Effective {\L}ojasiewicz gradient inequality and finite determinacy of non-isolated Nash function singularities

TL;DR

This paper establishes bounds on the Łojasiewicz gradient inequality exponent for Nash functions on semialgebraic sets, linking it to polynomial degrees, and explores finite determinacy of non-isolated Nash singularities.

Contribution

It provides a new estimation method for the Łojasiewicz exponent based on polynomial degrees and applies this to analyze finite determinacy of Nash singularities.

Findings

Derived bounds for the Łojasiewicz gradient inequality exponent.

Linked the exponent estimation to the degree of a defining polynomial.

Provided criteria for finite determinacy of non-isolated Nash singularities.

Abstract

Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the {\L}ojasiewicz gradient inequality $|\nabla f(x)|\ge C|f(x)|^\varrho$ for $x\in X$, $|f(x)|<\varepsilon$ for some constants $C,\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities

Equations

$$|F(x)|\ge C|f(x){|}^{\eta }in\; a\; neighbourhood\; ofa\in {\mathbb{R}}^{n}for\; some\; constantC>0.$$

$${\mathcal{L}}_a(F,f)\le D^{M^{c\ell }},$$

$$|\nabla f(x)|\ge C|f(x){|}^{\varrho }in\; a\; neighbourhood\; ofa\in {\mathbb{R}}^{n}for\; some\; constantC>0$$

$${\varrho }_0(f)\le 1-\frac{1}{(d-1{)}^n+1},$$

$${\varrho }_0(f)\le 1-\frac{1}{d(3d-3{)}^{n-1}},providedd\ge 2.$$

$${\varrho }_a(f)\le 1-\frac{1}{2(2d-1{)}^{3n+1}}.$$

$${\varrho }_a(f)\le 1-\frac{1}{d(3d-2{)}^n+1},providedd\ge 2.$$

$$|\nabla f(x)|\ge C\mathrm{dist}(x,f^{-1}(0){)}^{2(2d-1{)}^{3n+1}-1}in\; a\; neighbourhood\; ofa.$$

$$|\nabla f(x)|\ge C\mathrm{dist}(x,f^{-1}(0){)}^{d(3d-2{)}^n}in\; a\; neighbourhood\; ofa.$$

$$|\nabla f(x)|\ge C\mathrm{dist}(x,Z{)}^{k-1}in\; a\; neighbourhood\; of\; the\; origin.$$

$$P(x,f(x))=0forx\in U,$$

$$d=\mathrm{deg}P.$$

$$\mathcal{R}(n,d)=\max \{2d(2d-1),d(3d-2{)}^n\}+1.$$

$$\frac{\partial P}{\partial y}(x,f(x))\ne 0forx\in U.$$

$$|\nabla f(x)|\ge C|f(x){|}^{\varrho }for|x-a|<\epsilon ,|f(x)|<\epsilon .$$

$$\mathcal{S}(n,d)=2(2d-1{)}^{3n+1}.$$

$$|\nabla f(x)|\ge C|f(x){|}^{\varrho }forx\in X$$

$$|\nabla (\phi \circ f)(x)|\ge (1-\varrho )Cforx\in X\setminus V.$$

$$U_{X,f}=\left\{x\in \mathrm{Int}X:\frac{1}{C(1-\varrho )}|f(x){|}^{1-\varrho }<\mathrm{dist}(x,{\mathbb{R}}^n\setminus X)\right\}.$$

$$H(x)=-\mathrm{sign}f(x)\frac{\nabla f(x)}{|\nabla f(x)|}forx\in U_{X,f}\setminus V.$$

$$\mathrm{dist}(\gamma (0),V)\le \mathrm{length}\gamma \le \frac{1}{(1-\varrho )C}|f(\gamma (0)){|}^{1-\varrho }.$$

$$(\phi \circ f\circ \gamma {)}^{\prime }=⟨\nabla (\phi \circ f)\circ \gamma ,{\gamma }^{\prime }⟩=-|\nabla (\phi \circ f)\circ \gamma |,$$

