Generators for the $C^m$-closures of Ideals

TL;DR

This paper presents an algorithm to compute generators for the $C^m$-closure of ideals in the ring of real polynomials, extending algebraic methods to smooth function contexts.

Contribution

The paper introduces a novel algorithm for explicitly computing generators of the $C^m$-closure of polynomial ideals.

Findings

Algorithm successfully computes generators for $C^m$-closures

Extends algebraic ideal theory to smooth function spaces

Provides a practical tool for analysis in real algebraic geometry

Abstract

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $. In this paper we exhibit an algorithm to compute generators for $\left[ A_{1},\cdots ,A_{M};C^{m}\right] $.

Equations

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

$$L\vec{f}\left(x\right)=\sum _{|\alpha |\le s}\sum _{i=1}^N{a}_{\alpha }^i\left(x\right){\partial }^{\alpha }{f}_i\left(x\right)\text{ for }\vec{f}=\left(f_1,\cdots ,f_N\right)\text{,}$$

$$\sum _{i=1}^N{A}_{\nu }^i\left(x\right)P_i\left(x\right)=0\left(\nu =1,\cdots ,{\nu }_{\max }\right)$$

$$L\left(f_1,\cdots ,f_N\right)\left(x\right)=\sum _{|\alpha |=s}{\partial }^{\alpha }\left\{\sum _{i=1}^N{a}_{\alpha }^i{f}_i\right\}\left(x\right)+\tilde{L}\left(f_1,\cdots ,f_N\right)\left(x\right)\text{,}$$

$$L\left(Q\vec{P}\right)\left(x_0\right)=\sum _{i=1}^N{a}_{\gamma }^i\left(x_0\right)\cdot P_i\left(x_0\right)\text{.}$$

$$\sum _{i=1}^N{a}_{\gamma }^i{P}_i=0\text{ on }{\mathbb{R}}^n\text{ for }\left|\gamma \right|=s;$$

$$0=L\left(Q\vec{P}\right)=\sum _{|\alpha |=s}{\partial }^{\alpha }\left\{Q\sum _{i=1}^N{a}_{\alpha }^i{P}_i\right\}+\tilde{L}\left(Q\vec{P}\right)=\tilde{L}\left(Q\vec{P}\right)\text{.}$$

$$\vec{P}\in \mathcal{M}\left(\tilde{L}\right)\text{.}$$

$$L\left(f_1,\cdots ,f_N\right)\left(x,y\right)=\sum _{|\alpha |,|\beta |\le s}\sum _{i=1}^N{a}_{\alpha \beta }^i\left(x,y\right){\partial }_x^{\alpha }{\partial }_y^{\beta }{f}_i\left(x,y\right)$$

$$L^{\Gamma }\vec{f}\left(x,y\right)={\mathbb{I}}_{\Gamma }\left(x,y\right)\cdot L\vec{f}\left(x,y\right)\text{.}$$

$$L\left(Q\vec{P}\right)=0\text{ on }\Gamma \text{ for all }Q\in \mathcal{R}\text{.}$$

$$L_{\nu }\left(f_1,\cdots ,f_N\right)\left(x,y\right)=\sum _{|\beta |\le s}\sum _{i=1}^N{b}_{\nu \beta }^i\left(x,y\right){\partial }_y^{\beta }{f}_i\left(x,y\right)(\text{each }{b}_{\nu \beta }^i\in \mathcal{R}),$$

$$\mathcal{M}\left(L^{\Gamma }\right)=\mathcal{M}\left(L_1,\Gamma \right)\cap \cdots \cap \mathcal{M}\left(L_{{\nu }_{\max }},\Gamma \right)\text{.}$$

$${\Gamma }^{+}=\left\{\left(x,y,z\right):\left(x,y\right)\in \Gamma ,z_{\alpha \beta }^i=a_{\alpha \beta }^i\left(x,y\right)\right\}\text{;}$$

$$L^{+}\left(f_1,\cdots ,f_N\right)\left(x,y,z\right)=\sum _{|\alpha |,|\beta |\le s}\sum _{i=1}^N{z}_{\alpha \beta }^i{\partial }_x^{\alpha }{\partial }_y^{\beta }{f}_i\left(x,y,z\right)\text{.}$$

$$L^{+}\left(Q\vec{P}\right)=0\text{ on }{\Gamma }^{+}\text{ for all }Q\text{.}$$

$$\left\{\left(x_1,\cdots ,x_n,y_1,\cdots ,y_m\right)\in U:\Delta \left(x_1,\cdots ,x_n\right)\ne 0\right\}$$

$${\tilde{P}}_{\mu }\left(x_1,\cdots ,x_n,y_{\mu }\right)=a_{\mu }\left(x_1,\cdots ,x_n\right)y_{\mu }^{D_{\mu }}+\sum _{j<D_{\mu }}{b}_{j\mu }\left(x_1,\cdots ,x_n\right){\left(y_{\mu }\right)}^j$$

$${\left(\Delta \left(x\right)\right)}^{l}P\left(x_1,\cdots ,x_n,y_1,\cdots ,y_m\right)$$

$$\frac{P\left(x_1,\cdots ,x_n\right)}{{\left(\Delta \left(x\right)\right)}^p}\text{, for some polynomial }P\text{ and some integer power }p\text{.}$$

$$P\left(x_1,\cdots ,x_n,y_1,\cdots ,y_m\right)$$

$$R\left(y_1,\cdots ,y_m\right)=\sum _{{\alpha }_1<K{D}_1}\cdots \sum _{{\alpha }_i<K{D}_i}{R}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}\left(y_{i+1},\cdots ,y_m\right)y_1^{{\alpha }_1}\cdots y_i^{{\alpha }_i},$$

$$R_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}={\tilde{H}}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}\cdot {\left[{\tilde{P}}_{i+1}\left(x_1,\cdots ,x_n,y_{i+1}\right)\right]}^K+S_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}\left(x_1,\cdots ,x_n,y_{i+1},\cdots ,y_m\right)$$

$${\tilde{H}}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)},S_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}\in \mathcal{R}\left[y_{i+1},\cdots ,y_m\right]$$

$${\mathrm{deg}}_{y_{i+1}}{S}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}<K{D}_{i+1}.$$

$$\tilde{H}=\sum _{{\alpha }_1<K{D}_1}\cdots \sum _{{\alpha }_i<K{D}_i}{\tilde{H}}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}{y}_1^{{\alpha }_1}\cdots y_i^{{\alpha }_i}$$

$$S=\sum _{{\alpha }_1<K{D}_1}\cdots \sum _{{\alpha }_i<K{D}_i}{S}_{\left({\alpha }_1,\cdots ,{\alpha }_i\right)}{y}_1^{{\alpha }_1}\cdots y_i^{{\alpha }_i}\text{.}$$

$$P\left(x_1,\cdots ,x_n,y_1,\cdots ,y_m\right)$$

$${\tilde{P}}_{\mu }\left(y_{\mu }\right)=y_{\mu }^{D_{\mu }}+\sum _{i=1}^{D_{\mu }-1}{a}_{\mu i}{y}_{\mu }^i$$

$$\sum _{\mu =1}^m{H}_{\mu }\left(y_1,\cdots ,y_m\right){\tilde{P}}_{\mu }\left(y_{\mu }\right)$$

Full text

The main result of this paper is an algorithm to compute generators for $\mathcal{M}\left(L\right)$, given an arbitrary semialgebraic differential operator $L$. Once we know generators for each $\mathcal{M}\left(L_{\nu}\right)$ $\left(\nu=1,\cdots,K\right)$ as in Theorem 1, standard computational algebra [1, 6] allows us to compute generators for $\mathcal{M}\left(L_{1}\right)\cap\cdots\cap\mathcal{M}\left(L_{K}\right)$. Thanks to Theorem 1, the task of exhibiting generators for $\left[\boldsymbol{A};C^{m}\right]$ is thus reduced to the task of computing generators for $\mathcal{M}\left(L\right)$.

The problem of computing the $C^{0}$-closure $\left[\boldsymbol{A};C^{0}\right]$ was posed by Brenner [5], and Epstein-Hochster [8], and solved by Fefferman-Kollár [9] and Kollár [13]. See also [10]. So our results on $\left[\boldsymbol{A};C^{m}\right]$ are new only for $m\geq 1$.

To explain the ideas in our computation of $\mathcal{M}\left(L\right)$, we consider in turn several examples of increasing complexity.

Thus, $L$ is a $0^{th}$ order operator acting on scalar functions $f$.

Then $\mathcal{M}\left(L\right)$ is the ideal $I\left(E\right)$ of all polynomials that vanish on $E$. To get a hint of the issues that arise, let $E_{a}=\left\{\left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}-zy^{2}=0,z\leq a\right\}$ for all $a\in\mathbb{R}$.

If $a>0$, one checks that $I\left(E_{a}\right)$ is the principal ideal generated by $x^{2}-zy^{2}$. However, if $a<0$, then $E_{a}=\left\{\left(0,0,z\right):z\leq a\right\}$ and $I\left(E_{a}\right)$ is the ideal generated by $x$ and $y$.

Remarkably, there are no standard algorithms to compute generators for $I\left(E\right)$ given a semialgebraic set $E$. Safey El Din et al [14] produce an algorithm that does the job. Here, we introduce a simpler but less efficient algorithm than that of [14] to compute generators for $I\left(E\right)$, using a geometric idea to reduce matters to standard algorithms.

Our algorithm for $I\left(E\right)$ proceeds as follows. First, we partition $E$ into finitely many simple smooth pieces $E_{\nu}$ $\left(\nu=1,\cdots,\nu_{\max}\right)$.

To each $E_{\nu}\subset\mathbb{R}^{n}$ we associate a complex variety $V_{\nu}\mathbb{\subset\mathbb{C}}^{n}$ (the “complexification” of $E_{\nu}$) and show that the polynomials vanishing on $E_{\nu}$ are precisely those that vanish on $V_{\nu}$.

This reduces the computation of generators for $I\left(E_{\nu}\right)$ to the corresponding problem for complex affine varieties, which is well-understood (See [1, 6]). Because $E$ is the union of the $E_{\nu}$, we have $I\left(E\right)=I\left(E_{1}\right)\cap\cdots\cap I\left(E_{\nu_{\max}}\right)$. Once we know generators for each $I\left(E_{\nu}\right)$, standard algorithms [1, 6] produce generators for their intersection. So we can compute generators for $\mathcal{M}\left(L\right)$ in Example 1.

Then we can easily prove the following assertion.

[TABLE]

with polynomial coefficients $A_{\nu}^{i}$. Moreover, the $A_{\nu}^{i}$ may be computed from $L$.

Standard computational algebra [1, 6] produces generators for the $\mathscr{R}$-module of polynomial vectors satisfying (0.0.2).

To prove Claim 1 we proceed by induction on $s$, the order of the operator $L$. In the base case $s=0$ there is nothing to prove.

For the induction step, we fix $s\geq 1$, assume Claim 1 for operators of order less than $s$, and suppose $L$ is given by (0.0.1), with polynomial coefficients. We can then write $L$ in the form

[TABLE]

where $\tilde{L}$ has polynomial coefficients and order less than $s$.

By definition, a polynomial vector $\vec{P}=\left(P_{1},\cdots,P_{N}\right)$ belongs to $\mathcal{M}\left(L\right)$ if and only if $L\left(Q\vec{P}\right)=0$ on $\mathbb{R}^{n}$ for every polynomial $Q$.

For fixed $x_{0}\in\mathbb{R}^{n}$ and fixed multiindex $\gamma$ of order $\left|\gamma\right|=s$, we take $Q\left(x\right)=\frac{1}{\gamma!}\left(x-x_{0}\right)^{\gamma}$. Substituting into (0.0.3), we see that

[TABLE]

Therefore, if $\vec{P}=\left(P_{1},\cdots,P_{N}\right)$ belongs to $\mathcal{M}\left(L\right)$, then

[TABLE]

consequently, for any $Q\in\mathscr{R}$ we have

[TABLE]

Thus, if $\vec{P}$ belongs to $\mathcal{M}\left(L\right)$, then $\left(\ref{i-4}\right)$ holds, and

[TABLE]

Conversely, $\left(\ref{i-4}\right)$ and $\left(\ref{i-5}\right)$ obviously imply $\vec{P}\in\mathcal{M}\left(L\right)$.

So $\mathcal{M}\left(L\right)$ consists of all $\vec{P}\in\mathcal{M}\left(\tilde{L}\right)$ satisfying $\left(\ref{i-4}\right)$.

Claim 1 for $L$ now follows at once from Claim 1 for $\tilde{L}$, completing our induction on $s$.

So we can compute generators for $\mathcal{M}\left(L\right)$ in Example 2.

[TABLE]

with polynomial coefficients $a_{\alpha\beta}^{i}\left(x,y\right)$, and let

[TABLE]

Thus, $\mathcal{M}\left(L^{\Gamma}\right)$ is the $\mathscr{R}$-module of all polynomial vectors $\vec{P}=\left(P_{1},\cdots,P_{N}\right)$ (each $P_{i}\in\mathscr{R}$), such that

[TABLE]

We write $\mathcal{M}\left(L,\Gamma\right)$ to denote the $\mathscr{R}$-module of all $\vec{P}$ that satisfy (0.0.7).

This generalizes Example 2, which arises here as the case $p=0$.

The analogue of Claim 1 here is as follows.

[TABLE]

such that

[TABLE]

Moreover, we can compute the $L_{\nu}$ from $L$ and $\Gamma$.

The point is that the $L_{\nu}$ involve no $x$-derivatives.

By using our algorithm to compute $I\left(E\right)$ (see Example 1), we can reduce the computation of generators for each $\mathcal{M}\left(L_{\nu},\Gamma\right)$ to the study of a system of linear equations with polynomial coefficients, which can be treated by standard computational algebra. Thus, we are able to produce generators for $\mathcal{M}\left(L^{\Gamma}\right)$ in Example 3.

We reduce matters to Example 3 by introducing new variables $z_{\alpha\beta}^{i}$ to replace the coefficients $a_{\alpha\beta}^{i}$. We write $z$ to denote $\left(z_{\alpha\beta}^{i}\right)_{\begin{subarray}{c}i=1,\cdots,N\\ \left|\alpha\right|,\left|\beta\right|\leq s\end{subarray}}$ .

