}}^{\left(\underline{m}^{\prime}\right)}F\right)\in\underline{H}^{\text{bdd}}\left(x_{0},x_{1},\cdots,x_{k}\right)$ for each $\left(x_{1},\cdots,x_{k}\right)\in\underline{E}^{+}(x_{0})$.

If $x_{0}\in\hat{E}$, then $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$ by (4.0.61). Hence, by (4) and (4.0.70), we find that (4.0.48) holds for any $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}(x_{0})$, with $P_{i}:=J_{\underline{x}_{i}}^{\left(\bar{\bar{m}}\right)}F$ for $i=1,\cdots,k$.

(To see this, recall that (4) holds whenever ((4.0.36)) holds, and note that ((4.0.36)) holds whenever $x_{0},x_{1},\cdots,x_{k},z_{1},\cdots,z_{L},\underline{x}_{1},\cdots,\underline{x}_{k}$ are as in (4.0.48).) Thanks to ((4.0.51)), we therefore learn that

• (4.0.73)

$\left(P_{0},J_{\underline{x}_{1}}^{\left(\bar{\bar{m}}\right)}F,\cdots,J_{\underline{x}_{k}}^{\left(\bar{\bar{m}}\right)}F\right)\in\hat{H}^{\text{bdd}}\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)$ whenever $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$.

Hence, by (4.0.58), we have

• (4.0.75)

$\left(P_{0},J_{\underline{x}_{1}}^{\left(\bar{\bar{m}}\right)}F,\cdots,J_{\underline{x}_{k}}^{\left(\bar{\bar{m}}\right)}F\right)=\Pi_{\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)}\left(P_{0},J_{\underline{x}_{1}}^{\left(\bar{\bar{m}}\right)}F,\cdots,J_{\underline{x}_{k}}^{\left(\bar{\bar{m}}\right)}F\right)$ whenever $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$.

From ((4.0.62)) and ((4.0.75)), we obtain the following.

[TABLE]

whenever $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}(x_{0})$.

Now, ((4.0.36)) holds whenever

[TABLE]

Hence, for such points, (4) holds. From (4) and (4.0.70), we conclude that

[TABLE]

In (4.0.78), $x_{0}$ is fixed and $\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$ vary.

From (4.0.77) and (4.0.78), we see that

[TABLE]

Thus, we have verified (4.0.68c) for $x_{0}$, $P_{0}$, $F$. Recalling the equivalence of (4.0.68c), (4.0.68d), we now know that (4.0.68d) holds.

We have proven this under the assumption that $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$, and also assuming (4.0.70).

Thus, we have proven the following.

• (4.0.80)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Assume (4.0.70), and suppose that $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$. Then

[TABLE]

and

[TABLE]

for all $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$. (See ((4.0.73)).)

From ((4.0.11)), ((4.0.71)) and ((4.0.80)), we draw the following conclusion.

• (4.0.82)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Assume that

[TABLE]

Then for all $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$, we have

[TABLE]

and

[TABLE]

Conversely, let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Assume that for all $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$, we have

[TABLE]

[TABLE]

and

[TABLE]

We will prove (4.0.70) under the above assumptions.

To see this, first suppose $\underline{E}^{+}\left(x_{0}\right)=\emptyset.$ Then since ((4.0.44)) implies (4.0.38), we have

[TABLE]

where $x_{0}$ remains fixed and $z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$ vary.

Thus, (4.0.70) holds if $\underline{E}^{+}\left(x_{0}\right)=\emptyset$. We now suppose that $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$. Note that $x_{0}\in\hat{E}$. (See (4.0.61).)

From (4.0.85), (4.0.86), and the equivalence of (4.0.68c), (4.0.68d), we see that

[TABLE]

From (4.0.85), (4.0.86), and the definition of $\Pi_{\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k}\right)}$, we learn that

[TABLE]

for $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$.

Therefore, ((4.0.62)) and (4.0.87) yield the following:

[TABLE]

In (4.0.87) and (4.0.88), $x_{0}$ stays fixed, while the sup is taken over $\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}$.

When $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}(x_{0})$, we have ((4.0.36)); hence also (4.0.34) holds in this case.

From (4.0.34) and (4.0.88), we see that

[TABLE]

Whenever $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0})$ and $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$, we can pick $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$ as close as possible to $\left(x_{1},\cdots,x_{k}\right)$, and we will have

[TABLE]

unless

[TABLE]

Therefore, (4.0.89) implies the following.

[TABLE]

In (4.0.89) and (4.0.90), $x_{0}$ stays fixed; the $\sup$ is taken over the other variables.

Also, from ((4.0.15)), the equivalence of (4.0.20) to (4.0.21), and (4.0.84), we see that

[TABLE]

where again $x_{0}$ stays fixed. From (4.0.90) and (4.0.91), we learn that

[TABLE]

with $x_{0}$ fixed as usual. Thus, as promised, we have proven (4.0.70).

Recalling ((4.0.82)), we now see that we have established the following.

• (4.0.92)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Then

[TABLE]

remains bounded over all $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0})$ if and only if for all $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$, we have

[TABLE]

and

[TABLE]

From ((4.0.92)), we see easily that the conclusion of Proposition 2 holds for $E$, $E^{+}$, $E^{++}$, $Q$, $A$. This completes the induction step in our proof of Proposition 2.

We turn our attention to the induction step in the proof of Proposition 3.

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ be given. Assume that

[TABLE]

In our discussion of the induction step for Proposition 2, we already saw that (4.0.94) implies the following (see (4.0.77) and (4.0.79)).

[TABLE]

for

[TABLE]

[TABLE]

with $x_{0}$ held fixed in (4.0.95) and (4.0.96).

Now suppose $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$, i.e., $x_{0}\in\hat{E}$ (see (4.0.61)). Then (4.0.95), (4.0.96), and the equivalence of (4.0.69a), (4.0.69b), together show that

• (4.0.97)

$\left(P_{0},J_{x_{0}}^{\left(\hat{m}^{+}\right)}F\right)\in\check{H}^{\lim}\left(x_{0}\right)$ if and only if

[TABLE]

For $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$ we have ((4.0.36)) and therefore (4.0.34), (4). (See (4.0.59).)

Consequently, ((4.0.97)) implies the following.

• (4.0.99)

$\left(P_{0},J_{x_{0}}^{\left(\hat{m}^{+}\right)}F\right)\in\check{H}^{\lim}\left(x_{0}\right)$ if and only if

[TABLE]

Let $\delta>0$. If $\left(x_{0},\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)\in\hat{E}^{++}$ and

[TABLE]

and also

[TABLE]

it follows that

• (4.0.101)

$\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0})$, $\left(x_{1},\cdots,x_{k}\right)\not\in\underline{E}^{+}(x_{0})$, and $\left|x_{i}-x_{0}\right|<\delta$ (all $i$), $\left|z_{i}-x_{0}\right|<\delta$ (all $i$), see (4.0.59).

Conversely, suppose ((4.0.101)) holds and assume $(x_{0},\cdots,x_{0})\in\underline{E}^{+}(x_{0})$. Let $\left(\underline{x}_{1},\cdots,\underline{x}_{k}\right)\in\underline{E}^{+}\left(x_{0}\right)$ be as close as possible to $\left(x_{1},\cdots,x_{k}\right)$. Then by (4.0.59), $\left(\underline{x}_{1},\cdots,\underline{x}_{k},z_{1},\cdots,z_{L},x_{1},\cdots,x_{k}\right)$ belongs to $\hat{E}^{++}(x_{0});$ and we have

[TABLE]

In view of the above remarks, ((4.0.99)) implies the following.

• (4.0.103)

Suppose (4.0.94) holds, and suppose $\left(x_{0},\cdots,x_{0}\right)\in\underline{E}^{+}\left(x_{0}\right)$. Then $\left(P_{0},J_{x_{0}}^{\left(\hat{m}^{+}\right)}F\right)\in\check{H}^{\lim}\left(x_{0}\right)$ if and only if

[TABLE]

On the other hand, suppose (4.0.94) holds, $\underline{E}^{+}\left(x_{0}\right)\not=\emptyset$, and assume that $\left(x_{0},\cdots,x_{0}\right)\not\in\underline{E}^{+}\left(x_{0}\right)$.

Then we are in Case 1’; see ((4.0.44)). Thus, ((4.0.46)) holds.