$$\begin{array}{rl}\phi (f(\gamma (0)))-\phi (f(\gamma (s_1)))& =-s_1(\phi \circ f\circ \gamma {)}^{\prime }(t)=s_1|\nabla (\phi \circ f)\circ \gamma (t)|\\ & \ge (1-\varrho )C\mathrm{length}{\gamma }_{s_1}\end{array}$$

$$\mathrm{length}\gamma \le \frac{1}{(1-\varrho )C}(|f(\gamma (0)){|}^{1-\varrho }-\alpha )\le \frac{1}{(1-\varrho )C}|f(\gamma (0)){|}^{1-\varrho },$$

$$|f(x)|\ge {\left(C(1-\varrho )\right)}^{1/(1-\varrho )}\mathrm{dist}(x,V{)}^{1/(1-\varrho )},x\in U_{X,f},$$

$$|\nabla f(x)|\ge {\left(C(1-\varrho )\right)}^{\varrho /(1-\varrho )}\mathrm{dist}(x,V{)}^{\varrho /(1-\varrho )},x\in U_{X,f}.$$

$$|f(x)|\ge C\mathrm{dist}(x,V{)}^{2(2d-1{)}^{3n+1}},x\in X,$$

$$|\nabla f(x)|\ge C\mathrm{dist}(x,V{)}^{2(2d-1{)}^{3n+1}-1},x\in X.$$

$$|f(x)|\ge C\mathrm{dist}(x,V{)}^{d(3d-2{)}^n+1},x\in X,$$

$$|\nabla f(x)|\ge C\mathrm{dist}(x,V{)}^{d(3d-2{)}^n},x\in X.$$

Full text

1. Introduction

Łojasiewicz inequalities are important tools in various branches of mathematics: differential equations, singularity theory and optimization (for more detailed references, see for example [16], [18], [19], [22] and [34]). Quantitative aspects, like estimates (or exact computation), of these exponents are subject of intensive study in real and complex algebraic geometry (see for instance [18], [19], [20] and [33]). Our main goal is to give, in terms of the Łojasiewicz inequality, an effective sufficient condition for Nash function germs of non-isolated singularity at zero to be isotopical (Theorem 1.3). The main tool in the proof is an effective estimation of the exponent in the Łojasiewicz gradient inequality (Theorems 2.1 and 2.2).

Determinacy of jets of functions with isolated singularity at zero was investigated by many authors, including N. H. Kuiper [14], T. C. Kuo [15], J. Bochnak and S. Łojasiewicz [2] for real functions and S. H. Chang and Y. C. Lu [5], B. Teissier [40] and J. Bochnak and W. Kucharz [1] for complex functions. Similar investigations were also carried out for functions in a neighbourhood of infinity by P. Cassou-Noguès and H. H. Vui [4] (see also [35], [37]). The case of real jets with non-isolated singularities was studied among others by V. Grandjean [11] and X. Xu [41], and for complex functions by D. Siersma [36] and R. Pellikaan [30]. In the case of nondegenerate analytic functions $f$, $g$, a condition for topological triviality of deformations $f+tg$, $t\in[0,1]$ in terms of Newton polyhedra was obtained by J. Damon and T. Gaffney [8], and for blow analytic triviality by T. Fukui and E.  Yoshinaga [9]. Some algebraic conditions for finite determinacy of a smooth function jet were obtained by L. Kushner [21].

1.1. Łojasiewicz gradient inequality

Let $U\subset\mathbf{\mathbb{R}}^{n}$ be an open set and let $a\in U$. Let $f,F:U\to\mathbf{\mathbb{R}}$ be continuous semialgebraic functions such that $a\in F^{-1}(0)\subset f^{-1}(0)\subset U$. Then the following Łojasiewicz inequality holds:

[TABLE]

The lower bound of the exponents $\eta$ in (1.1) is called the Łojasiewicz exponent of the pair $(F,f)$ at $a$ and is denoted by ${\mathcal{L}}_{a}(F,f)$. It is known that ${\mathcal{L}}_{a}(F,f)$ is a rational number (see [3]) and the inequality (1.1) holds actually with $\eta={\mathcal{L}}_{a}(F,f)$ on some neighbourhood of the point $a$ for some positive constant $C$ (see for instance [39]). An asymptotic estimate for ${\mathcal{L}}_{a}(F,f)$ was obtained by Solernó [38]:

[TABLE]

where $D$ is a bound for the degrees of the polynomials involved in a description of $F$, $f$ and $U$; $M$ is the number of variables in these formulas; $\ell$ is the maximum number of alternating blocs of quantifiers in these formulas; and $c$ is an unspecified universal constant.