Instead of $\Gamma=\left\{\left(x,G\left(x\right)\right):x\in U\right\}$, we consider

[TABLE]

and instead of $L$, we consider

[TABLE]

The algorithm for Example 3 then produces generators for the module of all polynomial vectors $\vec{P}$ in $\left(x,y,z\right)$ such that

[TABLE]

We then have to understand which solutions $\vec{P}$ of (0.0.8) do not depend on $z$. We carry this out in Section 7 below, thus producing generators for the module $\mathcal{M}\left(L^{\Gamma}\right)$ in Example 4.

Finally, we compute generators for $\mathcal{M}\left(L\right)$ in the general case. Let $L$ be a semialgebaic differential operator. Standard algorithms allow us to partition $\mathbb{R}^{n}$ into semialgebraic sets $E_{\nu}$ $\left(\nu=1,\cdots,\nu_{\max}\right)$ on each of which (after a $\nu$-dependent linear change of coordinates) $L$ and $E_{\nu}$ may be brought to the form of Example 4, with $E_{\nu}$ playing the rôle of $\Gamma$. Therefore, $\mathcal{M}\left(L\right)$ is the intersection of finitely many $\mathscr{R}$-modules, for each of which we can produce generators. The computation of generators for $\mathcal{M}\left(L\right)$ may then be accomplished by standard computational algebra.

This concludes the introductory explanation of our algorithm to compute $\mathcal{M}\left(L\right)$. For full details, see Sections 1-7 below.

We have made no attempt here to estimate the number of computer operations needed to execute our algorithms. Surely an expert in computational semialgebraic geometry could make significant improvements in our algorithms, and estimate the complexity of the improved algorithms. We welcome such progress.

We are grateful to Matthias Aschenbrenner, Saugata Basu, Edward Bierstone, Jesús De Loera, Zeev Dvir, János Kollár, Pierre Milman, Wieslaw Pawłucki, Mohab Safey El Din, Ary Shaviv, Rekha Thomas, 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.

1. Preliminaries

We begin with a few elementary lemmas.

Lemma 1**.**

Suppose $V\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$ is a connected, real-analytic submanifold. Write $\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)$ to denote a point of $\mathbb{R}^{n}\times\mathbb{R}^{m}$, and suppose that $x_{1},\cdots,x_{n}$ are local coordinates on a nonempty relatively open set $U\subset V$. Let $\Delta\left(x_{1},\cdots,x_{n}\right)$ be a nonzero polynomial. Let $P\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)$ be a polynomial. If $\Delta\left(x_{1},\cdots,x_{n}\right)\cdot P\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)=0$ on $V$, then $P\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)=0$ on $V$.

Proof.

Since $\left(x_{1},\cdots,x_{n}\right)$ are local coordinates on $U$, we know that

[TABLE]

is dense in $U$, hence $P\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)=0$ on $U$. The lemma follows, since $P$ is a polynomial and $V$ is a connected real-analytic manifold. ∎

Lemma 2**.**

For each $\mu=1,\cdots,m$, let

[TABLE]

be a polynomial in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{\mu}\right]$, with $a_{\mu}\left(x_{1},\cdots,x_{n}\right)$ nonzero. Let $\Delta\left(x\right)=\prod_{\mu=1}^{m}a_{\mu}\left(x_{1},\cdots,x_{n}\right)$ for $x=\left(x_{1},\cdots,x_{n}\right)$. Let $P\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ be given and let $K\geq 1$. Then, for some $l\geq 0$, we can express

[TABLE]

where $H_{\mu}$ and $P^{\#}$ are polynomials, and $\deg_{y_{1},\cdots,y_{m}}P^{\#}\leq D$; here, $D$ is a constant that may be computed from the $D_{\mu}$ and $K$.

Proof.

Let $\mathscr{R}$ be the ring of rational functions of the form

[TABLE]

Each $\tilde{P}_{\mu}$ is a unit times a monic polynomial in $y_{\mu}$, when we work in the ring $\mathscr{R}\left[y_{1},\cdots,y_{m}\right]$. Hence, in $\mathscr{R}\left[y_{1},\cdots,y_{m}\right]$, we may divide polynomials by powers of $\tilde{P}_{\mu}$, obtaining a quotient and a remainder in the usual way.

By induction on $i\geq 0$ (with $i\leq m$), we show that we can write $P$ in the form

[TABLE]

with $H_{\mu},R\in\mathscr{R}\left[y_{1},\cdots,y_{m}\right]$ and $\deg_{y_{\mu}}R<KD_{\mu}$ for $1\leq\mu\leq i$.

In the base case, $i=0$, so (1) is trivial. We take $R=P$.

Suppose (1) holds for a given $i<m$. We will prove that $P$ may be expressed in the form (1), with $i+1$ in place of $i$.

Write

[TABLE]

with each $R_{(\alpha_{1},\cdots,\alpha_{i})}\in\mathscr{R}[y_{i+1},\cdots,y_{m}]$. Performing a division by $\left[\tilde{P}_{i+1}\left(x_{1},\cdots,x_{n},y_{i+1}\right)\right]^{K}$ in the ring $\mathscr{R}\left[y_{i+1},\cdots,y_{m}\right]$, we may write

[TABLE]

with

[TABLE]

and

[TABLE]

Now set

[TABLE]

and

[TABLE]

Then $\tilde{H}$, $S\in\mathscr{R}\left[y_{1},\cdots,y_{m}\right]$ and $\deg_{y_{\mu}}S<KD_{\mu}$ for $\mu=1,\cdots,i+1$. Moreover, $R=\tilde{H}\cdot\left[\tilde{P}_{i+1}\right]^{K}+S$. Therefore, by (1), we have

[TABLE]

proving (1) with $i+1$ in place of $i$.

This completes our induction, and establishes (1).

Now taking $i=m$ in (1), and clearing denominators by multiplying by a high power of $\left(\Delta\left(x\right)\right)$, we obtain the conclusion of Lemma 2. ∎

Lemma 3**.**

Let $\mathscr{R}$ be a ring, and let

[TABLE]

be monic polynomials with coefficients in $\mathscr{R}$ for $\mu=1,\cdots,m$.

Let $H_{\mu}(y_{1},\cdots,y_{m})$ be polynomials with coefficients in $\mathscr{R}$, and suppose that

[TABLE]

has degree $\leq D$, where $D\geq\max\left\{D_{1},\cdots,D_{m}\right\}$.

Then there exist polynomials $\tilde{H}_{\mu}\left(y_{1},\cdots,y_{m}\right)$ with coefficients in $\mathscr{R}$ such that

[TABLE]

with $\deg\tilde{H}_{\mu}\leq D-D_{\mu}$ for each $\mu$.

Proof.

Let $M=\max_{1\leq\mu\leq m}\left\{D_{\mu}+\deg H_{\mu}\right\}$. It is enough to show that, if $M>D$, then there exist $\tilde{H}_{\mu}$ satisfying (1.0.2) such that

[TABLE]

To show this, we proceed as follows.

Let $H_{\mu}=H_{\mu}^{o}+H_{\mu}^{err}$, where $H_{\mu}^{o}$ is homogeneous of degree $M-D_{\mu}$, and $\deg H_{\mu}^{err}<M-D_{\mu}$.

If $M>D$, then $\sum_{\mu}H_{\mu}^{o}y_{\mu}^{D_{\mu}}=0$, since $H_{\mu}\tilde{P}_{\mu}=H_{\mu}^{o}y_{\mu}^{D_{\mu}}+$(terms of degree $<M$).

Writing $H_{\mu}^{o}=\sum_{\left|\beta\right|=M-D_{\mu}}A_{\beta}^{\mu}y^{\beta}$ with $A_{\beta}^{\mu}\in\mathscr{R}$, and defining

[TABLE]

for multiindices $\gamma=\left(\gamma_{1},\cdots,\gamma_{m}\right)$, we conclude that

[TABLE]

for each $\left|\gamma\right|=M$. Here $\mathbb{I}_{\mu}=\left(a_{1},\cdots,a_{m}\right)$ with $a_{\mu}=1$ and $a_{i}=0$ when $i\not=\mu$.

We now use the following observation: Let $A_{\mu}\in\mathscr{R}$ for $\mu\in S$ (a finite set), and suppose that $\sum_{\mu\in S}A_{\mu}=0$. Then there exist $\lambda_{ij}\in\mathscr{R}$ for $i,j\in S$ distinct, such that

[TABLE]

Applying the observation to the $A_{\gamma-D_{\mu}\mathbb{I}_{\mu}}^{\mu}$ for fixed $\gamma$, we obtain $\lambda_{ij}^{\gamma}\in\mathscr{R}$ for $i,j\in S\left(\gamma\right)$ distinct, $\left|\gamma\right|=M$, such that

[TABLE]

Note that for $i,j\in S\left(\gamma\right)$ distinct, $\gamma-D_{i}\mathbb{I}_{i}-D_{j}\mathbb{I}_{j}$ is a multiindex (i.e., its components are non-negative).

For $\left|\gamma\right|=M$, let $H_{\mu}^{\gamma}=\sum_{i,j\in S\left(\gamma\right),i\not=j}\lambda_{ij}^{\gamma}y^{\gamma-D_{i}\mathbb{I}_{i}-D_{j}\mathbb{I}_{j}}\left(\tilde{P}_{j}\left(y_{j}\right)\delta_{\mu i}-\tilde{P}_{i}\left(y_{i}\right)\delta_{\mu j}\right)$.

Note that

[TABLE]

Moreover,

[TABLE]

whereas

[TABLE]

Setting $\tilde{H}_{\mu}=H_{\mu}-\sum_{\left|\gamma\right|=M}H_{\mu}^{\gamma}$, we conclude that

[TABLE]

The proof of the lemma is complete. ∎

Lemma 4**.**

For $\mu=1,\cdots,m$, let

[TABLE]

be a polynomial with $a_{\mu}$ nonzero.

Let $\Delta(x_{1},\cdots,x_{n})=\prod_{\mu=1}^{m}a_{\mu}(x_{1},\cdots,x_{n})$. Let $H_{\mu}(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m})$ be polynomials ($\mu=1,\cdots,m$), and suppose that

[TABLE]

where $D\geq\max\left\{D_{1},\cdots,D_{m}\right\}$.

Then there exist polynomials $H_{\mu}^{\#}$ such that for some $l$ we have

• •

$\left(\Delta\left(x\right)\right)^{l}\sum_{\mu=1}^{m}H_{\mu}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)$* *

$=\sum_{\mu=1}^{m}H_{\mu}^{\#}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)$* *

and

• •

$\deg_{y}H_{\mu}^{\#}\leq D-D_{\mu}$* for each $\mu=1,\cdots,m$.*

Proof.

Immediate from Lemma 3 with $\mathscr{R}$ taken to be the ring of rational functions of the form $\frac{P\left(x_{1},\cdots,x_{n}\right)}{\left(\Delta\left(x\right)\right)^{p}}$, for some polynomial $P$ and some integer power $p$. ∎

We recall a few elementary facts and definitions from algebraic geometry; see [12, 15].

An algebraic set in $\mathbb{C}^{n}$ is a set of the form $V=\{z\in\mathbb{C}^{n}:P_{1}(z)=\cdots=P_{k}(z)=0\}$ where $P_{1},\cdots,P_{k}$ are polynomials. $V$ is called irreducible if it cannot be expressed as the union of two algebraic sets $V=V_{1}\cup V_{2}$ with $V_{1},V_{2}\not=V$. An irreducible algebraic set is called an affine variety. Every algebraic set $V\subset\mathbb{C}^{n}$ may be expressed as the union of finitely many affine varieties $V_{1},\cdots,V_{p}$ with $V_{i}\not\subset V_{j}$ for $i\not=j$. These $V_{i}$ are the irreducible components of $V$. In any given affine variety $V$, the set $V_{\text{reg}}\subset V$ of regular points of $V$ is a connected complex analytic submanifold of $\mathbb{C}^{n}$. Moreover, $V_{\text{reg}}$ is dense in $V$.

Lemma 5**.**

Suppose $W=\{z\in\mathbb{C}^{n}:P_{\mu}(z)=0\text{ for }\mu=1,\cdots,m\}$ for polynomials $P_{1},\cdots,P_{m}$. Let $V_{1},\cdots,V_{p}$ be the irreducible components of $W$, and for each $j=1,\cdots,p$, let $V_{j,\text{reg}}$ be the set of all regular points of $V_{j}$.

Let $z_{0}\in W$ and suppose that the differentials $dP_{\nu}(z_{0})$ $(\nu=1,\cdots,m)$ are linearly independent.

Then there exist an index $j_{0}\in\{1,\cdots,p\}$ and a small ball $B(z_{0},r)\subset\mathbb{C}^{n}$ about $z_{0}$, such that

[TABLE]

and

[TABLE]

Proof.

See [15]. ∎

2. Background from Computational Algebraic Geometry

In this section, we present some known technology for computations involving semialgebraic sets in $\mathbb{R}^{n}$ and algebraic sets in $\mathbb{C}^{n}$. See the reference book [3].

We begin by describing our model of computation, copied from [10]. Our algorithms are to be run on an idealized computer with standard von Neumann architecture [16], 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 [3].

All arithmetic in [3] is performed within a subring $\Lambda$ of a real closed field $K$ (e.g. the integers sitting inside the reals). However, some algorithms in [3] 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 [3] 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

• (2.0.1)

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

• (2.0.3)

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

• (2.0.5)

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

for polynomials $P\in\mathbb{R}[x_{1},\cdots,x_{n}]$.

A given semialgebraic set may be specified as above in many different ways, but that won’t bother us.

A semialgebraic function $F:E\rightarrow\mathbb{R}^{m}$ is specified by specifying its graph $\{(x,F(x)):x\in E\}\subset E\times\mathbb{R}^{m}$.

We will also make computations with algebraic sets $V\subset\mathbb{C}^{n}$. To specify $V$, we exhibit polynomials $P_{1},\cdots,P_{k}\in\mathbb{C}[z_{1},\cdots,z_{n}]$, such that $V=\{z\in\mathbb{C}^{n}:P_{1}(z)=\cdots=P_{k}(z)=0\}$.

(We represent the coefficients of the $P_{j}$ in the obvious way, by specifying their real and imaginary parts.)