Consequently,

• (4.0.105)

$\underset{\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0})}{\lim_{x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\rightarrow x_{0}}}Q\left(x_{0},P_{0},x_{1},J_{x_{1}}^{\left(\bar{m}\right)}F,\cdots,x_{k},J_{x_{k}}^{\left(\bar{m}\right)}F,z_{1},\cdots,z_{L}\right)=0$ if and only if

[TABLE]

• (4.0.107)

We define

[TABLE]

to consist of all $\left(P_{0},P\right)$ such that

[TABLE]

Then, $H^{\prime}\left(x_{0}\right)\subset\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\oplus\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ is a vector space depending semialgebraically on $x_{0}$; see ((3.1.11)). Moreover, ((4.0.105)) now tells us the following.

• (4.0.109)

Suppose (4.0.94) holds, $\underline{E}\left(x_{0}\right)\not=\emptyset$, but $\left(x_{0},\cdots,x_{0}\right)\not\in\underline{E}\left(x_{0}\right)$. Then

[TABLE]

if and only if

[TABLE]

Next, suppose (4.0.94) holds and $\underline{E}\left(x_{0}\right)=\emptyset$. Then we are in Case 2 above; see (4.0.38)$,\cdots$,((4.0.44)).

In particular, ((4.0.44)) holds for $\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0})$; hence (4) and (4) hold. Therefore, as in our discussion of ((4.0.105))$\cdots$((4.0.109)), we learn from ((4.0.107)) that

[TABLE]

if and only if

[TABLE]

Together with ((4.0.109)), this tells us the following.

• (4.0.111)

Suppose (4.0.94) holds and $\left(x_{0},\cdots,x_{0}\right)\not\in\underline{E}^{+}\left(x_{0}\right)$. Then

[TABLE]

if and only if

[TABLE]

Next, exploiting ((4.0.1)), we apply our inductive hypothesis (Propositions 2 and 3 hold for $\dim E^{+}<\Delta$) to

[TABLE]

restricted to $\left\{\left(x_{1},\cdots,x_{k},z_{1},\cdots,z_{L}\right)\in E^{++}(x_{0}):\left(x_{1},\cdots,x_{k}\right)\in\underline{E}^{+}(x_{0})\right\}$. From Proposition 3 applied to this case, we obtain an integer $\underline{\underline{\hat{m}}}\geq\bar{m}$ and a computable family of vector spaces

[TABLE]

depending semialgebraically on $x_{0}\in E$, such that the following holds.

• (4.0.113)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$. Suppose that $\underline{E}^{+}(x_{0})\not=\emptyset$ (i.e., $x_{0}\in\underline{E}$), and

[TABLE]

Then

[TABLE]

if and only if

[TABLE]

From ((4.0.103)), ((4.0.111)), ((4.0.113)), we obtain the following.

• (4.0.115)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $F\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{I}\right)$ be given. Suppose that

[TABLE]

Then

• (a)

If $\left(x_{0},\cdots,x_{0}\right)\in\underline{E}^{+}\left(x_{0}\right)$, then

[TABLE]

if and only if

[TABLE]

and

[TABLE]

• (b)

If $\left(x_{0},\cdots,x_{0}\right)\not\in\underline{E}^{+}\left(x_{0}\right)$, then

[TABLE]

if and only if

[TABLE]

From ((4.0.115)), we obtain at once the conclusion of Proposition 3 for the data $E$, $E^{+}$, $E^{++}$, $Q$, $A$.

This concludes our inductive proof of Propositions 2 and 3. ∎

5. Bundles and Glaeser Refinements

5.1. Notation, Definitions, Preliminaries

Let $Q\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right);$ say $Q=\left(Q_{1},\cdots,Q_{D}\right)$ with each $Q_{i}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}\right)$. Let $P\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}\right)$.

For $x\in\mathbb{R}^{n}$, we define $P\odot_{x}Q:=\left(P\odot_{x}Q_{1},\cdots,P\odot_{x}Q_{D}\right)$, where $P\odot_{x}Q_{i}=J_{x}^{\left(m\right)}\left(PQ_{i}\right)$ is the product of $P$ and $Q_{i}$ as $m$-jets at $x$. The above multiplication $\odot_{x}$ makes $\mathscr{R}_{x}^{m}:=\mathcal{P}^{(m)}\left(\mathbb{R}^{n}\right)$ into a ring, and it also makes $\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ into an $\mathscr{R}_{x}^{m}$-module.

Let $E\subset\mathbb{R}^{n}$ be compact. Fix $m\geq 0$, $D\geq 1$. A bundle over $E$ is a family $\mathscr{H}=\left(H_{x}\right)_{x\in E}$ parameterized by points $x\in E$, where, for each $x$, the fiber $H_{x}$ is either the empty set or else has the form

[TABLE]

where $f\left(x\right)\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ and $I\left(x\right)\subset\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ is an $\mathscr{R}_{x}^{m}$-submodule.

We call $\mathscr{H}$ proper if each of its fibers is non-empty. Note that compactness of $E$ is part of the definition of a bundle. Let $\mathscr{H}=\left(H_{x}\right)_{x\in E}$ be a bundle, and let $F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. We say that $F$ is a section of $\mathscr{H}$ if $J_{x}^{\left(m\right)}F\in H_{x}$ for all $x\in E$. Clearly, this cannot happen unless $\mathscr{H}$ is proper.

Next, we define the strong Glaeser refinement of a bundle $\mathscr{H}=\left(H_{x}\right)_{x\in E}$, denoted by $\mathscr{G}\left(\mathscr{H}\right)=\left(\tilde{H}_{x}\right)_{x\in E}$. For any $x_{0}\in E$, the fiber $\tilde{H}_{x_{0}}$ consists of all $P_{0}=\left(P_{0,1},\cdots,P_{0,D}\right)\in H_{x_{0}}$ satisfying the following conditions, for a large enough $\bar{k}$ determined by $m,n,D$:

• (GR1)

For some finite constant $K$, we have

[TABLE]

for all $x_{1},\cdots,x_{\bar{k}}\in E$.

• (GR2)

The left-hand side of $\left(\ref{NDP1}\right)$ tends to zero as $\left(x_{1},\cdots,x_{\bar{k}}\right)\rightarrow\left(x_{0},\cdots,x_{0}\right)$ in $E^{\bar{k}}$.

This definition differs from the usual definition of Glaeser refinement in previous papers [6, 7, 9] on Whitney’s problem.

The fiber at $x_{0}$ of the standard Glaeser refinement of $\mathscr{H}$ is defined to consist of all $P_{0}\in H_{x_{0}}$ that satisfy (GR2); we do not require (GR1). This notion agrees with the definition of the Glaeser refinement in the previous papers on Whitney’s problems.

Note that the strong Glaeser refinement of $\mathscr{H}$ is a subbundle of the standard Glaeser refinement of $\mathscr{H}$, which in turn is a subbundle of $\mathscr{H}$.

(We say that $\mathscr{H}=\left(H_{x}\right)_{x\in E}$ is a subbundle of $\mathscr{H}^{\prime}=\left(H_{x}^{\prime}\right)_{x\in E}$ if $\mathscr{H}$ and $\mathscr{H}^{\prime}$ are bundles and $H_{x}\subseteq H_{x}^{\prime}$ for all $x\in E$.)

Moreover, Taylor’s theorem implies that any section of $\mathscr{H}$ is already a section of $\mathscr{G}(\mathscr{H})$, the strong Glaeser refinement of $\mathscr{H}$.

For any bundle $\mathscr{H}$, and for any integer $l\geq 0$, we define the $l^{th}$ iterated (strong) Glaeser refinement of $\mathscr{H}$ by the following induction:

$\mathscr{G}^{\left(0\right)}\mathscr{H}=\mathscr{H};\mathscr{G}^{\left(l+1\right)}\mathscr{H}=\mathscr{G}\left(\mathscr{G}^{\left(l\right)}\mathscr{H}\right)$. In principle, we can compute any given $\mathscr{G}^{(l)}(\mathscr{H})$ from $\mathscr{H}$.

By induction on $l$, we see that the sections of $\mathscr{H}$ are the same as the sections of $\mathscr{G}^{\left(l\right)}\mathscr{H}$.

The following result is therefore immediate from the corresponding assertion for the standard Glaeser refinement, proven in the papers [6, 9] for $m\geq 1$, and in [8] for $m=0$.

Theorem 4**.**

There exists $l_{\ast}$, depending only on $m$, $n$, $D$, for which the following holds. Let $\mathscr{H}$ be a bundle. Then $\mathscr{H}$ has a section if and only if $\mathscr{G}^{\left(l_{\ast}\right)}\mathscr{H}$ is a proper bundle.