In this paper, we consider the case when $F$ is equal to the gradient $\nabla f:=\left(\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}\right):U\to\mathbf{\mathbb{R}}^{n}$ of a Nash function $f$ in $x=(x_{1},\ldots,x_{n})$. Recall that semialgebraic and analytic functions are called Nash functions.

Our main goal is to obtain an effective estimate for the exponent $\varrho\in[0,1)$ in the following Łojasiewicz gradient inequality (see [23] or [24], cf. [40]):

[TABLE]

for an arbitrary Nash function $f:U\to\mathbf{\mathbb{R}}$, where $f(a)=0$, in terms of the degree of a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ describing the graph of $f$. We denote by $|\nabla f(x)|$ the Euclidean norm of $\nabla f(x)$, i.e. $|\nabla f(x)|^{2}=\left(\frac{\partial f}{\partial x_{1}}(x)\right)^{2}+\cdots+\left(\frac{\partial f}{\partial x_{n}}(x)\right)^{2}$.

The smallest exponent $\varrho$ in (Ł), denoted by $\varrho_{a}(f)$, is called the Łojasiewicz exponent in the gradient inequality at $a$. It is known that (Ł) holds with $\varrho=\varrho_{a}(f)$.

In the case of a polynomial function $f:\mathbf{\mathbb{R}}^{n}\to\mathbf{\mathbb{R}}$ of degree $d>0$ such that [math] is an isolated point of $f^{-1}(0)$, J. Gwoździewicz [12] (cf. [13]) proved that

[TABLE]

and in the general case of an arbitrary polynomial $f$, D. D’Acunto and K. Kurdyka [6] (cf. [7], [10] and [31]) showed that

[TABLE]

If $f$ is a rational function of the form $f={p}/{q}$, where $p,\,q\in\mathbf{\mathbb{R}}[x]$, $p(0)=0$ and $q(0)\neq 0$, then $\varrho_{0}(f)=\varrho_{0}(p)$, so (G2) and (DK) hold with $d=\deg p$.

The aim of this paper is to show generalizations of the above estimates for Nash functions (see Theorems 2.1 and 2.2 in Section 2). More precisely, let $U\subset\mathbf{\mathbb{R}}^{n}$ be a neighbourhood of $a\in\mathbf{\mathbb{R}}^{n}$ and let $f:U\to\mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solernó and D’Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality (Ł) in terms of the degree $d$ of a nonzero polynomial $P$ such that $P(x,f(x))=0$, $x\in U$. Namely, in Theorem 2.2 we obtain

[TABLE]

If additionally $n\geq 2$ and $\frac{\partial P}{\partial y}(x,f(x))\neq 0$ for $x\in U$, then in Theorem 2.1 we obtain

[TABLE]

The above estimates are comparable with the Solernó estimate (S), but our estimates are explicit.

As a corollary, we obtain the following inequality (see Corollary 3.6):

[TABLE]

If additionally $n\geq 2$ and $\frac{\partial P}{\partial y}(x,f(x))\neq 0$ for $x\in U$, then

[TABLE]

The inequalities (1.2), (1.3) are essential points in the effective estimate of the degree of sufficiency of non-isolated Nash function singularities given in the next section. The proof of these inequalities is based on Theorem 2.2 and estimates of the length of trajectories of the vector field $\nabla f$ in $U\setminus f^{-1}(0)$ (see Theorem 3.4).

1.2. Sufficiency of non-isolated Nash function singularities

Let ${\mathscr{C}}^{k}_{a}(n)$ denote the set of ${\mathscr{C}}^{k}$ real functions defined in neighbourhoods of $a\in\mathbf{\mathbb{R}}^{n}$.