Again, a given $V$ may be specified as above in many different ways, but that won’t bother us.

2.1. Known Algorithms

In this subsection we present several known algorithms from computational algebraic geometry.

We begin with two algorithms that deal with algebraic sets in $\mathbb{C}^{n}$.

Algorithm 1**.**

Given an algebraic set $V\subset\mathbb{C}^{n}$, we compute generators for the ideal of all $P\in\mathbb{C}[z_{1},\cdots,z_{n}]$ that vanish on $V$. (See [4]; see also [7] for a different algorithm.)

Algorithm 2**.**

Given an algebraic set $V\subset\mathbb{C}^{n}$, we compute its irreducible components $V_{1},\cdots,V_{p}$. (See [11].)

We will need several algorithms pertaining to semialgebraic sets $E\subset\mathbb{R}^{n}$.

Algorithm 3**.**

Given a semialgebraic set $E$, we compute its dimension. (See Algorithm 14.31 in [3].)

Algorithm 4**.**

Given a semialgebraic set $E$, we compute the connected components of $E$. In particular, if $E$ is zero-dimensional (and therefore finite), we compute a list of all the points of $E$. (See Algorithm 16.20 in [3].)

Algorithm 5**.**

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

• (2.1.1)

For every $x\in E_{1}$, there exists $y\in\mathbb{R}^{n_{2}}$ such that $(x,y)\in E$.

If ((2.1.1)) 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 [3].)

Algorithm 6**.**

Given a matrix of polynomials $\left[A_{ij}\right]_{i=1,\cdots,I;\text{ }j=1,\cdots,J}$ with each $A_{ij}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$, we produce a list of generators for the $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$-module of all solutions $\vec{P}=\left(P_{1},\cdots,P_{J}\right)$ (each $P_{j}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$) of the equations

[TABLE]

Moreover, given $\left[A_{ij}\right]$ as above, and given $\vec{Q}=\left(Q_{1},\cdots,Q_{I}\right)$ with each $Q_{i}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$, we decide whether the equations

[TABLE]

have a solution $\vec{P}=\left(P_{1},\cdots,P_{J}\right)$ with $P_{1},\cdots,P_{J}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$; if there is a solution $\vec{P}$, then we produce one. (See Sections 3.5-3.7 in [1].)

Algorithm 7**.**

(Graph Decomposition Algorithm) Given a semialgebraic set $E\subset\mathbb{R}^{q}$, we compute a partition of $E$ into finitely many semialgebraic sets $E_{\nu}$, for each of which there is an invertible linear map $T_{\nu}:\mathbb{R}^{q}\rightarrow\mathbb{R}^{q}$ such that $T_{\nu}E_{\nu}$ has the form

[TABLE]

where $U_{\nu}\subset\mathbb{R}^{n_{\nu}}$ is a semialgebraic open set and $G_{\mu\nu}:U_{\nu}\rightarrow\mathbb{R}$ is semialgebraic. Moreover, we compute the above $T_{\nu},U_{\nu}$, and $G_{\mu\nu}$. (See Algorithm 11.3 in [3].)

2.2. Elimination of Quantifiers

In this subsection, 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 [2, 3].

• •

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]

2.3.

We present the following trivial algorithm.

Algorithm 8**.**

Given a semialgebraic function $F:E\rightarrow\mathbb{R}$ defined on a semialgebraic set $E\subset\mathbb{R}^{n}$, we produce a nonzero polynomial $P(x,t)$ on $\mathbb{R}^{n}\times\mathbb{R}$ such that $P(x,F(x))=0$ for all $x\in E$.

Explanation.

$F$ is specified to us by expressing its graph $\Gamma=\{(x,F(x)):x\in E\}$ as a Boolean combination of sets of the form $\{P>0\}$ or $\{P=0\}$ for nonzero polynomials $P$ on $\mathbb{R}^{n}\times\mathbb{R}$. Therefore, we can trivially express $\Gamma$ as a disjoint union over $\nu=1,\cdots,N$ of nonempty sets of the form

[TABLE]

with each $i_{\max}(\nu),j_{\max}(\nu)\geq 0$. No $\Gamma_{\nu}$ can be open in $\mathbb{R}^{n}\times\mathbb{R}$, because ${\Gamma}_{\nu}\subset\Gamma$; hence $j_{\max}(\nu)\geq 1$ for each $\nu$. We take $P(x,t)=\prod_{\nu=1}^{N}P_{\nu 1}(x,t)$. ∎

2.4.

Next, we present several refinements of Algorithm 7. We explain these algorithms in detail, although experts in computational algebraic geometry will find them routine.

To prepare the way, we present the following algorithms.

Algorithm 9**.**

Given a nonempty open semialgebraic subset $U\subset\mathbb{R}^{n}$ and a semialgebraic function $F:U\rightarrow\mathbb{R}$, we compute semialgebraic sets $U_{junk}$, $U_{1},\cdots,U_{N}\subset\mathbb{R}^{n}$, with the following properties

• •

$U$* is the disjoint union of $U_{junk},U_{1},\cdots,U_{N}$.*

• •

$U_{junk}$* has dimension strictly less than $n$.*

• •

Each $U_{\nu}$ ($\nu=1,\cdots,N$) is open in $\mathbb{R}^{n}$.

• •

$F|_{U_{\nu}}$* is continuous, for each $\nu=1,\cdots,N$.*

Explanation.

We start by recalling a useful property of roots of polynomials. Fix $D\geq 1$. Given $a_{0},a_{1},\cdots,a_{D-1}\in\mathbb{R}$, we write

[TABLE]

to denote the real parts of the roots of the polynomial

[TABLE]

sorted in ascending order. Then we recall that the map $(a_{0},\cdots,a_{D-1})\mapsto\operatorname{Root}_{k}(a_{0},\cdots,a_{D-1})$ is continuous and semialgebraic, for each fixed $k=1,\cdots,D$.

Now let $U\subset\mathbb{R}^{n}$ be nonempty open and semialgebraic, and let $F:U\rightarrow\mathbb{R}$ be semialgebraic. We can find a nonzero polynomial $P(x_{1},\cdots,x_{n},z)$ such that

[TABLE]

for all $\left(x_{1},\cdots,x_{n}\right)\in U$. Note that $P\left(x_{1},\cdots,x_{n},z\right)$ cannot be independent of $z$, since $U$ is a nonempty open subset of $\mathbb{R}^{n}$. Thus, we may write

[TABLE]

for polynomials $a$, $b_{i}$ with $a$ nonzero. (We write $x$ to denote $\left(x_{1},\cdots,x_{n}\right)$.) We define $U_{junk,0}=\left\{x\in U:a\left(x\right)=0\right\}$; thus, $U_{junk,0}\subset U$ is semialgebraic and has dimension $\leq n-1$. On $U\setminus U_{junk,0}$,

[TABLE]

Each $a_{i}\left(\cdot\right)$ is a continuous semialgebraic (rational) function on $U\setminus U_{junk,0}$.

Thanks to (2.4.1), we have $F(x)=\operatorname{Root}_{k}(a_{0}(x),a_{1}(x),\cdots,a_{D-1}(x))$ for some $k$ (each $x\in U\setminus U_{junk,0}$).

For each $k=1,\cdots,D$, let

[TABLE]

Thus $U$ is the disjoint union of $U_{junk,0}$ and the $U_{1,k}$ ($k=1,\cdots,D$). Moreover, each $U_{1,k}$ is a semialgebraic subset of $\mathbb{R}^{n}$, and $F\left(x\right)=\operatorname{Root}_{k}\left(a_{0}\left(x\right),a_{1}\left(x\right),\cdots,a_{D-1}\left(x\right)\right)$ on $U_{1,k}$. Since the $a_{i}\left(x\right)$ are continuous on $U\setminus U_{junk,0}$, and since $\left(a_{0},\cdots,a_{D-1}\right)\mapsto\operatorname{Root}_{k}\left(a_{0},\cdots,a_{D-1}\right)$ is continuous, it follows that $F|_{U_{1,k}}$ is continuous.

We now compute semialgebraic sets $U_{k,junk},U_{k}$ with the following properties (see Section 2.1)

• •

$U_{1,k}$ is the disjoint union of $U_{k}$ and $U_{k,junk}$.

• •

$U_{k,junk}$ has dimension $\leq n-1$.

• •

$U_{k}$ is open in $\mathbb{R}^{n}$.

We define $U_{junk}=U_{junk,0}\cup\bigcup\limits_{k=1}^{D}U_{k,junk}$.

Thus, $U_{junk}$ is semialgebraic and has dimension $\leq n-1$; moreover, $U$ is the disjoint union of the $U_{junk}$, $U_{1},\cdots,U_{D}$. Also, $U_{k}\subset\mathbb{R}^{n}$ is open, and $F|_{U_{k}}$ is continuous, for each $k=1,\cdots,D$. This concludes our explanation of Algorithm 9. ∎

Algorithm 10**.**

Given a nonempty semialgebraic open set $U\subset\mathbb{R}^{n}$, and given a semialgebraic function $F:U\rightarrow\mathbb{R}$, we compute semialgebraic subsets $U_{junk},U_{1},\cdots,U_{N}\subset U$ with the following properties:

• •

$U$* is the disjoint union of $U_{junk},U_{1},\cdots,U_{N}$.*

• •

$U_{junk}$* has dimension $\leq n-1$.*

• •

For each $\nu$ ($1\leq\nu\leq N$), $U_{\nu}$ is a nonempty open subset of $\mathbb{R}^{n}$ and $F|_{U_{\nu}}$ is real-analytic.

Explanation.

Using Algorithm 9 we may easily reduce matters to the case in which $F$ is continuous on $U$. As in the explanation of Algorithm 9, we can produce a polynomial

[TABLE]

with $a\left(x\right)$ nonzero, such that

[TABLE]

We partition $U$ into the following semialgebraic sets:

[TABLE]

for $l=0,1,\cdots,D-1$;

[TABLE]

Taking $l^{\prime}=D$ in the definition of $\hat{U}_{final}$, we see that

[TABLE]

and therefore $\hat{U}_{final}$ has dimension $\leq n-1$. For fixed $l$ ($0\leq l\leq D-1$), let $Q_{l}(x,\xi)=\left(\partial_{\xi}\right)^{l}P\left(x,\xi\right)$. Then

[TABLE]

Applying an algorithm from Section 2.1, we partition $\hat{U}_{l}$ into a semialgebraic set $U_{l,junk}$ of dimension $\leq n-1$, and a semialgebraic open set $U_{l}$ (possibly empty). Thus, our original set $U$ is partitioned into the semialgebraic sets, $U_{l}$ ($l=0,1,\cdots,D-1$), and

[TABLE]

with $\dim U_{junk}\leq n-1$; the $U_{l}$ are open.

We now show that $F|_{U_{l}}$ is real-analytic for each $l=0,\cdots,D-1$. Fix $x^{0}\in U_{l}\subset\hat{U}_{l}$, and let $\xi^{0}=F(x^{0})$. By (2.4.2), we have $Q_{l}\left(x^{0},\xi^{0}\right)=0$ but $\partial_{\xi}Q_{l}\left(x^{0},\xi^{0}\right)\not=0$. Hence, thanks to the real analytic implicit function theorem, we know that for small enough $\varepsilon>0$, the set

[TABLE]

is contained in the graph of a real-analytic function $z=\psi\left(x\right)$ defined on some neighborhood of $x^{0}$.

For $x\in U_{l}\subset\hat{U}_{l}$, (2.4.2) tells us that $Q_{l}(x,F(x))=0$. Moreover, $F$ is continuous on $U$, hence on $U_{l}$. Therefore, for small enough $\delta>0$, we have

[TABLE]

Therefore, $F$ agrees with $\psi$ on the ball $B(x^{0},\delta)$. This proves that $F$ is real-analytic in a neighborhood of $x^{0}$. Since $x^{0}\in U_{l}$ was picked arbitrarily, we now know that $F$ is real-analytic on $U_{l}$.

We now delete any empty $U_{l}$ from our list of open sets $U_{0},\cdots,U_{D-1}$. The remaining list, together with the set $U_{junk}$ defined and computed above have all the properties asserted in Algorithm 10. The explanation of that algorithm is complete. ∎

Algorithm 11**.**

Given a nonempty semialgebraic set $U\subseteq\mathbb{R}^{n}$, and given semialgebraic functions $F_{1},\cdots,F_{K}:U\rightarrow\mathbb{R}$, we compute a partition of $U$ into semialgebraic subsets $U_{junk},U_{1},\cdots,U_{N}$ with the following properties

• •

$U_{junk}$* has dimension $\leq n-1$.*

• •

Each $U_{\nu}$ is a nonempty open subset of $\mathbb{R}^{n}$.

• •

The restriction of each $F_{k}$ to each $U_{\nu}$ is real-analytic.

Explanation.

We proceed by recursion on $K$, using Algorithm 10. If $K=1$, Algorithm 10 does the job. If $K>1$, then by applying Algorithm 10, we partition $U$ into $\hat{U}_{junk}$ and $\hat{U}_{i}$ ($i=1,\cdots,I$) (semialgebraic subsets), with the following properties: $\dim\hat{U}_{junk}\leq n-1$; each $\hat{U}_{i}$ is non-empty and open; $F_{K}|_{\hat{U}_{i}}$ is real-analytic. We recursively apply Algorithm 11, with $K$ replaced by $K-1$, to the non-empty open semialgebraic set $\hat{U}_{i}$ and the semialgebraic functions $F_{1}|_{\hat{U}_{i}},\cdots,F_{K-1}|_{\hat{U}_{i}}$. Thus, we partition $\hat{U}_{i}$ into semialgebraic sets $\hat{U}_{i,junk}$ and $\hat{U}_{i,\nu}$ ($\nu=1,\cdots,\nu_{\max}(i))$, with the following properties: $\hat{U}_{i,junk}$ has dimension $\leq n-1$; each $\hat{U}_{i,\nu}$ is non-empty and open; $F_{k}|_{\hat{U}_{i,\nu}}$ is real-analytic for $k=1,\cdots,K-1$.

In fact, $F_{k}|_{\hat{U}_{i,\nu}}$ is real-analytic for $k=1,\cdots,K$, since $F_{K}|_{\hat{U}_{i,\nu}}$ is real-analytic.