Let $\mathscr{H}=\left(f\left(x\right)+I\left(x\right)\right)_{x\in E}$ be a proper bundle. Then $\mathscr{H}=\left(\tilde{f}\left(x\right)+I\left(x\right)\right)_{x\in E}$ whenever $\tilde{f}\left(x\right)-f\left(x\right)\in I\left(x\right)$ for all $x\in E$. Thus, $\mathscr{H}$ uniquely determines $I\left(x\right)$, but it determines $f\left(x\right)$ only modulo $I\left(x\right)$. If $\varphi\in C^{\infty}_{0}\left(\mathbb{R}^{n}\right)$, then we define

[TABLE]

Thus, $\varphi\odot\mathscr{H}$ is again a proper bundle. Note that our definition of $\varphi\odot\mathscr{H}$ is independent of the choice of $f\left(x\right)$, i.e., it is unaffected by changing $f\left(x\right)$ to $\tilde{f}\left(x\right)$ when $f\left(x\right)-\tilde{f}\left(x\right)\in I\left(x\right)$. If $F$ is a section of $\mathscr{H}$, then $\varphi F$ is a section of $\varphi\odot\mathscr{H}$.

The operation $\mathscr{H}\mapsto\varphi\odot\mathscr{H}$ is related to the (strong) Glaeser refinement as follows:

Lemma 2**.**

Let $\mathscr{H}=(H_{x})_{x\in E}$ be a bundle with strong Glaeser refinement $\tilde{\mathscr{H}}=(\tilde{H}_{x})_{x\in E}$, and let $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n})$. If $P_{0}\in\tilde{H}_{x_{0}}$, then $J_{x_{0}}^{(m)}\varphi\odot_{x_{0}}P_{0}$ belongs to the fiber of $\mathscr{G}(\varphi\odot\mathscr{H})$ at $x_{0}$.

Proof.

We write $K_{1},K_{2},\cdots$ to denote constants determined by $m,n,D,\varphi,E,\mathscr{H},$ $P_{0}$.

Let $x_{1},\cdots,x_{\bar{k}}\in E$. Because (GR1) holds for $x_{0},P_{0},\mathscr{H}$, there exist $P_{1}\in H_{x_{1}},\cdots,P_{\bar{k}}\in H_{x_{\bar{k}}}$ such that

• (5.1.2)

$|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq K_{1}|x_{i}-x_{j}|^{m-|\alpha|}$ for $|\alpha|\leq m,i,j=0,\cdots,\bar{k}$.

(We adopt the convention that $0^{0}=0$ to deal with the degenerate case $|\alpha|=m,x_{i}=x_{j}$.)

Taking $j=0$, we see that

• (5.1.4)

$|\partial^{\alpha}P_{i}(x_{0})|\leq K_{2}$ for $|\alpha|\leq m,i=0,\cdots,\bar{k}$.

By expanding $P_{i}(y)$ in powers of $y-x_{0}$, we deduce from ((5.1.4))

• (5.1.6)

$|\partial^{\alpha}P_{i}(y)|\leq K_{3}$ for $|\alpha|\leq m,|y-x_{0}|\leq\text{diameter}(E),i=0,\cdots,\bar{k}$.

In particular,

• (5.1.8)

$|\partial^{\alpha}P_{i}(x_{j})|\leq K_{3}$ for $|\alpha|\leq m,i,j=0,\cdots,\bar{k}$.

For $k=0,\cdots,\bar{k}$, let

• (5.1.10)

$P_{k}^{\#}=J_{x_{k}}^{(m)}\varphi\odot_{x_{k}}P_{k}$. Thus,

• (5.1.12)

$P_{k}^{\#}$ belongs to the fiber of $\varphi\odot\mathscr{H}$ at $x_{k}$, for each $k=0,\cdots,\bar{k}$.

We estimate the derivatives of $P_{k}^{\#}-P_{k^{\prime}}^{\#}$. To do so, we write

[TABLE]

From ((5.1.2)), we have

• (5.1.15)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(1\right)}\left(x_{k}\right)\right|\leq K_{4}\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$.

From ((5.1.8)) and Taylor’s theorem for $\varphi$, we have

• (5.1.17)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(2\right)}\left(x_{k}\right)\right|\leq K_{5}\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$.

Moreover,

• (5.1.19)

$R_{kk^{\prime}}^{\left(3\right)}=J_{x_{k}}^{\left(m\right)}G_{k^{\prime}}-J_{x_{k^{\prime}}}^{\left(m\right)}G_{k^{\prime}}$, where $G_{k^{\prime}}=\left(J_{x_{k^{\prime}}}^{\left(m\right)}\varphi\right)\cdot P_{k^{\prime}}$.

Because $J_{x_{k^{\prime}}}^{\left(m\right)}\varphi$ and $P_{k^{\prime}}$ are polynomials of degree at most $m$, ((5.1.6)) yields the estimate

• (5.1.21)

$\left|\partial^{\alpha}G_{k^{\prime}}\left(y\right)\right|\leq K_{6}$ for $\left|\alpha\right|=m+1$ and $\left|y-x\right|\leq$ diameter $\left(E\right)$.

Consequently, ((5.1.19)) and Taylor’s theorem tell us that

• (5.1.23)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(3\right)}\left(x\right)\right|\leq K_{7}\left|x_{k}-x_{k^{\prime}}\right|^{m+1-|\alpha|}\leq K_{8}\left|x_{k}-x_{k^{\prime}}\right|^{m-|\alpha|}$ for $|\alpha|\leq m,k,k^{\prime}=0,\cdots,\bar{k}$.

Putting ((5.1.15)), ((5.1.17)), ((5.1.23)) into (5.1), we learn that

• (5.1.25)

$\left|\partial^{\alpha}\left(P_{k}^{\#}-P_{k^{\prime}}^{\#}\right)\left(x_{k}\right)\right|\leq K_{9}\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$, $k,k^{\prime}=0,\cdots,\bar{k}$.

Together with ((5.1.12)), this tells us that

• (5.1.27)

$P_{0}^{\#}=J_{x_{0}}^{\left(m\right)}\varphi\odot_{x_{0}}P_{0}$ satisfies (GR1) for the point $x_{0}$ and the bundle $\varphi\odot\mathscr{H}$.

Similarly, we establish (GR2) for $P_{0}^{\#}$, $x_{0}$, $\varphi\odot\mathscr{H}$.

We sketch the argument. Let $0<\varepsilon<1$ be given. Let $\delta>0$ be as in (GR2) for $P_{0}$, $x_{0}$, $\mathscr{H}$, and let $x_{1},\cdots,x_{\bar{k}}\in B\left(x_{0},\hat{\delta}\right)\cap E$, where $\hat{\delta}\in\left(0,\delta\right)$ will be picked below.

Then there exist $P_{1}\in H_{x_{1}},\cdots,P_{\bar{k}}\in H_{x_{\bar{k}}}$ such that

• (5.1.29)

$\left|\partial^{\alpha}\left(P_{i}-P_{j}\right)\left(x_{j}\right)\right|\leq\varepsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$, $i,j=0,\cdots,\bar{k}$.

Thus, ((5.1.4))$\cdots$((5.1.8)) hold as before. Defining $P_{k}^{\#}$ $\left(k=0,1,\cdots,\bar{k}\right)$ by ((5.1.10)), we again have ((5.1.12)) and (5.1).

Estimate ((5.1.29)) implies that

• (5.1.31)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(1\right)}\left(x_{k}\right)\right|\leq K_{10}\varepsilon\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m,$ $k,k^{\prime}=0,\cdots,\bar{k}$.

Estimate ((5.1.8)) and Taylor’s theorem for $\varphi$ yield

• (5.1.33)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(2\right)}\left(x_{k}\right)\right|\leq K_{11}\left|x_{k}-x_{k^{\prime}}\right|^{\left(m+1\right)-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m,k,k^{\prime}=0,\cdots,\bar{k}$.

Moreover, ((5.1.19)) and ((5.1.21)) hold, from which we have

• (5.1.35)

$\left|\partial^{\alpha}R_{kk^{\prime}}^{\left(3\right)}\left(x_{k}\right)\right|\leq K_{12}\left|x_{k}-x_{k^{\prime}}\right|^{\left(m+1\right)-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$, $k,k^{\prime}=0,\cdots,\bar{k}$.