By a $k$-jet at $a\in\mathbf{\mathbb{R}}^{n}$ in the class ${\mathscr{C}}^{\ell}$ we mean a family of functions $w\subset{\mathscr{C}}^{\ell}_{a}(n)$, called ${\mathscr{C}}^{\ell}$-realizations of this jet, possessing the same Taylor polynomial of degree $k$ at $a$. We also say that $f$ determines a $k$-jet at $a$ in ${\mathscr{C}}^{\ell}$ if $f$ is a ${\mathscr{C}}^{\ell}$-realization of this jet. For a function $f\in{\mathscr{C}}^{k}_{a}(n)$, we denote by $j^{k}f(a)$ the $k$-jet at $a$ (in ${\mathscr{C}}^{k}$) determined by $f$.

Let $Z\subset\mathbf{\mathbb{R}}^{n}$ be a set such that $0\in Z$ and let $k\in\mathbf{\mathbb{Z}}$, $k>0$. By a $k$-$Z$-jet in the class ${\mathscr{C}}^{k}$, or briefly a $k$-$Z$-jet, we mean an equivalence class $w\subset{\mathscr{C}}^{k}_{0}(n)$ of the following equivalence relation: $f\sim g$ iff for some neighbourhood $U\subset\mathbf{\mathbb{R}}^{n}$ of the origin, $j^{k}f(a)=j^{k}g(a)$ for $a\in Z\cap U$ (cf. [27], [41]). The functions $f\in w$ are called ${\mathscr{C}}^{k}$-$Z$-realizations of the jet $w$ and we write $w=j^{k}_{Z}f$. The set of all jets $j^{k}_{Z}f$ is denoted by $J^{k}_{Z}(n)$.

The $k$-$Z$-jet $w\in J^{k}_{Z}(n)$ is said to be ${\mathscr{C}}^{r}$-$Z$-sufficient (resp. $Z$-$v$-sufficient) in the class ${\mathscr{C}}^{k}$ if for every of its ${\mathscr{C}}^{k}$-$Z$-realizations $f$ and $g$ there exist sufficiently small neighbourhoods $U_{1},\,U_{2}\subset\mathbf{\mathbb{R}}^{n}$ of [math], and a ${\mathscr{C}}^{r}$ diffeomorphism $\varphi:U_{1}\to U_{2}$, such that $f\circ\varphi=g$ in $U_{1}$ (resp. there exists a homeomorphism $\varphi:[f^{-1}(0)\cup Z]\cap U_{1}\to[g^{-1}(0)\cup Z]\cap U_{2}$ with $\varphi(0)=0$ and $\varphi(Z\cap U_{1})=Z\cap U_{2}$).

The classical and significant result on sufficiency of jets is the following:

Theorem 1.1** (Kuiper, Kuo, Bochnak-Łojasiewicz).**

Let $w$ be a $k$-jet at $0\in\mathbf{\mathbb{R}}^{n}$ and let $f$ be its ${\mathscr{C}}^{k}$-realization. If $f(0)=0$ then the following conditions are equivalent:

• (a)

$w$* is ${\mathscr{C}}^{0}$-sufficient in ${\mathscr{C}}^{k}$,*

• (b)

$w$* is $v$-sufficient in ${\mathscr{C}}^{k}$,*

• (c)

$|\nabla f(x)|\geq C|x|^{k-1}$* in a neighbourhood of the origin for some $C>0$.*

The implication (c)$\Rightarrow$(a) was proved by N. H. Kuiper [14] and T. C. Kuo [15], (b)$\Rightarrow$(c) by J. Bochnak and S. Łojasiewicz [2], and (a)$\Rightarrow$(b) is obvious (cf. [29]).

Let us recall the notions of isotopy and topological triviality. Let $\Omega\subset\mathbf{\mathbb{R}}^{n}$ be a neighbourhood of $0\in\mathbf{\mathbb{R}}^{n}$ and let $Z\subset\mathbf{\mathbb{R}}^{n}$ with $0\in Z$.

A continuous mapping $H\colon\Omega\times[0,1]\to\mathbf{\mathbb{R}}^{n}$ is called an isotopy near $Z$ at zero if:

(a) $H_{0}(x)=x$ for $x\in\Omega$ and $H_{t}(x)=x$ for $t\in[0,1]$ and $x\in\Omega\cap Z$,

(b) for any $t$ the mapping $H_{t}:\Omega\to\mathbf{\mathbb{R}}^{n}$ is a homeomorphism onto $H_{t}(\Omega)$,

where $H_{t}(x)=H(x,t)$ for $x\in\Omega$, $t\in[0,1]$.