We now define $U_{junk}=\hat{U}_{junk}\cup\bigcup_{i=1}^{I}\hat{U}_{i,junk},$ and let $U_{1},\cdots,U_{N}$ be an enumeration of the $\hat{U}_{i,\nu}$ ($i=1,\cdots,I,\nu=1,\cdots,\nu_{\max}\left(i\right)$).

These sets are semialgebraic; $\dim U_{junk}\leq n-1$; each $U_{\nu}$ is nonempty and open; and $F_{k}|_{U_{\nu}}$ is real-analytic ($k=1,\cdots,K$) for each $\nu$. This concludes the explanation of Algorithm 11. ∎

Now we can present our refinements of Algorithm 7.

Algorithm 12**.**

Given a semialgebraic set $E\subset\mathbb{R}^{q}$ of dimension $\leq n$, and given semialgebraic functions $F_{1},\cdots,F_{K}:E\rightarrow\mathbb{R}$, we compute a decomposition of $E$ into semialgebraic sets $E_{junk}$, $E_{1},\cdots,E_{N}$ with the following properties:

• •

$\dim E_{junk}\leq n-1$**

• •

For each $\nu=1,\cdots,N$ there exists an invertible linear map $T_{\nu}:\mathbb{R}^{q}\rightarrow\mathbb{R}^{q}$ such that $T_{\nu}E_{\nu}=\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{q-n}:x\in U_{\nu},y=G_{\nu}(x)\}$, where $U_{\nu}\subset\mathbb{R}^{q-n}$ is a semialgebraic nonempty open set and $G_{\nu}$ is a semialgebraic map.

• •

For each $\nu=1,\cdots,N$ and each $\lambda=1,\cdots,K$, we have $F_{\lambda}\circ T_{\nu}^{-1}(x,y)=H_{\lambda\nu}(x)$ for all $x\in U_{\nu}$, $y=G_{\nu}(x)$ (i.e., for all $(x,y)\in T_{\nu}E_{\nu}$), where $H_{\lambda\nu}:U_{\nu}\rightarrow\mathbb{R}$ is a semialgebraic function.

• •

Moreover, we compute $T_{\nu},G_{\nu},H_{\lambda\nu}$ as above. Also, for each $\nu=1,\cdots,N$ and each $\mu=1,\cdots,q-n_{\nu}$, we compute a nonzero polynomial $P_{\mu\nu}(x_{1},\cdots,x_{n},y_{\mu})$ such that $P_{\mu\nu}(x_{1},\cdots,x_{n},y_{\mu})=0$ for all $(x_{1},\cdots,x_{n},y_{1},\cdots,y_{q-n})\in T_{\nu}E_{\nu}$.

• •

Finally, for each $\nu=1,\cdots,N$ and each $\lambda=1,\cdots,K$, we compute a nonzero polynomial $\hat{P}_{\lambda\nu}(x_{1},\cdots,x_{n},z_{\lambda})$ such that $\hat{P}_{\lambda\nu}(x_{1},\cdots,x_{n},z_{\lambda})=0$ for $(x_{1},\cdots,x_{n},y_{1},\cdots,y_{q-n})\in T_{\nu}E_{\nu}$ and $z_{\lambda}=F_{\lambda}\circ T_{\nu}^{-1}(x_{1},\cdots,x_{n},y_{1},\cdots,y_{q-n})$.

Explanation.

We first apply Algorithm 7 to $E$, and throw away all the $E_{\nu}$ with $\dim E_{\nu}<n$ into $E_{junk}$. Thus, $E_{junk}$ is a semialgebraic set of dimension $\leq n-1$. There remain the $E_{\nu}$ with $\dim E_{\nu}=n$. For each such an $E_{\nu}$, Algorithm 7 provides $T_{\nu},U_{\nu},G_{\nu}$ as described in Algorithm 12. Moreover, once we know $U_{\nu}$ and $G_{\nu}$, we compute a nonzero polynomial $P_{\mu\nu}(x_{1},\cdots,x_{n},y_{\mu})$ as in the statement of Algorithm 12.

Next, since $T_{\nu}E_{\nu}=\{(x,y):x\in U_{\nu},y=G_{\nu}(x)\}$ with $U_{\nu},G_{\nu}$ semialgebraic, we can compute semialgebraic functions $H_{\lambda\nu}:U_{\nu}\rightarrow\mathbb{R}$ as in the statement of Algorithm 12. Once we know $U_{\nu}$ and $H_{\lambda\nu}$, we can easily compute nonzero polynomials $\hat{P}_{\lambda\nu}$ as in the statement of Algorithm 12. This completes the explanation of Algorithm 12. ∎

Algorithm 13**.**

Given a semialgebraic set $E\subset\mathbb{R}^{q}$ of dimension $\leq n$, and given semialgebraic functions $F_{1},\cdots,F_{K}:E\rightarrow\mathbb{R}$, we compute objects $E_{junk},E_{1},\cdots,E_{N},T_{\nu},U_{\nu},G_{\nu},H_{\lambda\nu},P_{\mu\nu},\hat{P}_{\mu\nu}$ having all the properties asserted in Algorithm 12, but also satisfying the following:

• •

Each $U_{\nu}$ is connected.

• •

Each $G_{\nu},H_{\lambda\nu}$ is real analytic (on $U_{\nu}$).

Explanation.

First, we execute Algorithm 12. For each $\nu$, we write $G_{\mu\nu}(x)$ to denote the $\mu^{\text{th}}$ component of $G_{\nu}(x)\in\mathbb{R}^{q-n}$. We apply Algorithm 11 to the open set $U_{\nu}$ and the list of functions consisting of the $G_{\mu\nu}\quad(\mu=1,\cdots,q-n)$ and the $H_{\lambda\nu}\quad(\lambda=1,\cdots,K)$.

Thus, $U_{\nu}$ is partitioned into semialgebraic open sets $U_{\nu i}\quad(i=1,\cdots,I(\nu))$ and a set $U_{\nu,junk}$ of dimension $\leq n-1$; on each $U_{\nu i}$, all the $G_{\mu,\nu}$ and $H_{\lambda\nu}$ are real-analytic. Using Algorithm 4 we further partition $U_{\nu i}$ into its connected components $U_{\nu ij}$.

We now define:

[TABLE]

One checks easily that $E_{junk}^{\ast}$ is semialgebraic and has dimension $\leq n-1$; $E$ is partitioned into $E_{junk}^{\ast}$ and the $E_{\nu ij}^{\ast}$; each $E_{\nu ij}^{\ast}$ is semialgebraic; $T_{\nu ij}^{\ast}E_{\nu ij}^{\ast}=\left\{\left(x,y\right):x\in U_{\nu ij}^{\ast},y=G_{\nu ij}^{\ast}\left(x\right)\right\}$; $U_{\nu ij}^{\ast}\subset\mathbb{R}^{n}$ is nonempty, open, connected and semialgebraic; $G_{\nu ij}^{\ast}:U_{\nu ij}^{\ast}\rightarrow\mathbb{R}^{q}$ is a real-analytic semialgebraic map; $H^{\ast}_{\lambda\nu ij}:U^{\ast}_{\nu ij}\rightarrow\mathbb{R}$ is semialgebraic and real-analytic; $F_{\lambda}\circ\left(T_{\nu ij}^{\ast}\right)^{-1}\left(x,y\right)=H_{\lambda\nu ij}^{\ast}\left(x\right)$ for $x\in U_{\nu ij}^{\ast}$, $y=G_{\nu ij}^{\ast}\left(x\right)$; $P_{\mu,\nu ij}^{\ast}\left(x,y_{\mu}\right)=0$ for $\left(x,y_{1},\cdots,y_{q-n}\right)\in T_{\nu_{ij}}^{\ast}E_{\nu ij}^{\ast}$; and $\hat{P}_{\lambda\nu ij}^{\ast}\left(x,z_{\lambda}\right)=0$ for $\left(x,y\right)\in T_{\nu ij}^{\ast}E_{\nu ij}^{\ast}$ and $z_{\lambda}=F_{\lambda}\circ\left(T_{\nu}^{\ast}\right)^{-1}\left(x,y\right)$.

The above objects have all the properties asserted in Algorithm 13. This completes the explanation of Algorithm 13. ∎

Our main refinement of Algorithm 7 is the following.

Algorithm 14**.**

Given a semialgebraic set $E\subset\mathbb{R}^{q}$, and given semialgebraic functions $F_{1},\cdots,F_{K}:E\rightarrow\mathbb{R}$, we compute a partition $E_{1},\cdots,E_{N}$ of $E$, satisfying the following for each $\nu$:

• •

$E_{\nu}$* has the form*

[TABLE]

where $T_{\nu}:\mathbb{R}^{q}\rightarrow\mathbb{R}^{q}$ is an invertible linear map, $U_{\nu}\subset\mathbb{R}^{n_{\nu}}$ is a connected, open semialgebraic set, and $G_{\nu}:U_{\nu}\rightarrow\mathbb{R}^{q-n_{\nu}}$ is real-analytic and semialgebraic.

• •

For each $\lambda=1,\cdots,K$, we have

[TABLE]

where $H_{\lambda\nu}:U_{\nu}\rightarrow\mathbb{R}$ is real-analytic and semialgebraic.

Moreover, we compute the above $T_{\nu},n_{\nu},U_{\nu},G_{\nu},H_{\lambda\nu}$. In addition, we compute nonzero polynomials $P_{\mu\nu}\left(x,y_{\nu}\right)$ $\left(\mu=1,\cdots,q-n_{\nu}\right)$ and $\hat{P}_{\lambda\nu}\left(x,z_{\lambda}\right)$ $\left(\lambda=1,\cdots,K\right)$ such that

• •

$P_{\mu\nu}\left(x,y_{\mu}\right)=0$* for $x\in U_{\nu}$, $\left(y_{1},\cdots,y_{q-n_{\nu}}\right)=G_{\nu}\left(x\right)$; and $\hat{P}_{\lambda\nu}\left(x,H_{\lambda\nu}\left(x\right)\right)=0$ for $x\in U_{\nu}$.*

Explanation.

We proceed by recursion on $\dim E$. If $\dim E=0$, then $E$ consists of finitely many points, and our task is trivial.

Fix $n\geq 1$, and suppose we can carry out Algorithm 14 whenever $\dim E\leq n-1$. We explain how to carry out the algorithm when $\dim E=n$.

To do so, we apply Algorithm 13. The semialgebraic sets $E_{\nu}$ have the form desired for Algorithm 14, and we obtain also the corresponding $T_{\nu},U_{\nu},G_{\nu},P_{\mu\nu},\hat{P}_{\mu\nu}$ (with $n_{\nu}=n$).

However, we still have to deal with the semialgebraic set $E_{junk}$ arising from Algorithm 13. Since $\dim E_{junk}\leq n-1$, we may recursively apply Algorithm 14 to the set $E_{junk}$ and the functions $F_{1}|_{E_{junk}},\cdots,F_{K}|_{E_{junk}}$.

It is now trivial to carry out Algorithm 14 for the given $E,F_{1},\cdots,F_{K}$. ∎

2.5.

In this subsection, we explain how to compute generators for the ideal of polynomials that vanish on given semialgebraic set.

The strategy is to apply Algorithm 14, and then execute the following procedure.

Algorithm 15** (Complexify).**

Let $\Gamma=\left\{\left(x,G\left(x\right)\right):x\in U\right\}$ where $U\subset\mathbb{R}^{n}$ is an open, connected, nonempty semialgebraic set, and $G\left(x\right)=\left(G_{1}\left(x\right),\cdots,G_{m}\left(x\right)\right)$ is a real-analytic semialgebraic map from $U$ to $\mathbb{R}^{m}$. Suppose $P_{1}\left(x,t\right),\cdots,P_{m}\left(x,t\right)$ are nonzero (real) polynomials on $\mathbb{R}^{n}\times\mathbb{R}$ such that $P_{\mu}\left(x,G_{\mu}\left(x\right)\right)=0$ on $U$ for all $\mu=1,\cdots,m$. Given $U,G,P_{1},\cdots,P_{m}$ as above, we compute an affine variety $V\subset\mathbb{C}^{n}\times\mathbb{C}^{m}$ with the following property: Let $P\in\mathbb{C}\left[z_{1},\cdots,z_{n},w_{1},\cdots,w_{m}\right]$. Then $P$ vanishes everywhere on $\Gamma$ if and only if $P$ vanishes everywhere on $V$.

Explanation.

We first reduce to the case in which, for each $\mu=1,\cdots,m$, $\partial_{t}P_{\mu}\left(x,t\right)$ isn’t identically zero on the graph of $G_{\mu}:U\rightarrow\mathbb{R}$. To do so, note that if for some $\mu$, we have

[TABLE]

then we may simply replace $P_{\mu}$ by $\partial_{t}P_{\mu}$, preserving all the assumptions made in Algorithm 15. Note that we can decide whether (2.5.1) holds, and note that $\partial_{t}P_{\mu}$ is a nonzero polynomial. (Otherwise we would have $P_{\mu}\left(x,t\right)=P_{\mu}^{\#}\left(x\right)$ for a nonzero polynomial $P_{\mu}^{\#}$; our assumption $P_{\mu}\left(x,G_{\mu}\left(x\right)\right)=0$ would then imply that $P_{\mu}^{\#}\left(x\right)=0$ for all $x$ in the nonempty open set $U$, which is impossible.) Each time we replace $P_{\mu}$ by $\partial_{t}P_{\mu}$ for some $\mu$, the quantity $\sum_{\mu=1}^{m}\deg P_{\mu}$ decreases. Therefore, after finitely many steps, we arrive at the case in which, for each $\mu=1,\cdots,m$, $\partial_{t}P_{\mu}$ isn’t identically zero on the graph of $G_{\mu}$.