Taking $\hat{\delta}\leq\varepsilon$ so that $\left|x_{k}-x_{k^{\prime}}\right|^{\left(m+1\right)-\left|\alpha\right|}\leq\varepsilon\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$, we deduce from ((5.1.31)), ((5.1.33)), ((5.1.35)) and (5.1) that

• (5.1.37)

$\left|\partial^{\alpha}\left(P_{k}^{\#}-P_{k^{\prime}}^{\#}\right)\left(x_{k}\right)\right|\leq K_{13}\varepsilon\left|x_{k}-x_{k^{\prime}}\right|^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$, $k,k^{\prime}=0,\cdots,\bar{k}$.

From ((5.1.10)) and ((5.1.37)), we see that

• (5.1.39)

$P_{0}^{\#}=J_{x_{0}}^{\left(m\right)}\varphi\odot_{x_{0}}P_{0}$ satisfies (GR2) for the point $x_{0}$ and the bundle $\varphi\odot\mathscr{H}.$

Thanks to ((5.1.27)) and ((5.1.39)), we know that $J_{x_{0}}^{\left(m\right)}\varphi\odot_{x_{0}}P_{0}$ belongs to the fiber at $x_{0}$ of $\mathscr{G}\left(\varphi\odot\mathscr{H}\right)$.

The proof of Lemma 2 is complete. ∎

A homogeneous bundle is a bundle of the form $\mathscr{H}=\left(I\left(x\right)\right)_{x\in E}$ with each $I\left(x\right)\subset\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ an $\mathscr{R}_{x}^{m}$-submodule. (That is, we can take $f\equiv 0$ in $\mathscr{H}=\left(f\left(x\right)+I\left(x\right)\right)_{x\in E}$.)

The (strong) Glaeser refinement of a homogenous bundle is again a homogeneous bundle.

• (5.1.41)

Let $\mathscr{H}=\left(f\left(x\right)+I\left(x\right)\right)_{x\in E}$ be a proper bundle, and let $\left(\tilde{I}\left(x\right)\right)_{x\in E}$ be the (strong) Glaeser refinement of $\left(I\left(x\right)\right)_{x\in E}$. If for each $x\in E$, $g\left(x\right)$ belongs to the fiber at $x$ of the strong Glaeser refinement $\mathscr{G}\left(\mathscr{H}\right)$, then it follows that $\mathscr{G}\left(\mathscr{H}\right)=\left(g\left(x\right)+\tilde{I}\left(x\right)\right)_{x\in E}$.

Therefore, Lemma 2 implies the following.

Corollary 1**.**

Let $\mathscr{H}$ be a bundle, and suppose $\mathscr{G}\left(\mathscr{H}\right)$ is proper. Then for any $\varphi\in C^{\infty}_{0}\left(\mathbb{R}^{n}\right)$, we have $\mathscr{G}\left(\varphi\odot\mathscr{H}\right)=\varphi\odot\mathscr{G}\left(\mathscr{H}\right)$.

Proof.

For each $x\in E$, pick $f\left(x\right)$ in the fiber of $\mathscr{G}\left(\mathscr{H}\right)$ at $x$. In particular, $f\left(x\right)$ belongs to the fiber of $\mathscr{H}$ at $x$, so $\mathscr{H}=\left(f\left(x\right)+I\left(x\right)\right)_{x\in E}$ for a family of $\mathscr{R}_{x}^{m}$-submodules $I\left(x\right)\subset\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Let $\left(\tilde{I}\left(x\right)\right)_{x\in E}$ be the strong Glaeser refinement of $\left(I\left(x\right)\right)_{x\in E}$. We have $\mathscr{G}\left(\mathscr{H}\right)=\left(f\left(x\right)+\tilde{I}\left(x\right)\right)_{x\in E}$ by ((5.1.41)), hence $\varphi\odot\mathscr{G}\left(\mathscr{H}\right)=\left(J_{x}^{(m)}\varphi\odot_{x}f\left(x\right)+\tilde{I}\left(x\right)\right)_{x\in E};$ also $\varphi\odot\mathscr{H}=\left(J_{x}^{(m)}\varphi\odot_{x}f\left(x\right)+I\left(x\right)\right)_{x\in E}$, hence if $\mathscr{G}\left(\varphi\odot\mathscr{H}\right)$ is a proper bundle, then (by ((5.1.41))) the fiber at every $x\in E$ of $\mathscr{G}(\varphi\odot\mathscr{H})$ has the form

[TABLE]

for some $g\left(x\right)$. According to the preceding lemma, $\mathscr{G}(\varphi\odot\mathscr{H})$ is a proper bundle; and, with $g$ as in (5.1.43), we have $J_{x}^{(m)}\varphi\odot_{x}f\left(x\right)\in g\left(x\right)+\tilde{I}\left(x\right)$. Therefore, the fiber at $x$ of $\mathscr{G}\left(\varphi\odot\mathscr{H}\right)$ is equal to $J_{x}^{(m)}\varphi\odot_{x}f\left(x\right)+\tilde{I}\left(x\right)$. That’s also the fiber at $x$ of $\varphi\odot\mathscr{G}\left(\mathscr{H}\right)$, as we saw above. Thus, $\mathscr{G}\left(\varphi\odot\mathscr{H}\right)=\varphi\odot\mathscr{G}\left(\mathscr{H}\right)$, as claimed. ∎

6. Bundles Determined by Smooth Functions

Setup. Let $E\subset\mathbb{R}^{n}$ be compact, semialgebraic. For $x\in E$, let $I\left(x\right)\subset\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ be an $\mathscr{R}_{x}^{m}$-submodule, depending semialgebraically on $x$. For $l\geq 0$, let $\left(I^{\left(l\right)}\left(x\right)\right)_{x\in E}$ denote the $l^{th}$ (strong) Glaeser refinement of the homogeneous bundle $\left(I\left(x\right)\right)_{x\in E}$. Thus, $I^{\left(0\right)}\left(x\right)=I\left(x\right)$, and $\left(I^{\left(l+1\right)}\left(x\right)\right)_{x\in E}=\mathscr{G}\left(\left(I^{\left(l\right)}\left(x\right)\right)_{x\in E}\right)$. Note that $I^{(l)}(x)$ depends semialgebraically on $x$, by an obvious induction on $l$. Let $T\left(x\right):\mathcal{P}^{(\bar{m})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)\rightarrow\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ be a linear map, depending semialgebraically on $x\in E$. Here, $\bar{m}\geq m$. For $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, we write $\mathscr{H}_{f}$ to denote the bundle

[TABLE]

We make the following

Assumption:

[TABLE]

For $l\geq 0$, let $\mathscr{H}_{f}^{\left(l\right)}$ denote the $l^{th}$ (strong) Glaeser refinement of $\mathscr{H}_{f}$. Thus, $\mathscr{H}_{f}^{\left(0\right)}=\mathscr{H}_{f}$ and $\mathscr{H}_{f}^{\left(l+1\right)}=\mathscr{G}\left(\mathscr{H}_{f}^{\left(l\right)}\right)$.

Under the above assumptions, we will prove the following result:

Lemma 3** (Main Lemma on $\mathscr{H}_{f}$).**

For each $l\geq 0$, there exist an integer $\bar{\bar{m}}\geq\bar{m}$, a finite list of linear differential operators $L_{\nu}$ $\left(\nu=1,\cdots,\nu_{\max}\right)$, and a linear map

[TABLE]

with the following properties.

• •

Each $L_{\nu}$ has semialgebraic coefficients, and maps functions in $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ to scalar-valued functions on $\mathbb{R}^{n}$.

• •

Each $L_{\nu}$ has order at most $\bar{\bar{m}}$.

• •

$T^{\left(l\right)}\left(x\right)$* depends semialgebraically on $x.$*

• •

Let $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle (i.e., all of its fibers are non-empty) if and only if $L_{\nu}f=0$ on $E$ for each $\nu=1,\cdots,\nu_{\max}$.

• •

Let $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. If $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle, then $\mathscr{H}_{f}^{\left(l\right)}=\left(T^{\left(l\right)}\left(x\right)J_{x}^{\left(\bar{\bar{m}}\right)}f+I^{\left(l\right)}\left(x\right)\right)_{x\in E}$.

Proof.

We use induction on $l$. For $l=0$, we take $\bar{\bar{m}}=\bar{m}$, $\left\{L_{1},\cdots,L_{\nu_{\max}}\right\}=$ empty list $\left(\nu_{\max}=0\right)$, $T^{\left(0\right)}\left(x\right)=T\left(x\right).$ The conclusions of Lemma 3 (for $l=0)$ are immediate from our assumptions in Setup.

For the induction step, we fix $l\geq 1$ and assume that Lemma 3 on $\mathscr{H}_{f}$ holds with $l$ replaced by $l-1$. We then prove Lemma 3 on $\mathscr{H}_{f}$ for the given $l$.