Functions $f:\Omega_{1}\to\mathbf{\mathbb{R}}$, $g:\Omega_{2}\to\mathbf{\mathbb{R}}$, where $\Omega_{1},\Omega_{2}\subset\mathbf{\mathbb{R}}^{n}$ are neighbourhoods of $0\in\mathbf{\mathbb{R}}^{n}$, are called isotopical near $Z$ at zero if there exists an isotopy near $Z$ at zero, $H:\Omega\times[0,1]\to\mathbf{\mathbb{R}}^{n}$, with $\Omega\subset\Omega_{1}\cap\Omega_{2}$, such that $f(H_{1}(x))=g(x)$, $x\in\Omega$.

A deformation $f+tg$ is called topologically trivial near $Z$ along $[0,1]$ if there exists an isotopy near $Z$ at zero, $H:\Omega\times[0,1]\to\mathbf{\mathbb{R}}^{n}$, with $\Omega\subset\Omega_{1}\cap\Omega_{2}$, such that $f(H(t,x))+tg(H(t,x))$ does not depend on $t$.

Theorem 1.1 concerns the case of an isolated singularity of $f$ at [math], i.e. [math] is an isolated zero of $\nabla f$. In the case of a non-isolated singularity of $f$ at [math], from [27, Theorems 1.3 and 1.4] (cf. [41]) we have the following criterion for sufficiency of jets.

Theorem 1.2**.**

Let $f\in{\mathscr{C}}_{0}^{k}(n)$ be a ${\mathscr{C}}^{k}$-$Z$-realization of a $k$-$Z$-jet $w\in J^{k}_{Z}(n)$, where $k>1$ and $Z=f^{-1}(0)$, $0\in Z$, and suppose $(\nabla f)^{-1}(0)\subset Z$. Then the following conditions are equivalent:

(a)* The $k$-$Z$-jet $w$ is ${\mathscr{C}}^{0}$-$Z$-sufficient in ${\mathscr{C}}^{k}$.*

(b)* For any ${\mathscr{C}}^{k}$-$Z$-realizations $f_{1},\,f_{2}$ of $w$, the deformation $f_{1}+t(f_{2}-f_{1})$, $t\in\mathbf{\mathbb{R}}$, is topologically trivial along $[0,1]$.*

(c)* Any two ${\mathscr{C}}^{k}$-$Z$-realizations of $w$ are isotopical at zero.*

(d)* The $k$-$Z$-jet $w$ is $Z$-$v$-sufficient in ${\mathscr{C}}^{k}$.*

(e)* There exists a positive constant $C$ such that*

[TABLE]

Let $f:U\to\mathbf{\mathbb{R}}$ be a Nash function, where $U\subset\mathbf{\mathbb{R}}^{n}$ is a neighbourhood of the origin, let $Z=f^{-1}(0)$, and suppose $0\in Z$.

The main result of this paper is the following corollary from Theorem 1.2 and inequality (1.2).

Theorem 1.3**.**

Let $k=2(2d-1)^{3n+1}$, where $d=\deg_{0}f$, and let $w\in J^{k}_{Z}(n)$ be the $k$-$Z$-jet for which $f$ is a ${\mathscr{C}}^{k}$-$Z$-realization. Then:

(a)* The $k$-$Z$-jet $w$ is ${\mathscr{C}}^{0}$-$Z$-sufficient in ${\mathscr{C}}^{k}$.*

(b)* For any ${\mathscr{C}}^{k}$-$Z$-realizations $f_{1},\,f_{2}$ of $w$, the deformation $f_{1}+t(f_{2}-f_{1})$, $t\in\mathbf{\mathbb{R}}$, is topologically trivial along $[0,1]$.*

(c)* Any two ${\mathscr{C}}^{k}$-$Z$-realizations of $w$ are isotopical at zero.*

(d)* The $k$-$Z$-jet $w$ is $Z$-$v$-sufficient in ${\mathscr{C}}^{k}$.*

Under additional assumption on $f$, from Theorem 1.2 and inequality (1.3), we obtain

Theorem 1.4**.**

Assume that there exists a nonzero polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\neq 0$ for $x\in U$. Then the assertion of Theorem 1.3 holds with $k=d(3d-2)^{n}+1$, where $d=\deg P$.