From now on, we assume we are in this case. The functions $U\ni x\mapsto\partial_{t}P_{\mu}\left(x,G_{\mu}\left(x\right)\right)$ $\left(\mu=1,\cdots,m\right)$ are real-analytic and nonzero. Hence, there exists $x^{0}\in U$ such that

[TABLE]

We can compute such an $x^{0}\in U$, by standard algorithms in Section 2.1. Let $y^{0}=\left(y_{1}^{0},\cdots,y_{m}^{0}\right)=G\left(x^{0}\right)$. Thus, $\left(x^{0},y^{0}\right)\in\Gamma$, and $\left(x^{0},y^{0}\right)$ belongs to the algebraic set

[TABLE]

Moreover, thanks to (2.5.2), we know that

• (2.5.4)

The differentials $dP_{\mu}\left(x^{0},y^{0}\right)$ $\left(\mu=1,\cdots,m\right)$ are linearly independent, where we regard each $P_{\mu}\left(x,w_{\mu}\right)$ as a polynomial in $\left(x,w_{1},\cdots,w_{n}\right)$.

Hence we may apply the holomorphic implicit function theorem and Lemma 5 in Section 1. From the holomorphic implicit function theorem we obtain a ball $B_{1}\subset\mathbb{C}^{n}$ centered at $x^{0}$, an open ball $B_{2}\subset\mathbb{C}^{n}\times\mathbb{C}^{m}$ centered at $\left(x^{0},y^{0}\right)$, and a holomorphic map $G^{\mathbb{C}}:B_{1}\rightarrow\mathbb{C}^{m}$ such that $G^{\mathbb{C}}\left(x^{0}\right)=y^{0}$, and

[TABLE]

We may suppose $B_{1}\cap\mathbb{R}^{n}\subset U$. Note that $\Gamma\subset W$, hence (2.5.6) gives $G\left(x\right)=G^{\mathbb{C}}\left(x\right)$, provided $x\in B_{1}\cap\mathbb{R}^{n}$, and

[TABLE]

Moreover, both $G$ and $G^{\mathbb{C}}$ are continuous, and we have $\left(x^{0},G\left(x^{0}\right)\right)=\left(x^{0},G^{\mathbb{C}}\left(x^{0}\right)\right)=\left(x^{0},y^{0}\right)\in B_{2}$. Therefore, (2.5.7) holds for all $x\in U$ sufficiently close to $x^{0}$. Consequently,

[TABLE]

Returning to (2.5.3) and ((2.5.4)), we now apply Lemma 5 in Section 1. Let $V_{1},\cdots,V_{p}$ be the irreducible components of $W$. (We can compute them, see Algorithm 4.) Let $V_{1,\text{reg}},\cdots,V_{p,\text{reg}}$ denote the set of regular points of $V_{1},\cdots,V_{p}$, respectively. Then Lemma 5 produces an index $j_{0}$ and a small ball $\tilde{B}\subset\mathbb{C}^{n}\times\mathbb{C}^{m}$ about $\left(x^{0},y^{0}\right)$, for which  the following hold.

[TABLE]

In particular, $j_{0}$ is the one and only $j\in\left\{1,\cdots,p\right\}$ such that $\left(x^{0},y^{0}\right)\in V_{j}$. Hence, we can compute $j_{0}$. Now let $P\left(z,w\right)$ be any polynomial on $\mathbb{C}^{n}\times\mathbb{C}^{m}$. We claim that

• (2.5.10)

$P$ vanishes everywhere on $\Gamma$ if and only if $P$ vanishes everywhere on $V_{j_{0}}$.

If ((2.5.10)) holds, then we can take $V$ in Algorithm 15 to be $V_{j_{0}}$, and we are done. Thus, the explanation of Algorithm 15 is reduced to the task of proving ((2.5.10)).

Suppose $P=0$ everywhere on $\Gamma$. Then $P\left(x,G\left(x\right)\right)=0$ for all $x$ in a small neighborhood of $x^{0}$ in $\mathbb{R}^{n}$. Because $G^{\mathbb{C}}$ is holomorphic in a small neighborhood of $x^{0}$ in $\mathbb{C}^{n}$, it follows from (2.5.8) that $P\left(z,G^{\mathbb{C}}\left(z\right)\right)=0$ for all $z$ in a small neighborhood of $x^{0}\in\mathbb{C}^{n}$. Therefore, by (2.5.6), we have

• (2.5.12)

$P\left(z,w\right)=0$ for all $\left(z,w\right)\in W$ sufficiently close to $\left(x^{0},y^{0}\right)$.

Thanks to (2.5.9) and ((2.5.12)), we have $\left(x^{0},y^{0}\right)\in V_{j_{0},\text{reg}}$, and $P\left(z,w\right)=0$ for all $\left(z,w\right)\in V_{j_{0},\text{reg}}$ sufficiently close to $\left(x^{0},y^{0}\right)$. Because $P$ is a polynomial and $V_{j_{0},\text{reg}}$ is a connected complex-analytic submanifold of $\mathbb{C}^{n}$, it follows that $P\left(z,w\right)=0$ for all $\left(z,w\right)\in V_{j_{0},\text{reg}}$. Because $P$ is continuous and $V_{j_{0},\text{reg}}$ is dense in $V_{j_{0}}$, it follows that $P\left(z,w\right)=0$ for all $\left(z,w\right)\in V_{j_{0}}$. Thus, we have shown that if $P=0$ everywhere on $\Gamma$, then $P=0$ everywhere on $V_{j_{0}}$.

Conversely, suppose $P=0$ everywhere on $V_{j_{0}}$. Then, in particular, $P=0$ everywhere on $V_{j_{0},\text{reg}}$. Applying (2.5.9), we have that $P=0$ on a small neighborhood of $\left(x^{0},y^{0}\right)$ in $W$. Thanks to (2.5.6) and the continuity of $G^{\mathbb{C}}$, this yields $P\left(z,G^{\mathbb{C}}\left(z\right)\right)=0$ for all $z\in\mathbb{C}^{n}$ sufficiently close to $x^{0}$. Consequently, by (2.5.8), we have $P\left(x,G\left(x\right)\right)=0$ for all $x\in U$ sufficiently close to $x^{0}$. However, we assume in Algorithm 15 that $G$ is real-analytic and $U$ is connected. Therefore, $P\left(x,G\left(x\right)\right)=0$ for all $x$ $\in U$ . Thus, $P=0$ everywhere on $\Gamma$. This completes the proof of ((2.5.10)) and the explanation of Algorithm 15. ∎

Algorithm 16**.**

Given a semialgebraic set $E\subset\mathbb{R}^{q}$, we compute generators for the ideal of all polynomials $P\in\mathbb{R}\left[x_{1},\cdots,x_{q}\right]$ that vanish on $E$.

Explanation.

Applying Algorithm 14 (with, say, a single $F_{\lambda}\equiv 1$), we obtain a partition of $E$ into finitely many semialgebraic subsets $E_{\nu}$ $(\nu=1,\cdots,N)$, together with invertible linear maps $T_{\nu}:\mathbb{R}^{q}\rightarrow\mathbb{R}^{q}$, such that each $T_{\nu}E_{\nu}$ is given in the form assumed in Algorithm 15. For each $\nu$, Algorithm 14 also produces polynomials $P_{\mu\nu}$ $\left(\mu=1,\cdots,q-n_{\nu}\right)$ that play the rôle of the $P_{\mu}$ in Algorithm 15. From Algorithm 15, we therefore obtain for each $\nu$ an affine variety $V_{\nu}\subset\mathbb{C}^{q}$ such that for any $P\in\mathbb{C}\left[z_{1},\cdots,z_{q}\right]$, $P|_{T_{\nu}E_{\nu}}=0$ if and only if $P|_{V_{\nu}}=0$. Algorithm 1 produces generators $P_{\nu,1},\cdots,P_{\nu,G\left(\nu\right)}\in\mathbb{C}\left[z_{1},\cdots,z_{q}\right]$ for the ideal of polynomials in $\mathbb{C}\left[z_{1},\cdots,z_{q}\right]$ that vanish on $V_{\nu}$. Thus, $P_{\nu,1},\cdots,P_{\nu,G\left(\nu\right)}$ generate the ideal of polynomials in $\mathbb{C}\left[z_{1},\cdots,z_{q}\right]$ that vanish on $T_{\nu}E_{\nu}\subset\mathbb{R}^{q}$. Consequently, the real and imaginary parts of the $P_{\nu,g}$ $\left(g=1,\cdots,G\left(\nu\right)\right)$ generate the ideal of all polynomials in $\mathbb{R}\left[x_{1},\cdots,x_{q}\right]$ that vanish on $T_{\nu}E_{\nu}$. It is now trivial to produce generators for the ideal $\mathcal{I}_{\nu}$ of all polynomials in $\mathbb{R}\left[x_{1},\cdots,x_{q}\right]$ that vanish on $E_{\nu}$. Because $E$ is the union of the $E_{\nu}$ the ideal of all polynomials in $\mathbb{R}\left[x_{1},\cdots,x_{q}\right]$ that vanish on $E$ is equal to $\mathcal{I}_{1}\cap\cdots\cap\mathcal{I}_{N}$. We have produced generators for each $\mathcal{I}_{\nu}$. Using Algorithm 6, we can now produce generators for their intersection. Thus, we compute generators for the ideal of all real polynomials that vanish on $E$. This completes our explanation of Algorithm 16. ∎

2.6.

We conclude this section with an elementary algorithm.

Algorithm 17** (Find Critical $l$).**

Given matrices $\left(A_{ij}\right)_{i=1,\cdots,I;j=1,\cdots J}$ and $\left(B_{ik}\right)_{i=1,\cdots I;k=1,\cdots,K}$ with polynomial entries $A_{ij},B_{ik}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$; and given a nonzero polynomial $\Delta\left(x_{1},\cdots,x_{n}\right)$, we produce an integer $l_{0}\geq 0$ with the following property:

Let $P_{1},\cdots,P_{K}\in\mathbb{R}[x_{1},\cdots,x_{n}]$. Suppose there exist $Q_{1},\cdots,Q_{J}\in\mathbb{R}[x_{1},\cdots,x_{n}]$ and an integer $l\geq 0$ such that

[TABLE]

Then there exist $Q_{1}^{0},\cdots,Q_{J}^{0}\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$ such that

[TABLE]

Explanation.

We write $\vec{P}$ to denote a vector $(P_{1},\cdots,P_{K})$ of polynomials, and we write $\mathcal{M}_{l}$ to denote the $\mathbb{R}[x_{1},\cdots,x_{n}]$-module of all $\vec{P}$ for which (2.6.1) admits a polynomial solution $Q_{1},\cdots,Q_{J}$.

Given any $l\geq 0$, we can compute generators $\vec{P}^{g,l}$ for $\mathcal{M}_{l}$ ($g=1,\cdots,G_{l}$).

We have $\mathcal{M}_{0}\subseteq\mathcal{M}_{1}\subseteq\cdots$. Since $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$ is Noetherian, it follows that

[TABLE]

We can test a given $l$ to decide whether $\mathscr{M}_{l}=\mathscr{M}_{l+1},$ by checking whether the generators for $\mathscr{M}_{l+1}$ belong to the module $\mathscr{M}_{l}$, for which we have computed generators.

Therefore, we may successively test $l=0,l=1,l=2$, etc., to determine whether $\mathcal{M}_{l}=\mathcal{M}_{l+1}$. If so, we stop.

We know that the above algorithm will eventually terminate, thus producing an integer $l_{0}\geq 0$ for which

[TABLE]

For any $l\geq l_{0}$, we have

[TABLE]

and

[TABLE]

From (2.6.2), (2.6.3), (2.6.4), we conclude that $\mathscr{M}_{l}=\mathscr{M}_{l+1}$ for $l\geq l_{0}$. Thus, we have computed $l_{0}$ for which

[TABLE]

This completes the explanation of Algorithm 17. ∎

3. Solutions of Differential Equations I

In this section, we suppose we are given a semialgebraic, connected, real-analytic submanifold $V\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$. We assume that

[TABLE]

where each $\tilde{P}_{\mu}$ is a nonzero polynomial.

We assume that $x_{1},\cdots,x_{n}$ serve as local coordinates on some non-empty relatively open subset of $V$.

We fix an integer $J\geq 1$, and we write $\vec{P}$ to denote a vector

[TABLE]

of polynomials $P_{j}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

We suppose we are given a linear partial differential operator $L$ of order $M$ with polynomial coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

We suppose that $L$ acts on vectors $\vec{P}$ and produces scalars, i.e., $L\vec{P}$ consists of a single polynomial.

We assume that $L$ involves only $y_{j}$-derivatives; it involves no $x_{i}$-derivatives.

In this section, we present the following

Algorithm 18**.**

Given $V,\tilde{P}_{\mu}$, $L$ as above, we produce generators for the $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$-module $\mathscr{M}$ consisting of all $\vec{P}$ such that

[TABLE]

on $V$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

Explanation.

We compute generators $S_{1},\cdots,S_{\nu_{\max}}$ for the ideal of polynomials

[TABLE]

that vanish on $V$. (See Algorithm 16.) Write

[TABLE]

with $a_{\mu}$ nonzero.

(Note that $\tilde{P}_{\mu}$ cannot be independent of $y_{\mu}$, since $\tilde{P}_{\mu}=0$ on $V$, $\tilde{P}_{\mu}$ is not identically zero, and $\left(x_{1},\cdots,x_{n}\right)$ serve as local coordinates on a neighborhood in $V$.)

Set $\Delta\left(x_{1},\cdots,x_{n}\right)=\prod_{\mu=1}^{m}a_{\mu}\left(x_{1},\cdots,x_{n}\right)$. Thus $\Delta$ isn’t identically zero.

Now suppose $\vec{P}$ satisfies (3.0.1). Applying Lemma 2, we can write

[TABLE]

where $\deg_{y}\vec{P}^{\#}\leq D_{1}^{*}$. Here, $D_{1}^{*}$ may be computed from $D_{1},\cdots,D_{m},M$.

Since $\vec{P}\in\mathscr{M}$, we have $\left(\Delta\left(x_{1},\cdots,x_{n}\right)\right)^{l}\vec{P}\in\mathscr{M}$.

Also, since $L$ is $M^{th}$-order and $\left[\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)\right]^{M+1}$ vanishes to $\left(M+1\right)^{rst}$-order at $V$, we have

[TABLE]

Hence, (3.0.2) tells us that $\vec{P}^{\#}\in\mathscr{M}$. Thus, $L\left(QP^{\#}\right)=0$ on $V$ for each polynomial $Q$. In particular, for each multiindex $\gamma$ of order $\left|\gamma\right|\leq M$, we have

[TABLE]

and therefore

[TABLE]

for polynomials $\dot{A}_{\nu}^{\gamma}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

Applying Lemma 2 to $\dot{A}_{\nu}^{\gamma}$, we see that we can express

[TABLE]

where the $\hat{A}_{\nu}^{\gamma}$ are polynomials, the $\tilde{H}_{\mu}^{\gamma}$ are polynomials, and $\deg_{y}\hat{A}_{\nu}^{\gamma}\leq D_{2}^{*}$. Here, $D_{2}^{*}$ may be computed from $D_{1}^{*}$ and the $\tilde{P}_{\mu}$.