By our inductive assumption, there exist $\hat{m}\geq m;$ linear differential operators

[TABLE]

of order at most $\hat{m}$, with semialgebraic coefficients; and a linear map $T^{old}\left(x\right)$ mapping $\mathcal{P}^{(\hat{m})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)\rightarrow\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ such that the following hold.

• (6.0.3)

Let $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle if and only if $L_{\nu}^{old}f=0$ on $E$ for each $\nu=1,\cdots,\nu^{old}_{\max}$.

• (6.0.5)

Let $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. If $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle, then $\mathscr{H}_{f}^{\left(l-1\right)}=\left(T^{old}\left(x\right)J_{x}^{\left(\hat{m}\right)}f+I^{\left(l-1\right)}\left(x\right)\right)_{x\in E}$.

Suppose $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ are given; and assume that $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle.

We investigate whether $P_{0}$ belongs to the fiber at $x_{0}$ of $\mathscr{H}_{f}^{\left(l\right)}=\mathscr{G}\left(\mathscr{H}_{f}^{\left(l-1\right)}\right)$. We recall the definition (GR1) and (GR2) of the (strong) Glaeser refinement in Section 5.1.

For $x_{1},\cdots,x_{\bar{k}}\in E$, and for $P_{1},\cdots,P_{\bar{k}}\in\mathcal{P}^{(\hat{m})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, we define $Q\left(x_{0},P_{0},x_{1},P_{1},\cdots,x_{\bar{k}},P_{\bar{k}}\right)$ to be the minimum of the quantity

[TABLE]

over all $P_{1}^{\#}=\left(P_{1,1}^{\#},\cdots,P_{1,D}^{\#}\right),\cdots,P_{\bar{k}}^{\#}=\left(P_{\bar{k},1}^{\#},\cdots,P_{\bar{k},D}^{\#}\right)$ such that

[TABLE]

Here, we take $P_{0}^{\#}=(P_{0,1}^{\#},\cdots,P_{0,D}^{\#})$ to equal $P_{0}$.

By linear algebra, $Q\left(x_{0},P_{0},x_{1},P_{1},\cdots,x_{\bar{k}},P_{\bar{k}}\right)$ is a positive semidefinite quadratic form in $P_{0},P_{1},\cdots,P_{\bar{k}}$, for each fixed $x_{0},x_{1},\cdots,x_{\bar{k}}\in E$. Moreover, since $x\mapsto T^{old}\left(x\right)$ and $x\mapsto I^{(l-1)}(x)$ are semialgebraic, it follows that

[TABLE]

is a semialgebraic function of $\left(x_{0},P_{0},\cdots,x_{\bar{k}},P_{\bar{k}}\right)$.

If we define $A\left(x_{0},\cdots,x_{\bar{k}}\right)$ to be the norm of this quadratic form, i.e., the least $A\left(x_{0},\cdots,x_{\bar{k}}\right)$ such that

[TABLE]

for all $P_{0},P_{1},\cdots,P_{\bar{k}}$, then $A\left(x_{0},\cdots,x_{\bar{k}}\right)$ is a nonnegative semialgebraic function of $\left(x_{0},\cdots,x_{\bar{k}}\right)$.

Therefore, we have all the conditions assumed in the setup in Section 3.1. (The number of $z$’s there is zero.)

From Propositions 2 and 3, we have the following conclusions.

There exist $\bar{\bar{m}}\geq\hat{m};$ and families of vector subspaces

[TABLE]

and

[TABLE]

with the following properties.

• (6.0.7)

$H^{\text{bdd}}\left(x_{0},\cdots,x_{\bar{k}}\right)$ depends semi-algebraically on $\left(x_{0},x_{1},\cdots,x_{\bar{k}}\right)\in E\times\cdots\times E$.

• (6.0.9)

$H^{\lim}\left(x_{0}\right)$ depends semi-algebraically on $x_{0}\in E$.

• (6.0.11)

Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then

[TABLE]

is bounded if and only if $\left(P_{0},J_{x_{1}}^{\left(\bar{\bar{m}}\right)}f,\cdots,J_{x_{\bar{k}}}^{\left(\bar{\bar{m}}\right)}f\right)\in H^{\text{bdd}}\left(x_{0},x_{1},\cdots,x_{\bar{k}}\right)$ for all $x_{1},\cdots,x_{\bar{k}}\in E$.

• (6.0.13)

Let $x_{0}\in E$, $\mathcal{P}_{0}\in\mathcal{P}^{\left(m\right)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Suppose that

[TABLE]

is bounded. Then

[TABLE]

if and only if $\left(P_{0},J_{x_{0}}^{\left(\bar{\bar{m}}\right)}f\right)\in H^{\lim}\left(x_{0}\right)$.

Comparing our definition of $Q(\cdots)$ with (GR1), and (GR2) in the definition of the (strong) Glaeser refinement, we see the following.

Let $\mathscr{H}^{(l-1)}=\left(T^{old}\left(x\right)J_{x}^{\left(\hat{m}\right)}f+I^{\left(l-1\right)}\left(x\right)\right)_{x\in E}$, where $f\in C^{\infty}_{0}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ is given. Let $x_{0}\in E$, and let $P_{0}$ belong to the fiber of $\mathscr{H}^{(l-1)}$ at $x_{0}$. Then (GR1) holds for $\left(x_{0},P_{0}\right)$ and $\mathscr{H}^{(l-1)}$, if and only if

[TABLE]

is bounded. Moreover, (GR2) holds for $\left(x_{0},P_{0}\right)$ and $\mathscr{H}^{(l-1)}$, if and only if

[TABLE]

Recalling ((6.0.5)), ((6.0.11)), ((6.0.13)), we obtain the following result.

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, and let $P_{0}$ belong to the fiber of $\mathscr{H}_{f}^{\left(l-1\right)}$ at $x_{0}$. Then $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}=\mathscr{G}\left(\mathscr{H}_{f}^{\left(l-1\right)}\right)$ at $x_{0}$, if and only if $\left(P_{0},J_{x_{0}}^{\left(\bar{\bar{m}}\right)}f\right)\in H^{\lim}\left(x_{0}\right)$ and $\left(P_{0},J_{x_{1}}^{\left(\bar{\bar{m}}\right)}f,\cdots,J_{x_{\bar{k}}}^{\left(\bar{\bar{m}}\right)}f\right)\in H^{\text{bdd}}\left(x_{0},x_{1},\cdots,x_{\bar{k}}\right)$ for all $x_{1},\cdots,x_{\bar{k}}\in E$.

Together with ((6.0.3)) and ((6.0.5)), this immediately implies that there exists a family of vector spaces

• (6.0.15)

$H^{GR}\left(x_{0},x_{1},\cdots,x_{\bar{k}}\right)\subset\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)\oplus\sum_{i=0}^{\bar{k}}\mathcal{P}^{(\bar{\bar{m}})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ depending semialgebraically on

$x_{0},x_{1},\cdots,x_{\bar{k}}\in E,$

for which the following holds.

• (6.0.17)

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Then $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0}$ if and only if

[TABLE]

for all $x_{1},\cdots,x_{\bar{k}}\in E$.

By linear algebra (and basic facts about semialgebraic sets), there exist finitely many linear functionals

[TABLE]

depending semialgebraically on $x_{0},x_{1},\cdots,x_{\bar{k}}\in E$, such that the following holds.

Let $x_{0},x_{1},\cdots,x_{\bar{k}}\in E$. Let $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ and let $P_{0}^{\#},\cdots,P_{\bar{k}}^{\#}\in\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then

[TABLE]

if and only if

[TABLE]

Hence, $\left(\ref{B9}\right)$ yields the following result.

• (6.0.19)

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Then $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0}$ if and only if

[TABLE]

for $x_{1},\cdots,x_{\bar{k}}\in E$ and $\nu=1,\cdots,\nu_{\max}^{\#}$.

It may happen that $x_{i}=x_{j}$ in $\left(\ref{B10}\right)$ for some $i\not=j$. In that case, $\lambda_{i,\nu}$ and $\lambda_{j,\nu}$ play redundant roles. We may easily rewrite $\left(\ref{B10}\right)$ as follows.

For each integer $k$ $\left(0\leq k\leq\bar{k}\right)$, there are linear functionals

[TABLE]

and

[TABLE]

$i=0,\cdots,k$, $\nu=1,\cdots,\nu_{\max}^{\#}\left(k\right)$, depending semialgebraically on $\left(x_{0},\cdots,x_{{k}}\right)\in E\times\cdots\times E$, for which we have the following.