Remark 1.5**.**

If $f$ is a polynomial of degree $d>1$ or a rational function $f={p}/{q}$, where $p(0)=0$, $q(0)\neq 0$ and $d=\deg p$, then from Theorem 1.2 and by (DK), the assertion of Theorem 1.3 holds with $k=d(3d-3)^{n-1}$. If additionally the origin is an isolated zero of $f$, then by (G2) the assertion of Theorem 1.3 holds with $k=(d-1)^{n}+1$.

2. Łojasiewicz gradient inequality

Let $f:U\to\mathbf{\mathbb{R}}$, where $U\subset\mathbf{\mathbb{R}}^{n}$ is a connected neighbourhood of $a\in\mathbf{\mathbb{R}}^{n}$, be a Nash function. Let $P\in\mathbf{\mathbb{R}}[x,y]$ be the unique irreducible real polynomial such that

[TABLE]

and let

[TABLE]

We will call this number $d$ the degree of the Nash function $f$ at $a$ and denote it by $\deg_{a}f$. Obviously $d=\deg_{a}f>0$ is uniquely determined. For $d=1$, the function $f$ is linear and (Ł) holds with $\varrho=0$, so we will assume that $d>1$. We will also assume that $\nabla f(a)=0$, because in the opposite case (Ł) holds with $\varrho=0$.

Put

[TABLE]

The main result of this section is the following theorem.

Theorem 2.1**.**

Let $f:U\to\mathbf{\mathbb{R}}$ be a nonzero Nash function such that $f(a)=0$ and $\nabla f(a)=0$. Assume that for the unique polynomial $P$ satisfying (2.1) we have

[TABLE]

Then $\varrho_{a}(f)\leq 1-\frac{1}{\mathcal{R}(n,d)}$. Moreover, for $\varrho=1-\frac{1}{\mathcal{R}(n,d)}$ and some constants $C,\varepsilon>0$,

[TABLE]

Without the assumption (2.2), we have a somewhat weaker estimation of the exponent $\varrho_{a}(f)$ than in Theorem 2.1. Namely, let

[TABLE]

Theorem 2.2**.**

Let $f:U\to\mathbf{\mathbb{R}}$ be a nonzero Nash function such that $f(a)=0$ and $\nabla f(a)=0$ and let $P$ be the unique polynomial satisfying (2.1). Then $\varrho_{a}(f)\leq 1-\frac{1}{\mathcal{S}(n,d)}$. Moreover, (2.3) holds actually with $\varrho=1-\frac{1}{\mathcal{S}(n,d)}$.

Theorems 2.1 and 2.2 are generalizations for Nash functions of the above mentioned results by J. Gwoździewicz and D. D’Acunto and K. Kurdyka in the polynomial function case. They are also comparable with Solernó’s estimate (S), but our estimates are explicit. In the case of Nash functions with isolated singularity at zero, a similar result was obtained in [17].

We give the proofs of Theorems 2.1 and 2.2 in Section 5.

3. Łojasiewicz inequality

Let $X\subset\mathbf{\mathbb{R}}^{n}$ be a compact semialgebraic set and let $f:X\to\mathbf{\mathbb{R}}$ be a Nash function. Then $f$ is defined in a neighbourhood of $X$. So, there exists a compact semialgebraic set $Y\subset\mathbf{\mathbb{R}}^{n}$ such that $X\subset\operatorname{Int}Y$ and $f$ is defined on $Y$.

The degree of $f$ is defined to be $\sup\{\deg_{a}f:a\in X\}$ and is denoted by $\deg_{X}f$. In fact, $\deg_{X}f=\max\{\deg_{a}f:a\in X\}$. Moreover, one can assume that $Y$ was chosen in such a manner that $\deg_{X}f=\deg_{Y}f$.