We now conclude from (3) that $\sum_{\mu=1}^{m}\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)\tilde{H}_{\mu}^{\gamma}\left(x_{1},\cdots,x_{n}\right)$ has degree at most $D_{3}^{*}$ in $\left(y_{1},\cdots,y_{m}\right)$, where $D_{3}^{*}$ may be computed from from $L$, $D_{1}^{*},M,D_{2}^{*}$ and the $S_{\nu}$.

Therefore, by Lemma 4, we have (for some $\tilde{l}\geq 0$):

[TABLE]

with $\deg_{y}H_{\mu}^{\gamma\#}\leq D_{4}^{*}$; here, $D_{4}^{*}$ may be computed from $D_{3}^{*}$ and the $\tilde{P}_{\mu}$.

Substituting (3.0.4) into (3), we learn that

[TABLE]

with $\deg_{y}\vec{P}^{\#}\leq D_{1}^{*}$, $\deg_{y}\check{A}_{\nu}^{\gamma}\leq D_{2}^{*}$, and $\deg_{y}H_{\mu}^{\gamma\#}\leq D_{4}^{*}$.

In view of these bounds on the degrees of $\vec{P}^{\#}$, $\check{A}_{\nu}^{\gamma}$, $H_{\mu}^{\gamma\#}$ in $\left(y_{1},\cdots,y_{m}\right)$, we may view (3) as a system of linear equations with polynomial coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$; here, we use the fact that $L$ differentiates only in $\left(y_{1},\cdots,y_{m}\right),$ not in $\left(x_{1},\cdots,x_{n}\right)$.

We now perform Algorithm 17. Thus, from $L$, $S_{1},\cdots,S_{\nu},P_{1},\cdots,P_{m},D_{1}^{*},D_{2}^{*},D_{4}^{*}$, we produce an integer $l_{0}\geq 0$ such that (3) implies that we may express

[TABLE]

with $\deg_{y}A_{\nu}^{\gamma}\leq D_{2}^{*}$ and $\deg_{y}H_{\mu}^{\gamma}\leq D_{4}^{*}$.

We can compute generators $\vec{P}_{1}^{\#},\cdots,\vec{P}_{K}^{\#}$ for the $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$-module of all $\vec{P}^{\#}$ with $\deg_{y}\vec{P}^{\#}\leq D_{1}^{*}$, for which (3) admits a polynomial solution $\left(A_{\nu}^{\gamma}\right)_{\left|\gamma\right|\leq M,\nu=1,\cdots,\nu_{\max}}$, $\left(H_{\mu}^{\gamma}\right)_{\left|\gamma\right|\leq M,\mu=1,\cdots,m}$ with $\deg_{y}A_{\nu}^{\gamma}\leq D_{2}^{*}$ and $\deg_{y}H_{\mu}^{\gamma}\leq D_{4}^{*}$.

Thus, for some polynomials $A_{\nu,k}^{\gamma}$, $H_{\mu,k}^{\gamma}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$, we have

[TABLE]

and

[TABLE]

Substituting this last equation into (3.0.2), we find that

[TABLE]

for some polynomials $\vec{H}_{\mu}$, $G_{k}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$. (We forget that the $G_{k}$ depend only on $x_{1},\cdots,x_{n}$.)

This holds for some $l$. Using Algorithm 17, we can compute from the $\tilde{P}_{\mu}$ and $\vec{P}_{k}^{\#}$ (and $M$) an integer $l_{1}\geq 0$ such that (3) implies the existence of $\vec{H}_{\mu}^{\ast}$ and $G_{k}^{\ast}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ satisfying

[TABLE]

To summarize: Suppose $\vec{P}$ belongs to the module $\mathscr{M}$, i.e., suppose $L\left(Q\vec{P}\right)=0$ on $V$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$. Then there exist $\vec{H}_{\mu}^{\ast}$ and $G_{k}^{\ast}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ satisfying (3). Here, the integer $l_{1}$ and the polynomial vectors $\vec{P}_{k}^{\#}$ ($k=1,\cdots,K$) have been computed from $V$, $\tilde{P}_{\mu}$ ($\mu=1,\cdots,m$), and $M$, $L$.

Conversely, suppose (3) holds. We will prove that $\vec{P}$ belongs the module $\mathscr{M}$, i.e., $L\left(Q\vec{P}\right)=0$ on $V$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

We know that $\left[\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)\right]^{M+1}\vec{H}_{\mu}^{\ast}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)$ belongs to $\mathscr{M}$, since $L$ is of $M^{th}$-order and $\left[\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)\right]^{M+1}$ vanishes to $\left(M+1\right)^{\text{rst}}$ order on $V$.

We will check that

[TABLE]

Once we prove (3.0.10), we will know that $G_{k}^{\ast}\vec{P}_{k}^{\#}\in\mathscr{M}$. In view of (3), that will tell us that $\left(\Delta\left(x_{1},\cdots,x_{n}\right)\right)^{l_{1}}\vec{P}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)\in\mathscr{M}$, i.e.,

[TABLE]

On the other hand, since $\Delta=\Delta\left(x_{1},\cdots,x_{n}\right)$ and $L$ involves no $x$-derivatives, we have

[TABLE]

and thus

[TABLE]

By Lemma 1, we will know that $L\left(Q\vec{P}\right)=0$ on $V$ for each $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$, completing the proof that $\vec{P}\in\mathscr{M}$ as claimed.

Thus, once we check (3.0.10), we will know that $\vec{P}\in\mathscr{M}$ if and only if (3) admits a polynomial solution $\left(H_{\mu}^{\ast},G_{k}^{\ast}\right)$.

To prove (3.0.10), we return to (3.0.7). We know that the polynomials $S_{\nu}$, $\tilde{P}_{\mu}$ vanish on $V$. Hence, (3.0.7) tells us that

[TABLE]

Again applying Lemma 1, we see that

[TABLE]

(Here, we use the fact that $L$ involves no $x$-derivatives; multiplying $\vec{P}_{k}^{\#}$ by $x^{\beta}$ has a trivial effect.)

Now, let $Q$ be any polynomial, and let $\left(x^{o},y^{o}\right)\in V$. We expand $Q$ about $(x^{o},y^{o})$ and thus write

[TABLE]

for coefficients $Q_{\beta\gamma}$ and a polynomial $Q_{low}$, with $\deg\left(Q_{low}\right)\leq M$.

We have already seen that $L\left(Q_{low}\vec{P}_{k}^{\#}\right)=0$ on $V$ (see (3.0.11)), hence $L\left(Q_{low}\vec{P}_{k}^{\#}\right)$ vanishes at $\left(x^{o},y^{o}\right)$.

On the other hand, since $L$ is of $M^{th}$-order, we have also

[TABLE]

Therefore, $L\left(Q\vec{P}_{k}^{\#}\right)=0$ at $\left(x^{o},y^{o}\right)$ for any $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$, completing the proof of (3.0.10).

We now know that $\vec{P}\in\mathscr{M}$ if and only if there exist polynomials $H_{\mu}^{\ast}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)$ ($\mu=1,\cdots,m$) and $G_{k}^{\ast}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right)$ ($k=1,\cdots,K$) satisfying (3).

Since $l_{1}$, $\tilde{P}_{\mu}$, and $\vec{P}_{k}^{\#}$ in (3) have all been computed, we may now produce generators for the $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$-module of solutions $\left(\vec{P},\left(\vec{H}_{\mu}^{\ast}\right)_{\mu=1,\cdots,m}\left(G_{k}^{\ast}\right)_{k=1,\cdots,K}\right)$ of (3).

The $\vec{P}$-components of these generators are generators for the module $\mathscr{M}$.

This completes our explanation of Algorithm 18. ∎

4. Solutions of Differential Equations II

In this section, we assume that

[TABLE]

where $\tilde{P}_{\mu},\hat{P}_{\lambda}$ are nonzero polynomials.

We assume that $V\subset\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}$ is a semialgebraic real-analytic connected manifold, and that $\left(x_{1},\cdots,x_{n}\right)$ serve as local coordinates on a nonempty (relatively) open subset of $V$.

We fix an integer $J\geq 1$, and we write $\vec{P}$ to denote a vector $\left(P_{1},\cdots,P_{J}\right)$ of polynomials

[TABLE]

We suppose we are given an $M^{th}$-order linear partial differential operator $L$ with polynomial coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$.

We suppose $L$ maps vectors $\vec{P}$ to scalars (i.e., $L\vec{P}$ is a single polynomial).

We assume that $L$ contains no $x$-derivatives, although it may involve both $y$ and $z$-derivatives.

We write $\vec{P}\left(x,y\right)$ to denote a vector of polynomials $\left(P_{1},\cdots,P_{J}\right)$, with each

[TABLE]

(i.e., the $P_{j}$ do not depend on $z_{1},\cdots,z_{p}$).

In this section, we present the following

Algorithm 19**.**

Given $V,\tilde{P}_{\mu},\hat{P}_{\lambda}$, $L$ as above, we compute generators for the $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ module consisting of all vectors $\vec{P}\left(x,y\right)$ such that

[TABLE]

Explanation.

Applying Algorithm 18 (with $z_{1},\cdots,z_{p}$ regarded as $y_{m+1},\cdots,y_{m+p}$), we obtain generators $\vec{P}_{1},\cdots,\vec{P}_{K}$ for the $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$-module of all $\vec{P}$ such that $L\left(Q\vec{P}\right)=0$ on $V$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$.

We write $D_{1}^{*},D_{2}^{*},\cdots$ to denote constants that can be computed from $V,\tilde{P}_{\mu},\hat{P}_{\lambda},L$.

For each $\lambda=1,\cdots,p$, we write

[TABLE]

with $a_{\lambda}\left(x_{1},\cdots,x_{n}\right)\not\equiv 0$. (Note that $\hat{P}_{\lambda}$ cannot be independent of $z_{\lambda}$, since $\hat{P}_{\lambda}=0$ on $V$, yet $\hat{P}_{\lambda}\not\equiv 0$, and $\left(x_{1},\cdots,x_{n}\right)$ serve as local coordinates on a nonempty neighborhood in the real-analytic manifold $V$.)

Let $\Delta\left(x_{1},\cdots,x_{n}\right)=\prod_{\lambda=1}^{p}a_{\lambda}\left(x_{1},\cdots,x_{n}\right)$; thus $\Delta$ is a nonzero polynomial.

Now suppose $\vec{P}\left(x,y\right)$ satisfies (4.0.1). Then for polynomials

[TABLE]

($k=1,\cdots,K$), we have

[TABLE]

Note that the $A_{k}^{\left(0\right)}$ and $\vec{P}_{k}$ may depend on $z_{1},\cdots,z_{p}$, although $\vec{P}\left(x,y\right)$ does not. By Lemma 2, there exist $l\geq 0$ and polynomials $A_{k}^{\left(1\right)}$, $B_{k\lambda}\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$ such that

[TABLE]

for each $k=1,\cdots,K$; and $\deg_{z}A_{k}^{\left(1\right)}\leq D_{1}^{*}$. Substituting this equation into (4.0.2), we see that

[TABLE]

for some vector of polynomials $\vec{H}_{\lambda}^{\left(1\right)}$ in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$.

Thus, (4.0.3) holds, with $\deg_{z}A_{k}^{\left(1\right)}\leq D_{1}^{*}$ for all $k$.

All terms in (4.0.3) except for the sum on $\lambda$ have degree $\leq D_{2}^{*}$ in $\left(z_{1},\cdots,z_{p}\right)$.

Therefore, (4.0.3) tells us that

[TABLE]

Hence, by Lemma 4, there exist an integer $l^{\prime}\geq 0$ and vectors of polynomials $\vec{H}_{\lambda}^{\left(2\right)}$in

[TABLE]

satisfying

[TABLE]

and $\deg_{\left(z_{1},\cdots,z_{p}\right)}\vec{H}_{\lambda}^{\left(2\right)}\leq D_{3}^{*}$.

Substituting the above equation into (4.0.3), we find that for some $l^{\prime\prime}\geq 0$:

[TABLE]

Here, the $A_{k}^{\left(2\right)}$, and the components of the vectors $\vec{H}_{\lambda}^{\left(2\right)}$, are polynomials in $x_{1},\cdots,x_{n}$, $y_{1},\cdots,y_{m}$, $z_{1},\cdots,z_{p}$; and these polynomials have degree at most $D_{4}^{*}$ in $\left(z_{1},\cdots,z_{p}\right)$ (thanks to the above bound for the degrees of $A_{k}^{\left(1\right)}$, $\vec{H}_{\lambda}^{\left(2\right)}$ in $\left(z_{1},\cdots,z_{p}\right)$).

We may regard (4.0.4) as a system of linear equations with coefficients in

[TABLE]

Applying Algorithm 17, we now produce an integer $l_{0}\geq 0$ such that (4.0.4) implies that

[TABLE]

for some $A_{k}$, $\vec{H}_{\lambda}$; where $A_{k}$ and the components of $\vec{H}_{\lambda}$ are polynomials in $x_{1},\cdots,x_{n}$, $y_{1},\cdots,y_{m}$, $z_{1},\cdots,z_{p}$ having degree at most $D_{4}^{*}$ in $\left(z_{1},\cdots,z_{p}\right)$.

To summarize: We have computed $l_{0},\Delta,\vec{P}_{k},\hat{P}_{\lambda}$. Moreover, whenever $\vec{P}(x,y)$ satisfies $L\left(Q\vec{P}\right)=0$ on $V$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$, then (4.0.5) holds with $A_{k}$ and $\vec{H}_{\lambda}$ having degree at most $D_{4}^{*}$ in $z$.