• (6.0.21)

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Then $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0}$ if and only if, for each $0\leq k\leq\bar{k}$, for each $x_{1},\cdots,x_{k}\in E\setminus\{x_{0}\}$ distinct, and for each $\nu=1,\cdots,\nu_{\max}^{\#}\left({k}\right)$, we have

[TABLE]

Now let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, and suppose $P_{0}$ belongs to the fiber at $x_{0}$ of $\mathscr{H}_{f}^{\left(l\right)}$.

Let $\varphi$ be a scalar-valued $C_{0}^{\infty}$ function on $\mathbb{R}^{n}$. Then $\mathscr{H}_{\varphi f}=\varphi\odot\mathscr{H}_{f}$ (see assumption (6.0.2)), hence by Corollary 1, we see that $\mathscr{H}_{\varphi f}^{\left(l-1\right)}=\varphi\odot\mathscr{H}_{f}^{\left(l-1\right)}$. In particular, $\mathscr{H}_{\varphi f}^{\left(l-1\right)}$ is a proper bundle, so $\left(\ref{B11}\right)$ applies, with $\varphi f$ in place of $f$, and with $J_{x_{0}}^{\left(m\right)}\varphi\odot_{x_{0}}P_{0}$ in place of $P_{0}$. Note that $J_{x_{0}}^{\left(m\right)}\varphi\odot_{x_{0}}P_{0}$ belongs to the fiber at $x_{0}$ of $\mathscr{H}_{\varphi f}^{\left(l\right)}=\mathscr{G}\left(\mathscr{H}_{\varphi f}^{\left(l-1\right)}\right)=\mathscr{G}\left(\varphi\odot\mathscr{H}_{f}^{\left(l-1\right)}\right)$, thanks to Lemma 2.

Therefore, from ((6.0.21)), we obtain the following result.

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$.

Suppose $P_{0}$ belongs to the fiber at $x_{0}$ of $\mathscr{H}_{f}^{\left(l\right)}$.

Then, for any $\varphi\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$, for each $0\leq k\leq\bar{k}$, for each $x_{1},\cdots,x_{k}\in E\setminus\{x_{0}\}$ distinct, and for each $\nu=1,\cdots,\nu_{\max}^{\#}\left(k\right)$, we have

[TABLE]

Let $0\leq k\leq\bar{k}$, $x_{1},\cdots,x_{k}\in E$ distinct (also distinct from $x_{0}$), and $\nu=1,\cdots,\nu_{\max}^{\#}\left(k\right)$ be given. For each $i=0,\cdots,k$, we may pick $\varphi\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ so that $\varphi\equiv 1$ in a neighborhood of $x_{i}$, but $\varphi\equiv 0$ in a neighborhood of each $x_{j}$ $\left(j\not=i\right)$. (That’s why we were careful to arrange that $x_{0},\cdots,x_{k}$ are distinct.)

If $i=0$, then (6.0.23) tells us that

[TABLE]

If $1\leq i\leq{k}$, then $\left(\ref{B12}\right)$ tells us that

[TABLE]

Thus, we have learned the following:

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$.

• (6.0.26)

If $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0}$, then $\left(\ref{B13}\right)$ holds for $0\leq k\leq\bar{k}$, $x_{1},\cdots,x_{{k}}\in E\setminus\left\{x_{0}\right\}$ distinct, and $\nu=1,\cdots,\nu_{\max}^{\#}\left(k\right);$ and also $\left(\ref{B14}\right)$ holds for $1\leq k\leq\bar{k}$, $x_{1},\cdots,x_{{k}}\in E\setminus\left\{x_{0}\right\}$ distinct, $i=1,\cdots,{k}$, and $\nu=1,\cdots,\nu_{\max}^{\#}\left(k\right)$.

The converse of ((6.0.26)) is immediate from ((6.0.21)). Therefore, we obtain the following result, since the $\mu_{\nu}^{k}\left(x_{0},\cdots,x_{k}\right)$ and $\lambda_{i,\nu}^{k}\left(x_{0},\cdots,x_{k}\right)$ are semialgebraic.

There exist families of vector subspaces

[TABLE]

and

[TABLE]

with the following properties.

• (6.0.28)

$H^{GR,0}\left(x_{0}\right)$ depends semialgebraically on $x_{0}$; $H^{GR,1}\left(x_{0},x\right)$ depends semialgebraically on $x_{0}$, $x$.

• (6.0.30)

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Let $x_{0}\in E$, $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Then $P_{0}$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0}$ if and only if

[TABLE]

and

[TABLE]

For each $x_{0}\in E$, there exist a vector subspace $H^{GR,00}\left(x_{0}\right)\subset\mathcal{P}^{(\bar{\bar{m}})}(\mathbb{R}^{n},\mathbb{R}^{M})$ and a linear map

[TABLE]

both depending semialgebraically on $x_{0}$, such that the following holds:

• (6.0.32)

Let $x_{0}\in E$ and let $P^{\#}\in\mathcal{P}^{\left(\bar{\bar{m}}\right)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then there exists $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ such that $\left(P_{0},P^{\#}\right)\in H^{GR,0}\left(x_{0}\right)$, if and only if $P^{\#}\in H^{GR,00}\left(x_{0}\right)$. Moreover, if such a $P_{0}$ exists, then $\left(T^{new}\left(x_{0}\right)P^{\#},P^{\#}\right)\in H^{GR,0}\left(x_{0}\right)$.

Indeed, ((6.0.32)) follows from linear algebra and standard properties of semialgebraic sets and functions.

Now let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. According to $\left(\ref{B17}\right)$, $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle if and only if

• •

$J_{x}^{\left(\bar{\bar{m}}\right)}f\in H^{GR,1}\left(x_{0},x\right)$ for all $x_{0},x\in E$ with $x_{0}\not=x$, and

• •

For each $x_{0}\in E$ there exists $P_{0}\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ such that $\left(P_{0},J_{x_{0}}^{\left(\bar{\bar{m}}\right)}f\right)\in H^{GR,0}\left(x_{0}\right)$.

Therefore, ((6.0.30)) and ((6.0.32)) tell us the following.

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Then $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle if and only if

• •

$J_{x}^{\left(\bar{\bar{m}}\right)}f\in H^{GR,1}\left(x_{0},x\right)$ for all $x_{0},x\in E$ with $x_{0}\not=x$, and

• •

$J_{x_{0}}^{\left(\bar{\bar{m}}\right)}f\in H^{GR,00}\left(x_{0}\right)$ for all $x_{0}\in E$.

Moreover, if $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle, then, for each $x_{0}\in E$, $T^{new}\left(x_{0}\right)J_{x_{0}}^{\left(\bar{\bar{m}}\right)}f$ belongs to the fiber of $\mathscr{H}_{f}^{\left(l\right)}$ at $x_{0};$ consequently if $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle, then $\mathscr{H}_{f}^{\left(l\right)}=\left(T^{new}\left(x\right)J_{x}^{\left(\bar{\bar{m}}\right)}f+I^{\left(l\right)}\left(x\right)\right)_{x\in E}.$ (Recall that $\left(I^{\left(l\right)}\left(x\right)\right)_{x\in E}$ is the $l^{th}$ (strong) Glaeser refinement of the homogeneous bundle $\left(I\left(x\right)\right)_{x\in E}$ introduced in the setup of this section.)

The above remarks imply that there exists a family of vector subspaces $H^{new}\left(x\right)\mathcal{\subset\mathcal{P}}^{(\bar{\bar{m}})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, depending semialgebraically on $x\in E$, such that the following holds.

• (6.0.34)

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, and suppose $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle. Then $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle if and only if $J_{x}^{\left(\bar{\bar{m}}\right)}f\in H^{new}\left(x\right)$ for all $x\in E$. Moreover, if $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle, then $\mathscr{H}_{f}^{\left(l\right)}=\left(T^{new}\left(x\right)J_{x}^{\left(\bar{\bar{m}}\right)}f+I^{\left(l\right)}\left(x\right)\right)_{x\in E}$.

We can now produce linear functionals $\lambda_{\nu}^{new}\left(x\right):\mathcal{P}^{(\bar{\bar{m}})}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)\rightarrow\mathbb{R}$, $\nu=1,\cdots,\nu_{\max}^{new}$, depending semialgebraically on $x\in E$ such that

[TABLE]

Setting $L_{\nu}^{new}f\left(x\right):=\lambda_{\nu}^{new}\left(x\right)\left[J_{x}^{\left(\bar{\bar{m}}\right)}f\right]$, we obtain the following from ((6.0.34)), (6.0.36):

• •

Each $L_{\nu}^{new}$ is a linear differential operator, with semialgebraic coefficients, carrying functions in $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ to scalar-valued functions.