Let $\operatorname{dist}(x,V)$ denote the distance of a point $x\in\mathbf{\mathbb{R}}^{n}$ to a set $V\subset\mathbf{\mathbb{R}}^{n}$ in the Euclidean norm (with $\operatorname{dist}(x,V)=1$ if $V=\emptyset$).

3.1. Global gradient £ojasiewicz inequality

Theorems 2.1 and 2.2 have a local character. From these theorems we obtain a global Łojasiewicz gradient inequality.

Corollary 3.1**.**

Let $d=\deg_{X}f$. If $(\nabla f)^{-1}(0)\subset f^{-1}(0)$ then for some positive constant $C$,

[TABLE]

with $\varrho=1-\frac{1}{\mathcal{S}(n,d)}$. If additionally there exists a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\neq 0$ for $x\in X$ and $d_{1}=\deg P$, then (3.1) holds with $\varrho=1-\frac{1}{\mathcal{R}(n,d_{1})}$.

Denote by $\varrho_{X}(f)$ the smallest exponent $\varrho$ for which (3.1) holds. We call it the Łojasiewicz exponent in the gradient inequality on $X$. It is known that the inequality (3.1) holds with $\varrho=\varrho_{X}(f)$. So, from Corollary 3.1 we obtain

Corollary 3.2**.**

$\varrho_{X}(f)\leq 1-\frac{1}{S(n,d)}$.

3.2. Length of trajectory

Let $f:X\to\mathbf{\mathbb{R}}$ be a nonzero Nash function such that $(\nabla f)^{-1}(0)\subset f^{-1}(0)$, let $\varrho\in(0,1)$ and $C>0$ be such that the global inequality (3.1) in Corollary 3.1 holds in $X$, and let $V=f^{-1}(0)$. Then $\nabla f(x)\neq 0$ for $x\in X\setminus V$.

Let $\varphi(t)=|t|^{1-\varrho}$ for $t\in\mathbf{\mathbb{R}}$. By the same argument as in the proof of [18, Proposition 1] we obtain (cf. [16])

Proposition 3.3** (Kurdyka-Łojasiewicz inequality).**

Under the above notations,

[TABLE]

We will also assume that $\overline{\operatorname{Int}X\setminus V}=X$. Let

[TABLE]

Then $U_{X,f}\subset X$ is a neighbourhood of $(\operatorname{Int}X)\cap V$.

Take a global trajectory $\gamma:[0,s)\to U_{X,f}\setminus V$ of the vector field

[TABLE]

Then the function $f\circ\gamma$ is monotonic, so the limit $\lim_{t\to s}f\circ\gamma(t)$ exists.

Let $\operatorname{length}\gamma$ denote the length of $\gamma$. Since $|\gamma^{\prime}(t)|=1$, we have $\operatorname{length}\gamma=s$.

The following generalization of [18, Theorem 1] has a similar proof.

Theorem 3.4**.**

The limit $\lim_{t\to s}\gamma(t)$ exists and belongs to $V$. Moreover,

[TABLE]

Proof.

Let $s_{1}\in[0,s)$ and $\gamma_{s_{1}}=\gamma|_{[0,s_{1}]}$. Then $\operatorname{length}\gamma_{s_{1}}=s_{1}$. Since $\frac{\nabla f}{|\nabla f|}\,{\rm sign}\,f(x)=\frac{\nabla(\varphi\circ f)}{|\nabla(\varphi\circ f)|}$ for $x\in U\setminus V$, it follows that

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in $\mathbf{\mathbb{R}}^{n}$, and Proposition 3.3 gives

[TABLE]

for some $t\in[0,s_{1}]$. Then, letting $s_{1}\to s$, from the definition of $\varphi$ we have

[TABLE]

where $\alpha=\lim_{s_{1}\to s}|f(\gamma(s_{1}))|^{1-\varrho}\geq 0$.

Since $\gamma(0)\in U_{X,f}$, we see that $\operatorname{length}\gamma<\operatorname{dist}(\gamma(0),\mathbf{\mathbb{R}}^{n}\setminus X)$, so the limit $\lim_{t\to s}\gamma(t)$ certainly exists and belongs to $U_{X,f}$. Consequently, $\lim_{t\to s}\gamma(t)\in V$ and $\operatorname{length}\gamma\geq\operatorname{dist}(\gamma(0),V)$. This gives the assertion. ∎

From Theorem 3.4 we have

Corollary 3.5**.**

Under the assumptions and notations of Theorem 3.4,

[TABLE]

and

[TABLE]

Similarly to [18], we obtain a version of the above corollary in the complex case with the same formulation.