Conversely, suppose (4.0.5) holds. Let $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$ be given. By the defining property of the $\vec{P}_{k}$, we have

[TABLE]

Moreover,

[TABLE]

since $L$ is of order $M$ and $\left[\hat{P}_{\lambda}\left(x_{1},\cdots,x_{n},z_{\lambda}\right)\right]^{M+1}$ vanishes to $\left(M+1\right)^{rst}$-order on $V$.

Therefore, (4.0.5) yields

[TABLE]

However, since $L$ involves no $x$-derivatives, we have

[TABLE]

and therefore

[TABLE]

Applying Lemma 1, we conclude that $L\left(Q\vec{P}\right)=0$ on $V$. This holds for arbitrary $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$.

Thus, $\vec{P}\left(x,y\right)$ satisfies (4.0.1) if and only if it satisfies (4.0.5) with $A_{k}$ and $\vec{H}_{\lambda}$ having degree at most $D_{4}^{*}$ in $z$.

However, using the standard algorithms in Section 2.1, we can compute generators for the $\mathbb{R[}x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}]$-module of all solutions of (4.0.5) with $A_{k}$ and $\vec{H}_{\lambda}$ having degree at most $D_{4}^{*}$ in $z$.

This concludes our explanation of Algorithm 19. ∎

5. Solutions of Differential Equations III

In this section, we suppose we are given a semialgebraic connected real analytic submanifold $V$ of $\mathbb{R}^{n}\times\mathbb{R}^{m}$, with

[TABLE]

where each $\tilde{P}_{\mu}$ is a nonzero polynomial.

We assume that $\left(x_{1},\cdots,x_{n}\right)$ serve as local coordinates on some nonempty relatively open subset $U$ of $V$.

We fix $J\geq 1$ and write $\vec{P}$ to denote a vector $\left(P_{1},\cdots,P_{J}\right)$ of polynomials

[TABLE]

We suppose we are given a linear differential operator $L$ of order $\leq M$, with polynomial coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

We assume that $L$ acts on vectors $\vec{P}$ and produces scalars (i.e., $L\vec{P}$ consists of a single polynomial).

We do not assume that $L$ involves only $y$-derivatives. Our $L$ may involve differentiations in any (or all) of the variables $x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}$.

The result of this section is the following.

Algorithm 20**.**

Given $V,\tilde{P}_{\mu},L$, we compute finitely many linear differential operators $L_{1},\cdots,L_{N}$, with the following properties

• •

Each $L_{\nu}$ maps vectors $\vec{P}$ to single polynomials.

• •

The coefficients of each $L_{\nu}$ are polynomials in $\mathbb{R}[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}]$.

• •

Each $L_{\nu}$ involves only $y$-differentiations (i.e., no $x$-differentiations appear in $L_{\nu}$).

• •

Let $\vec{P}$ be given. Then

[TABLE]

if and only if

[TABLE]

Explanation.

As noted (twice) before, $\tilde{P}_{\mu}\left(x_{1},\cdots,x_{n},y_{\mu}\right)$ cannot be independent of $y_{\mu}$, since $\tilde{P}_{\mu}=0$ on $V$ but $\tilde{P}_{\mu}$ is not identically zero, and $\left(x_{1},\cdots,x_{n}\right)$ serve as local coordinates in a neighborhood in $V$.

We begin by possibly modifying the $\tilde{P}_{\mu}$ as in our explanation of Algorithm 15, so that (for each $\mu$) the polynomial

[TABLE]

does not vanish everywhere on $V$.

Now let

[TABLE]

Thus, $\Delta$ is a polynomial and $\Delta$ is nonzero somewhere on $V$.

We introduce some useful vector fields on $\mathbb{R}^{n}\times\mathbb{R}^{m}$: For $j=1,\cdots,n$, define

[TABLE]

Note that $X_{j}$ has polynomial coefficients. More precisely,

[TABLE]

where the $b_{j\mu}$ are polynomials and $Y_{j}$ involves only $y$-derivatives.

Note also that $X_{j}\tilde{P}_{\mu^{\prime}}\left(x_{1},\cdots,x_{n},y_{\mu^{\prime}}\right)=0$ for each $\mu^{\prime}$ and each $j$, since

[TABLE]

It follows that the $X_{j}$ are tangent to $V$ at all points of $U\setminus\left\{\Delta=0\right\}$. (See the beginning of the section for the definition of $U$.) Since $V$ is a connected real-analytic manifold, and since $\Delta|_{V}$ is real-analytic and not identically zero, we conclude that $V\setminus\left\{\Delta=0\right\}$ is dense in $V$ and $U\setminus\left\{\Delta=0\right\}$ is dense in $U$. Therefore, since $V$ and $X_{j}$ are real-analytic, it follows that $X_{j}$ is tangent to $V$ everywhere on $U$, hence everywhere on $V$.

We record the observation:

[TABLE]

We will be commuting multiplication by $\Delta^{s}$ past products $X_{j_{1}}\cdots X_{j_{t}}$. An easy induction on $t$ gives

[TABLE]

where $L_{s,j_{1}\cdots j_{t}}$ is a differential operator with polynomial coefficients and order less than $t$. To see this, we write

[TABLE]

so that

[TABLE]

Since $L$ is a linear differential operator of order at most $M$ with polynomial coefficients, we may write

[TABLE]

where we write $L_{low}$ to denote a differential operator with polynomial coefficients of order $<M$. The symbol $L_{low}$ will be used to denote several such operators, i.e., $L_{low}$ may denote different operators in different occurrences.

In (5.0.5), $\vec{a}_{\alpha\beta}$ is a vector, and each component of $\vec{a}_{\alpha\beta}$ is a polynomial in $x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}$. Also, in (5.0.5), we use “$\cdot$” to denote the dot product of vectors.

From (5.0.5), we obtain

[TABLE]

We express each $\left(X_{1}-Y_{1}\right)^{\alpha_{1}}\cdots\left(X_{n}-Y_{n}\right)^{\alpha_{n}}$ as a sum of terms, each of which is a product of factors $X_{j}$ or $Y_{j}$.

Modulo operators $L_{low}$, we may move all $X_{j}$ in each term to the left of all the $Y_{j}$ in that term. Therefore,

[TABLE]

with the $\vec{\phi}$’s (vector valued) polynomials in $x_{1},\cdots,x_{n}$, $y_{1},\cdots,y_{m}$.

Let $D$ be a large enough integer constant, to be picked below.

Then (5.0.6) yields

[TABLE]

thanks to (5.0.4).

Now $L_{low}$ in (5) satisfies all the assumptions made on $L$, except that $L_{low}$ has order $\leq M-1$, whereas $L$ has order $\leq M$. Therefore, just as we proved (5) for $L$, we can now prove

[TABLE]

Here, the $\vec{\psi}$’s are (vector-valued) polynomials in $x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}$, and $L_{lower}$ is a linear differential operator of order $\leq M-2$ with polynomial coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$. Also, we point out that $L_{low}$ in (5) is the same as $L_{low}$ in (5).

Again, there is a formula for $\left[\Delta\left(x,y\right)\right]^{\left(D-2M\right)-2\left(M-1\right)}$ $L_{lower}\left(\vec{P}\right),$ analogous to (5) and (5).

Continuing in this way, we eventually reach $L_{lowest}$, a differential operator of order [math].

We simply pick $D$ large enough so that all the powers of $\Delta\left(x,y\right)$ appearing in the above formulas (5), (5) $\cdots$ are non-negative.

From the formulas (5), (5) $\cdots$ , we conclude that

[TABLE]

where the coefficients $\vec{\theta}_{k_{1}\cdots k_{b},\gamma}^{j_{1}\cdots j_{a}}$ are vectors whose components are polynomials in $x_{1},\cdots,x_{n}$, $y_{1},\cdots,y_{m}$.

Moreover, everything we did in deriving (5.0.9) was effective; hence, we may compute the integer $D$ and the polynomials $\vec{\theta}_{k_{1}\cdots k_{b},\gamma}^{j_{1}\cdots j_{a}}$ in (5.0.9).

It is convenient to express (5.0.9) in a different notation. First of all, for fixed $j_{1}\leq\cdots\leq j_{a}$, we define

[TABLE]

Thus, $L_{j_{1}\cdots j_{a}}$ is a linear differential operator involving no $x$-derivatives and having coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$. Thus, (5.0.9) becomes

[TABLE]

For a multiindex $\alpha=\left(\alpha_{1},\cdots,\alpha_{n}\right)$, we define $X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$. (The $X_{j}$ needn’t commute.)

Then the above formula for $[\Delta(x,y)]^{D}L(\vec{P})$ takes the form

[TABLE]

where each $L_{\alpha}$ is a linear differential operator involving no $x$-derivatives and having coefficients in $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$.

Moreover,

[TABLE]

We now prove the following

**Claim. **Let $\vec{P}=(P_{1},\cdots,P_{J})$ with each $P_{j}\in\mathbb{R}[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}]$. Then

[TABLE]

if and only if

[TABLE]

Once we have established the claim, we can simply take $L_{1},\cdots,L_{N}$ in Algorithm 20 to be a list of all the $L_{\alpha}$. Thus, to complete our explanation of that algorithm, it is enough to prove the above claim.

First suppose $\vec{P}$ satisfies $L_{\alpha}\left(Q\vec{P}\right)=0$ on $V$ for all $Q$ and all $\left|\alpha\right|\leq M$. Recalling from (5.0.3) that each $X_{j}$ is tangent to $V$, and recalling the definition $X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$ for $\alpha=\left(\alpha_{1},\cdots,\alpha_{n}\right)$, we conclude that $X^{\alpha}L_{\alpha}\left(Q\vec{P}\right)=0$ on $V$ for all $Q$, and for all $\left|\alpha\right|\leq M$. Summing over $\alpha$ and recalling (5.0.10), we conclude that

[TABLE]

We have seen that $V\setminus\left\{\left(x,y\right)\in V:\Delta\left(x,y\right)=0\right\}$ is dense in $V$. Therefore, for each $Q$, $L\left(Q\vec{P}\right)=0$ on a dense subset of $V$. Since $Q$, the components of $\vec{P}$, and the coefficients of $L$ are all polynomials in $x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}$, we conclude that $L\left(Q\vec{P}\right)=0$ on $V$. Thus, we have proven that (5.0.13) implies (5.0.12).

Next, we show that (5.0.12) implies (5.0.13).

For all $M_{\ast}\leq M$, we prove that

[TABLE]

implies

[TABLE]

Once we know that (5.0.14) implies (5.0.15), we just take $M_{\ast}=M$; then (5.0.14) follows from (5.0.12) , (5.0.15) becomes (5.0.13), and the proof of Claim will be complete. Thus, it remains to prove that (5.0.14) implies (5.0.15). We proceed by induction on $M_{\ast}$.

For $M_{\ast}=0$, (5.0.14) and (5.0.15) are obviously equivalent, since there is only one multiindex $\alpha$ with $|\alpha|\leq 0$.

For the induction step, fix $M_{\ast}\geq 1$, and suppose that (5.0.14)$\implies$(5.0.15) holds with $M_{\ast}-1$ in place of $M_{\ast}$. We will prove (5.0.14)$\implies$(5.0.15) for the given $M_{\ast}$.

In fact, suppose $\vec{P}$ satisfies (5.0.14). Fix $(x^{0},y^{0})\in V$, and $|\alpha|=M_{\ast}$. Recalling the form (5.0.2) of $X_{j}$, we see that

[TABLE]

where $L_{\beta}^{err}$ differentiates at most $\left|\beta\right|-1$ times in $x$, and perhaps many times in $y$. Therefore, for any smooth function $A\left(x,y\right)$, we have

[TABLE]

and

[TABLE]

For any $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$, we have from (5.0.14)

[TABLE]

However, $L_{\beta}$ involves no $x$-derivatives, and therefore

[TABLE]

Hence,

[TABLE]

Taking $A\left(x,y\right)=L_{\beta}\left(Q\cdot\vec{P}\right)$ in (5.0.16), (5.0.17), we now conclude that

[TABLE]

This holds for any $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$, any $\left(x^{0},y^{0}\right)\in V$ and any $\left|\alpha\right|=M_{\ast}$. Thus, $\left(\Delta\left(x,y\right)\right)^{M_{\ast}}L_{\alpha}\left(Q\cdot\vec{P}\right)=0$ on $V$, for $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ and $\left|\alpha\right|=M_{\ast}$.

Recalling that $V\setminus\{(x,y)\in V:\Delta(x,y)=0\}$ is dense in $V$, we now conclude that

[TABLE]

for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$ and $\left|\alpha\right|=M_{\ast}$. Recalling that the $X_{j}$ are tangent to $V$, we conclude that

[TABLE]

Together with (5.0.14), this tells us that

[TABLE]

for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right]$. Hence, by induction hypothesis ((5.0.14)$\implies$(5.0.15) with $M_{\ast}$ replaced by $M_{\ast}-1$), we have

[TABLE]

Together with (5.0.18) this completes the proof that (5.0.14)$\implies$(5.0.15) for the given $M_{\ast}$. This also completes our induction on $M_{\ast}$, the proof of Claim, and the explanation of Algorithm 20. ∎

6. Solutions of Differential Equations IV

In this section, we suppose we are given a semialgebraic, connected, real-analytic submanifold

[TABLE]

We write $\left(x,y,z\right)$ to denote a point of $\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}$, with $x=\left(x_{1},\cdots,x_{n}\right)$, $y=\left(y_{1},\cdots,y_{m}\right)$, and $z=\left(z_{1},\cdots,z_{p}\right)$.

We assume that $x_{1},\cdots,x_{n}$ serve as local coordinates in some nonempty relatively open subset of $V$.

We suppose we are given nonzero polynomials $\tilde{P}_{\mu}\left(x,y_{\mu}\right)$ ($\mu=1,\cdots,m$) and $\hat{P}_{\lambda}\left(x,z_{\lambda}\right)$ ($\lambda=1,\cdots p$) that vanish on $V$.

We write $\mathbb{R}\left[x,y,z\right]$ to denote the ring $\mathbb{R}\left[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m},z_{1},\cdots,z_{p}\right]$; similarly for $\mathbb{R}\left[x,y\right]$. We write $\vec{P}$ to denote a vector $\left(P_{1},\cdots,P_{J}\right)$ with components in $\mathbb{R}\left[x,y,z\right]$, and we write $\vec{P}\left(x,y\right)$ to denote a vector $\left(P_{1},\cdots,P_{J}\right)$ with components in $\mathbb{R}\left[x,y\right]$.