• •

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Then $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle if and only if $\mathscr{H}_{f}^{\left(l-1\right)}$ is a proper bundle and $L_{\nu}^{new}f=0$ for $\nu=1,\cdots,\nu_{\max}^{new}$.

• •

Let $f\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Suppose $\mathscr{H}_{f}^{\left(l\right)}$ is a proper bundle. Then

[TABLE]

Recall that $T^{new}\left(x\right)$ depends semialgebraically on $x\in E$. The above bullet points and ((6.0.3)) together imply the conclusions of Lemma 3.

This completes our induction on $l$, proving Lemma 3. ∎

7. Proof of Theorem 2

Proof.

In proving Theorem 2, we may suppose that $f$ has compact support, simply because $E$ is compact.

Thus, let $E\subset\mathbb{R}^{n}$ be a compact semialgebraic set, and let $M,N\geq 1$. Suppose we are given a matrix $\left(A_{ij}\right)_{1\leq i\leq N,1\leq j\leq M}$ of semialgebraic functions on $E$. Suppose further we are given $f_{i}\in C_{0}^{\infty}(\mathbb{R}^{n})$, $i=1,\cdots,N$.

We want to solve the equations

[TABLE]

for unknown functions $F_{1},\cdots,F_{M}\in C^{m}\left(\mathbb{R}^{n}\right)$.

We write $f=\left(f_{1},\cdots,f_{N}\right)\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$.

We reinterpret (7.0.1) in terms of bundles. For $x\in E$, let $I\left(x\right)\subset\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ be the set of all $P=\left(P_{1},\cdots,P_{M}\right)\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ such that

[TABLE]

Note that (7.0.2) involves only the values of the $P_{j}$ at $x$, and thus discards the higher-order information encoded in $P_{j}$.

Note also that $I(x)$ depends semialgebraically on $x$.

Observe that $I\left(x\right)$ is an $\mathscr{R}_{x}^{m}$-submodule of $\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$. Indeed, if $P=\left(P_{1},\cdots,P_{M}\right)\in I\left(x\right)$, and if $Q\in\mathscr{R}_{x}^{m}$, then

[TABLE]

and

[TABLE]

Therefore, condition (7.0.2) for $P$ implies condition (7.0.2) for $Q\odot_{x}P$.

• (7.0.3)

For each $x\in E$, let $\Pi\left(x\right)$ denote the orthogonal projection from $\mathbb{R}^{N}$ onto the range of the matrix $\left(A_{ij}\left(x\right)\right)$.

• (7.0.5)

For $\left(\xi_{1},\cdots,\xi_{N}\right)\in\mathbb{R}^{N}$, let $T^{o}\left(x\right)\left[\xi_{1},\cdots,\xi_{N}\right]$ denote the vector $\left(\eta_{1},\cdots,\eta_{M}\right)\in\mathbb{R}^{M}$ that solves the equation

[TABLE]

with the (Euclidean) norm of $\left(\eta_{1},\cdots,\eta_{M}\right)$ as small as possible.

Thus, $\Pi\left(x\right)$ and $T^{o}\left(x\right)$ are matrices that depend semialgebraically on $x\in E$.

• (7.0.7)

For any $P=\left(P_{1},\cdots,P_{N}\right)\in\mathcal{P}^{(m)}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$, and any $x\in E$, let $T\left(x\right)\left[P\right]$ be the vector of constant polynomials given by $T^{o}\left(x\right)\left[P_{1}\left(x\right),\cdots,P_{N}\left(x\right)\right]$.

We will prove the following two facts.

Lemma 4**.**

Let $f=\left(f_{1},\cdots,f_{N}\right)\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ be given. A function $F=\left(F_{1},\cdots,F_{M}\right)\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ solves equations $\left(\ref{S1}\right)$ if and only if it is a section of the bundle

[TABLE]

and $\left(I-\Pi\left(x\right)\right)f\left(x\right)=0$ for all $x\in E$.

Lemma 5**.**

For $f=\left(f_{1},\cdots,f_{N}\right)\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$, let $\mathscr{H}_{f}$ be defined by $\left(\ref{S6}\right)$. Then for $\varphi\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ and $f=\left(f_{1},\cdots,f_{N}\right)\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ we have

[TABLE]

Proof of Lemma 4.

If $\left(I-\Pi\left(x\right)\right)f\left(x\right)\not=0$ for some $x\in E$, then $\left(f_{1}\left(x\right),\cdots,f_{N}\left(x\right)\right)$ doesn’t belong to the range of $\left(A_{ij}\left(x\right)\right)$, so obviously equations (7.0.1) have no solution. Hence, we may assume that

[TABLE]

for all $x\in E$.

We then have (by ((7.0.5))):

[TABLE]

for each $x\in E$, hence

[TABLE]

Therefore, for $F=\left(F_{1},\cdots,F_{M}\right)\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, we have

[TABLE]

if and only if

[TABLE]

[TABLE]

(see the definition (7.0.2) of $I\left(x\right)$).

That is, whenever $\left(I-\Pi\left(x\right)\right)f\left(x\right)=0$ for all $x\in E$, we find that $F$ solves $\left(\ref{S1}\right)$ if and only if

[TABLE]

if and only if $F$ is a section of $\mathscr{H}_{f}$; see $\left(\ref{S6}\right)$. The proof of Lemma 4 is complete. ∎

Proof of Lemma 5.

We have

[TABLE]

so

[TABLE]

by definition of $\odot$. Also, by definition,

[TABLE]

To establish Lemma 5, we must therefore show that

[TABLE]

By the definition of $I\left(x\right)$, this means that

[TABLE]

We will check the stronger result that the expression in square brackets is zero.

In fact, $T\left(x\right)J_{x}^{\left(m\right)}f$ is the vector of constant polynomials T^{o}\left(x\right)\left(\begin{array}[]{c}f_{1}\left(x\right)\\ \vdots\\ f_{N}\left(x\right)\end{array}\right) by definition $\left(\ref{S5}\right)$.

Hence, \varphi\left(x\right)\left\{T\left(x\right)J_{x}^{\left(m\right)}f\right\}\left(x\right)=\varphi\left(x\right)\cdot T^{o}\left(x\right)\left(\begin{array}[]{c}f_{1}\left(x\right)\\ \vdots\\ f_{N}\left(x\right)\end{array}\right).

Another application of definition $\left(\ref{S5}\right)$ yields

[TABLE]

Consequently, the expression in square brackets in $\left(\ref{S7}\right)$ is equal to

[TABLE]

which equals zero, as promised. The proof of Lemma 5 is complete. ∎

Thanks to Lemma 5, our bundle $\mathscr{H}_{f}$ in $\left(\ref{S6}\right)$ satisfies the assumptions made in Section 6.

Hence, we may apply Lemma 3.

We take $l$ in Lemma 3 to equal the large constant $l_{\ast}$ from Theorem 4.

We can then argue as follows.

Let $f=\left(f_{1},\cdots,f_{N}\right)\in C_{0}^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ be given. Then, by Lemma 4, the equations (7.0.1) admit a $C^{m}$ solution $\left(F_{1},\cdots,F_{M}\right)$ if and only if

[TABLE]

for all $x\in E$ and $\mathscr{H}_{f}$ has a section.

By Theorem 4, this holds if and only if $\left(I-\Pi\left(x\right)\right)f\left(x\right)=0$ for all $x\in E$ and $\mathscr{G}^{\left(l_{\ast}\right)}\mathscr{H}_{f}$ is a proper bundle.

By Lemma 3, this in turn holds if and only if $\left(I-\Pi\left(x\right)\right)f\left(x\right)=0$ for all $x\in E$ and $L_{\nu}f=0$ on $E$ for $\nu=1,\cdots,\nu_{\max}$, where each $L_{\nu}$ is a linear partial differential operator with semialgebraic coefficients, mapping functions in $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ to scalar-valued functions on $\mathbb{R}^{n}$.

Since the equation $\left(I-\Pi\left(x\right)\right)f\left(x\right)=0$ on $E$ is also a system of such linear partial differential equations (of order 0), the proof of Theorem 2 is complete. ∎

8. Proof of Theorem 1

Proof.

Let $U$ be a bounded open semialgebraic subset of $\mathbb{R}^{n}$ and let $\left(A_{ij}(x)\right)_{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$.