We suppose we are given a linear differential operator $L$ with polynomial coefficients in $\mathbb{R}\left[x,y,z\right]$. The operator $L$ acts on vectors $\vec{P}$ and produces scalars (i.e., $L\vec{P}$ has just one component).

We present the following

Algorithm 21**.**

Given $V$, $\tilde{P}_{\mu}$, $\hat{P}_{\lambda}$, $L$, we find generators for the $\mathbb{R}\left[x,y\right]$-module of all polynomial vectors $\vec{P}\left(x,y\right)$ such that $L\left(Q\cdot\vec{P}\right)=0$ on $V$ for all $Q\in\mathbb{R}\left[x,y,z\right]$.

Explanation.

Regarding $z_{1},\cdots,z_{p}$ as $y_{m+1},\cdots,y_{m+p}$, we apply Algorithm 20, to produce linear differential operators $L_{1},\cdots,L_{N}$ with the following properties.

• •

Each $L_{\nu}$ maps vectors $\vec{P}$ to scalars (i.e., $L_{\nu}\vec{P}$ has only one component).

• •

Each $L_{\nu}$ has polynomial coefficients in $\mathbb{R}[x,y,z]$.

• •

Each $L_{\nu}$ involves only $y$ and $z$-derivatives (no $x$-derivatives).

• •

For any $\vec{P}$, we have $L(Q\vec{P})=0$ on $V$ for all $Q\in\mathbb{R}[x,y,z]$ if and only if $L_{\nu}(Q\vec{P})=0$ on $V$ for all $Q\in\mathbb{R}[x,y,z]$ and all $\nu=1,\cdots,N$.

For each $\nu=1,\cdots,N$, we apply Algorithm 19 to produce generators for the $\mathbb{R}[x,y]$-module

[TABLE]

Thanks to the last bullet point above, the $\mathbb{R}\left[x,y\right]$-module

[TABLE]

is equal to $\mathscr{M}_{1}\cap\cdots\cap\mathscr{M}_{N}$. Since we have computed generators for each $\mathscr{M}_{\nu}$, we can compute generators for their intersection. This completes our explanation of the algorithm. ∎

7. Solutions of Differential Equations V

In this section, we work in $\mathbb{R}^{n}$ and we write $x=\left(x_{1},\cdots,x_{n}\right)$ to denote a point of $\mathbb{R}^{n}$. We fix $J\geq 1$, and we write $\vec{F}=\left(F_{1},\cdots,F_{J}\right)$ to denote a vector of smooth functions on $\mathbb{R}^{n}$.

We suppose we are given a linear differential operator $L$, acting on vectors $\vec{F}$, and producing scalar-valued functions $L\vec{F}$.

We assume that $L$ has semialgebraic coefficients.

Algorithm 22** (Main Algorithm for Differential Equations).**

Given $L$ as above, we compute generators for the $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$-module $\mathcal{M}(L)$ consisting of all polynomial vectors $\vec{P}=\left(P_{1},\cdots,P_{J}\right)$ such that $L\left(Q\vec{P}\right)=0$ on $\mathbb{R}^{n}$ for all $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$.

Explanation.

Let $A_{1}\left(x\right)$, $A_{2}\left(x\right),\cdots,A_{K}\left(x\right)$ be a list of all the coefficients of $L$. The $A_{k}$ are semialgebraic functions.

Applying Algorithm 14 to the semialgebraic set $E=\mathbb{R}^{n}$ and the list of functions $A_{1},\cdots,A_{K}$, we obtain the following:

• •

A partition of $\mathbb{R}^{n}$ into finitely many semialgebraic sets $E_{\nu}\quad(\nu=1,\cdots,N)$.

• •

For each $\nu$, an invertible linear map $T_{\nu}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$.

• •

We guarantee that, for each $\nu$, $T_{\nu}E_{\nu}$ has the form

• –

$T_{\nu}E_{\nu}=\left\{\left(x^{\prime},x^{\prime\prime}\right)\in\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}:x^{\prime}\in U_{\nu},x^{\prime\prime}=G_{\nu}\left(x^{\prime}\right)\right\}$, where

• –

$U_{\nu}\subset\mathbb{R}^{n_{\nu}}$ is an open, connected semialgebraic set and

• –

$G_{\nu}:U_{\nu}\rightarrow\mathbb{R}^{n-n_{\nu}}$ is real-analytic and semialgebraic.

• –

Moreover, for each $\lambda=1,\cdots,K$, we have $A_{\lambda}\circ T_{\nu}^{-1}\left(x^{\prime},G_{\nu}\left(x^{\prime}\right)\right)=H_{\lambda\nu}\left(x^{\prime}\right)$ for all $x^{\prime}\in U_{\nu}$, where $H_{\lambda\nu}:U_{\nu}\rightarrow\mathbb{R}$ is a real-analytic semialgebraic function.

Algorithm 14 computes the above objects $E_{\nu},T_{\nu},U_{\nu},G_{\nu},H_{\lambda\nu}$ as well as nonzero polynomials

• •

$P_{\mu\nu}\left(x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{\mu}\right)\quad\left(\mu=1,\cdots,n-n_{\nu},\nu=1,\cdots,N\right)$ and

• •

$\hat{P}_{\lambda\nu}\left(x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},z_{\lambda}\right)\quad\left(\lambda=1,\cdots,K,\nu=1,\cdots,N\right)$, such that

[TABLE]

and

[TABLE]

We define $L_{\nu}\vec{F}=\left[L\left(\vec{F}\circ T_{\nu}\right)\right]\circ T_{\nu}^{-1}$. Thus, $L\vec{F}=0$ if and only if $L_{\nu}\left(F\circ T_{\nu}^{-1}\right)=0$ on $T_{\nu}\left(E_{\nu}\right)$ for each $\nu=1,\cdots,N$.

Therefore our module $\mathcal{M}(L)$ is equal to the intersection over all $\nu=1,\cdots N$ of the $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$-modules

[TABLE]

• (7.0.1)

$\mathcal{M}_{\nu}^{\#}$ is the $\mathbb{R}\left[x_{1},\cdots,x_{n}\right]$-module consisting of all polynomial vectors $\vec{P}=\left(P_{1},\cdots,P_{J}\right)$ such that

[TABLE]

on $\Gamma_{\nu}=\left\{\left(x^{\prime},x^{\prime\prime}\right)\in\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}:x^{\prime}\in U_{\nu},x^{\prime\prime}\in G_{\nu}\left(x^{\prime}\right)\right\}$ for all $Q\in\mathbb{R}[x^{\prime},x^{\prime\prime}]$.

We will compute generators for each $\mathcal{M}_{\nu}^{\#}$. From those, we can compute generators for $\mathcal{M}_{\nu}$, then for $\mathcal{M}$.

The coefficients of $L$ are the functions $A_{1},\cdots,A_{K}$; and $L_{\nu}$ arose from $L$ by a known linear coordinate change $T_{\nu}$.

• (7.0.3)

Therefore, the coefficients of $L_{\nu}$ are equal on $\Gamma_{\nu}$ to known linear combinations of the functions $H_{\lambda\nu}(x^{\prime})\quad(\lambda=1,\cdots,K)$.

We now introduce the semialgebraic set

• (7.0.5)

$V_{\nu}=\left\{\left(x^{\prime},x^{\prime\prime},z_{1},\cdots,z_{K}\right):x^{\prime}\in U_{\nu},x^{\prime\prime}=G_{\nu}\left(x^{\prime}\right),z_{\lambda}=H_{\lambda\nu}\left(x^{\prime}\right)\text{ for }\lambda=1,\cdots,K\right\}$.

Since $U_{\nu}$ is connected, open and semialgebraic in $\mathbb{R}^{n_{\nu}}$, and since $G_{\nu},H_{\lambda\nu}$ are real-analytic and semialgebraic on $U_{\nu}$, we see that $V_{\nu}$ is a semialgebraic, connected, real-analytic manifold in $\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}\times\mathbb{R}^{K}$ , on which the rectangular coordinates $x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime}$ of $x^{\prime}$ serve as global real-analytic coordinates.

Moreover, our nonzero polynomials $P_{\mu\nu},\hat{P}_{\lambda\nu}$ satisfy

[TABLE]

for $\left(x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}},z_{1},\cdots,z_{k}\right)\in V_{\nu}$  and

[TABLE]

for $\left(x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}},z_{1},\cdots,z_{k}\right)\in V_{\nu}$.

Thus, the $V_{\nu},P_{\mu\nu},\hat{P}_{\lambda\nu}$ (for fixed $\nu$) are as in the setup for Section 6.

Moreover, we may lift $L_{\nu}$ in a natural way to a differential operator $L_{\nu}^{\#}$, acting on vectors of smooth functions on $\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}\times\mathbb{R}^{K}$. To define $L_{\nu}^{\#}$, and to see its relationship to $L_{\nu}$, we recall the remark ((7.0.3)). Thus, for a finite list of coefficients $\left(\Omega_{\alpha\beta\nu}^{\lambda j}\right)$, we have

[TABLE]

for $\left(x^{\prime},x^{\prime\prime}\right)\in\Gamma_{\nu}$. The $\Omega_{\alpha\beta\nu}^{\lambda j}$ are real numbers, which we can compute.

Recalling ((7.0.5)), we are led to define

[TABLE]

for vectors $(F_{1},\cdots,F_{J})$ of smooth functions defined on $\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}\times\mathbb{R}^{K}$.

The relationship between $L_{\nu}$ and $L_{\nu}^{\#}$ is as follows: Suppose $\vec{F}=\left(F_{1},\cdots,F_{J}\right)$, where the functions $F_{j}:\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}\times\mathbb{R}^{K}\rightarrow\mathbb{R}$ do not depend on the last $K$ coordinates. Then we may regard $\vec{F}$ either as a vector-valued function on $\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}\times\mathbb{R}^{K}$ or on $\mathbb{R}^{n_{\nu}}\times\mathbb{R}^{n-n_{\nu}}$. We have $L_{\nu}\vec{F}=0$ on $\Gamma_{\nu}$ if and only if $L_{\nu}^{\#}\vec{F}=0$ on $V_{\nu}$.

Accordingly, our definition ((7.0.1)) of the module $\mathcal{M}_{\nu}^{\#}$ is equivalent to the following

• (7.0.8)

$\mathcal{M}_{\nu}^{\#}$ is the $\mathbb{R}[x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}}]$-module of all polynomial vectors

[TABLE]

with each $P_{j}\in\mathbb{R}[x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}}]$, such that

[TABLE]

for all $Q\in\mathbb{R}[x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}}]$.

Note that $L_{\nu}^{\#}$ is a linear differential operator with polynomial coefficients in

[TABLE]

Thus, $L_{\nu}^{\#}$ is as in the setup for Section 6.

We are now in position to apply Algorithm 21. Thus, we compute generators of the

[TABLE]

• (7.0.10)

$\mathcal{M}_{\nu}^{\#\#}$ consisting of all polynomial vectors $\vec{P}=(P_{1},\cdots,P_{J})$ with each

[TABLE]

such that $L_{\nu}^{\#}(Q\vec{P})=0$ on $V_{\nu}$ for all $Q\in\mathbb{R}[x_{1}^{\prime},\cdots,x_{n_{\nu}}^{\prime},y_{1},\cdots,y_{n-n_{\nu}},z_{1},\cdots,z_{K}]$.

Note that the $Q$ allowed in ((7.0.10)) are more general than the $Q$ allowed in ((7.0.8)).

We now show that

• (7.0.12)

$\mathcal{M}_{\nu}^{\#}=\mathcal{M}_{\nu}^{\#\#}$.

Once we know this, we will have computed generators for each $\mathcal{M}_{\nu}^{\#}$; as we explained earlier, this allows us to compute generators for the module $\mathcal{M}(L)$ introduced in the statement of Algorithm 22. Thus, our task is reduced to proving ((7.0.12)).

Trivially, $\mathcal{M}_{\nu}^{\#\#}\subset\mathcal{M}_{\nu}^{\#}$. Our task is to show that $\mathcal{M}_{\nu}^{\#}\subset\mathcal{M}_{\nu}^{\#\#}$.

Let $\vec{P}=\left(P_{1},\cdots,P_{J}\right)$ belong to $\mathcal{M}_{\nu}^{\#}$, and let $\left(x^{o},y^{o},z^{o}\right)\in V$, where $x^{o}=\left(x_{1}^{o},\cdots,x_{n_{\nu}}^{o}\right)$, $y^{o}=\left(y_{1}^{o},\cdots,y_{n-n_{\nu}}^{o}\right)$, $z^{o}=\left(z_{1}^{o},\cdots,z_{K}^{o}\right)$.

Thus, each $P_{j}$ is a polynomial in the variables $x_{1},\cdots,x_{n_{\nu}}$, $y_{1},\cdots,y_{n-n_{\nu}}$ (not involving $z_{1},\cdots,z_{K}$), and we have

[TABLE]

for any $Q^{o}\in\mathbb{R}\left[x_{1},\cdots,x_{n_{\nu}},y_{1},\cdots,y_{n-n_{\nu}}\right]$.

Let $Q\in\mathbb{R}[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n-n_{\nu}},z_{1},\cdots,z_{K}]$ be given. Define

[TABLE]

Thus

[TABLE]

hence

[TABLE]

However, since $L_{\nu}^{\#}$ (see (7.0.7)) involves no derivatives in the $z_{1},\cdots,z_{K}$, we have

[TABLE]

Therefore, $L_{\nu}^{\#}\left(Q\cdot\vec{P}\right)|_{\left(x^{o},y^{o},z^{o}\right)}=0$.

Since $\left(x^{o},y^{o},z^{o}\right)\in V$ and $Q\in\mathbb{R}\left[x_{1},\cdots,x_{n_{\nu}},y_{1},\cdots,y_{n-n_{\nu}},z_{1},\cdots,z_{K}\right]$ are arbitrary, we conclude from ((7.0.10)) that $P\in\mathcal{M}_{\nu}^{\#\#}$.

Thus $\mathcal{M}_{\nu}^{\#}\subset\mathcal{M}_{\nu}^{\#\#}$, proving ((7.0.12)) and completing the explanation of Algorithm 22. ∎