According to Theorem 2, there exist linear partial differential operators $L_{\nu}$ $(\nu=1,\cdots,\nu_{\max})$ with semialgebraic coefficients, such that given $f\in C^{\infty}(\mathbb{R}^{n},\mathbb{R}^{N})$ there exists $F\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{M})$ such that $\sum_{j=1}^{M}A_{ij}(x)F_{j}(x)=f_{i}(x)$ $(i=1,\cdots,N)$, all $x\in U^{\text{closure}}$ if and only if $L_{\nu}f=0$ on $U^{\text{closure}}$, all $\nu$.

Let $\bar{m}$ be greater than or equal to the order of each $L_{\nu}$.

Now, suppose $F\in C_{loc}^{m}\left(U,\mathbb{R}^{M}\right)$, $f\in C^{\infty}\left(U,\mathbb{R}^{N}\right)$, and $\sum_{j=1}^{M}A_{ij}\left(x\right)F_{j}\left(x\right)=f_{i}\left(x\right)$ $\left(i=1,\cdots,N\right)$ on $U$.

Fix $x_{0}\in U$, and let $\theta\in C_{0}^{\infty}\left(U\right)$.

Then $\theta F\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$, $\theta f\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$, and $\sum_{j=1}^{M}A_{ij}\left(x\right)\left(\theta F_{j}\right)\left(x\right)=\left(\theta f_{i}\right)\left(x\right)$ $\left(i=1,\cdots,N\right)$ on $U^{\text{closure}}$.

Consequently, $L_{\nu}\left(\theta f\right)(x_{0})=0$ for all $\nu$.

Given any $P\in\mathcal{P}^{\left(\bar{m}\right)}$ there exists $\theta\in C_{0}^{\infty}\left(U\right)$ such that $J_{x_{0}}^{\left(\bar{m}\right)}\theta=P$, hence $L_{\nu}\left(\theta f\right)\left(x_{0}\right)=L_{\nu}\left(Pf\right)\left(x_{0}\right)$.

Thus, $L_{\nu}\left(Pf\right)=0$ on $U$, all $P\in\mathcal{P}^{\left(\bar{m}\right)}$, i.e.,

• (8.0.1)

$L_{\nu}\left(x^{\gamma}f\right)=0$ on $U$ for all $\left|\gamma\right|\leq\bar{m}$ and each $\nu$.

Thus, if $f\in C^{\infty}\left(U,\mathbb{R}^{N}\right)$ admits a solution $F\in C_{loc}^{m}\left(U,\mathbb{R}^{M}\right)$ of $\sum_{j=1}^{M}A_{ij}F_{j}=f_{i}$ ($i=1,\cdots,N$) on $U$, then ((8.0.1)) holds.

Conversely, suppose ((8.0.1)) holds on $U$. Then $L_{\nu}\left(Pf\right)\left(x_{0}\right)=0$ for any $x_{0}\in U$, $P\in\mathcal{P}^{\left(\bar{m}\right)},$ $\nu=1,\cdots,\nu_{\max}$.

Let $\theta\in C^{\infty}\left(U\right)$. Then $L_{\nu}\left(\theta f\right)\left(x_{0}\right)=L_{\nu}\left(\left[J_{x_{0}}^{\left(\bar{m}\right)}\theta\right]f\right)\left(x_{0}\right)=0,$ for any $x_{0}\in U$; i.e., $L_{\nu}\left(\theta f\right)=0$ on $U$.

Lemma 6**.**

Given $f_{1},\cdots,f_{N}\in C^{\infty}\left(U\right)$, there exists $\theta\in C^{\infty}\left(U\right)$ such that $\theta>0$ on $U$, $\mathbb{I}_{U}\theta f_{i}\in C^{\infty}\left(\mathbb{R}^{n}\right)$ for each $i$, and $J_{x}^{\left(\bar{m}\right)}\left(\mathbb{I}_{U}\theta f_{i}\right)=0$ for each $x\in\partial U$. Here $\mathbb{I}_{U}$ is the indicator function of $U$.

Proof.

Fix cutoff functions $\varphi_{\nu}\left(x\right)$ $\left(\nu=1,2,3,\cdots\right)$ with the following properties.

• •

Each $\varphi_{\nu}$ is a nonnegative $C_{0}^{\infty}$ function on $\mathbb{R}^{n}$.

• •

Supp $\left(\varphi_{\nu}\right)\subset\subset U$ for each $\nu$.

• •

For any $x\in U$ we have $\varphi_{\nu}\left(x\right)>0$ for some $\nu$.

For instance, we may take $\left\{\varphi_{\nu}\right\}$ to be the Whitney partition of unity, associated to the decomposition of $U$ into Whitney cubes. (See [16].)

We then fix a sequence of positive numbers $\tau_{\nu}$ $\left(\nu=1,2,3,\cdots\right)$ with the following properties.

• •

$\tau_{\nu}\cdot\left|\partial^{\alpha}\varphi_{\nu}\left(x\right)\right|\leq 2^{-\nu}$ for $\left|\alpha\right|\leq\nu,$ $x\in\mathbb{R}^{n}$.

• •

$\tau_{\nu}\cdot\left|\partial^{\alpha}\left(\varphi_{\nu}f_{i}\right)\left(x\right)\right|\leq 2^{-\nu}$ for $\left|\alpha\right|\leq\nu,$ $x\in\mathbb{R}^{n}$, $i=1,\cdots,N$.

Such $\tau_{\nu}$ exist because $\varphi_{\nu}$, $f_{i}\in C^{\infty}\left(U\right)$ and supp $\varphi_{\nu}\subset\subset U$.

One checks easily that

[TABLE]

has all the properties asserted in the lemma. ∎

Picking $\theta$ as in Lemma 6, we see that

• (8.0.3)

$L_{\nu}\left(\mathbb{I}_{U}\theta f_{i}\right)=0$ on $U^{\text{closure}}$.

Indeed ((8.0.3)) holds in $U$, and it holds on $\partial U$ because $J_{x}^{\left(\bar{m}\right)}\left(\mathbb{I}_{U}\theta f_{i}\right)=0$ for $x\in\partial U$, while $L_{\nu}$ has order $\leq\bar{m}$.

Because $\mathbb{I}_{U}\theta f$ belongs to $C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{N}\right)$ and is annihilated by the $L_{\nu}$ on $U^{\text{closure}}$, there exists $\tilde{F}=\left(\tilde{F}_{1},\cdots,\tilde{F}_{M}\right)\in C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{M}\right)$ such that $\sum_{j=1}^{M}A_{ij}\tilde{F}_{j}=\mathbb{I}_{U}\theta f_{i}$ on $U^{\text{closure}}$ ($i=1,\cdots,N$). Setting $F_{j}=\tilde{F}_{j}/\theta$ on $U$ (recall, $\theta>0$ on $U,$ $\theta\in C^{\infty}\left(U\right)$), we have $F_{j}\in C_{loc}^{m}\left(U\right)$ and $\sum_{j=1}^{M}A_{ij}F_{j}=f_{i}$ on $U$ ($i=1,\cdots,N$).

So we have proven the following corollary of Theorem 2.

Corollary 2**.**

Let $U$ be a bounded open semialgebraic subset of $\mathbb{R}^{n}$, and let $\left(A_{ij}\left(x\right)\right)_{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $U$. Then there exist linear partial differential operators $L_{1},\cdots,L_{\nu_{\max}}$ with semialgebraic coefficients such that

• •

Each $L_{\nu}$ maps vectors $f$ of smooth functions to scalar-valued functions.

• •

Let $f=\left(f_{1},\cdots,f_{N}\right)\in C^{\infty}\left(U,\mathbb{R}^{N}\right)$. Then the equations

[TABLE]

admit a solution $F_{1},\cdots,F_{N}\in C_{loc}^{m}\left(U\right)$ if and only if $L_{\nu}f=0$ on $U$ for $\nu=1,\cdots,\nu_{\max}$.

Finally, note that $\mathbb{R}^{n}$ is semialgebraically diffeomorphic to the open unit cube $U=\left(-1,1\right)^{n}$ via the map

[TABLE]

Theorem 1 now follows at once from Corollary 2. (Recall that our notation has changed; the function space called $C^{m}(\mathbb{R}^{n},\mathbb{R}^{M})$ in the Introduction is now called $C^{m}_{loc}(\mathbb{R}^{n},\mathbb{R}^{M})$.) ∎

As promised, all the semialgebraic sets and functions introduced in Sections 3,…,8 above can be computed using the results and techniques presented in Sections 2.5 and 2.6. Therefore, in principle, we can compute the partial differential operators appearing in Theorem 1.