From Corollaries 3.1, 3.5 and Theorem 2.2, we immediately obtain

Corollary 3.6**.**

Let $d=\deg_{X}f$. Then there exists a positive constant $C$ such that

[TABLE]

and

[TABLE]

If additionally $n\geq 2$ and there exists a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\neq 0$ for $x\in X$, and $d=\deg P$, then

[TABLE]

*and *

[TABLE]

Proof.

Take a compact semialgebraic set $Y\subset\mathbf{\mathbb{R}}^{n}$ such that $X\subset\operatorname{Int}Y$ and $Y\subset\{x\in\mathbf{\mathbb{R}}^{n}:\operatorname{dist}(x,X)<\varepsilon\}$. If $\varepsilon$ is sufficiently small, then we can consider the function $f$ on $Y$. Then we may assume that $\deg_{Y}f=\deg_{X}f$ and $(\nabla f)^{-1}(0)\subset f^{-1}(0)$ after extending $f$ onto $Y$. So, the assertions of Theorem 3.4 and Corollary 3.5 hold with $\varrho=1-\frac{1}{S(n,d)}$ on the set $U_{Y,f}$. Hence the assertions hold for $x\in X\cap U_{Y,f}$. By the definition of $U_{Y,f}$, we see that $X\setminus U_{Y,f}$ is a compact set and $\min\{|x-y|:x\in V,\;y\in X\setminus U_{Y,f}\}>0$. So, diminishing $C$ if necessary, we obtain the first part of the assertion. The second part is proved analogously. ∎

3.3. Łojasiewicz exponent

Corollary 3.5 implies the known fact that the exponents $\alpha>0$ in the inequality

[TABLE]

for some positive constant $C$, are bounded below. The inequality (3.2) is called the Łojasiewicz inequality for $f$ on $X$ and the lower bound of the exponents $\alpha>0$ is the Łojasiewicz exponent of $f$ on $X$, denoted by ${\mathcal{L}}_{X}(f)$. It is known that (3.2) holds with $\alpha={\mathcal{L}}_{X}(f)$ and some positive constant $C$.

From Theorem 3.4 we obtain

Corollary 3.7**.**

${\mathcal{L}}_{X}(f)\leq\frac{1}{1-\varrho_{X}(f)}$.

Corollary 3.5 implies

Corollary 3.8**.**

If $d=\deg_{X}f$, then ${\mathcal{L}}_{X}(f)\leq 2(2d-1)^{3n+1}$.

For $n\geq 4$ the above estimate is sharper than the one given in [20] for continuous semialgebraic functions: ${\mathcal{L}}_{X}(f)\leq d(6d-3)^{n+r-1}$, where $r\leq\frac{n(n+1)}{2}$ is the degree of complexity of $f$, equal to the number of inequalities necessary to define the graph of $f$, and $d$ is the maximal degree of polynomials describing the graph of $f$. Consequently, this gives the estimate ${\mathcal{L}}_{X}(f)\leq d(6d-3)^{n+n(n+1)/2-1}$ in terms of the degree only. So, the estimate in Corollary 3.8 is more exact than the one above for $n\geq 4$.

4. Total degree of algebraic sets

Let $\mathbf{\mathbb{C}}[x]$ denote the ring of complex polynomials in $x=(x_{1},\ldots,x_{n})$.

Let $f=(f_{1},\ldots,f_{r}):\mathbf{\mathbb{C}}^{n}\to\mathbf{\mathbb{C}}^{r}$ be a polynomial mapping with $\deg f_{i}>0$ for $i=1,\ldots,r$. Let $V=f^{-1}(0)\subset\mathbf{\mathbb{C}}^{n}$.

The total degree of $V$ is the number

[TABLE]

where $V=V_{1}\cup\cdots\cup V_{s}$ is the decomposition into irreducible components (see [25]).

We have the following useful fact (see [25]).

Fact 4.1**.**

If $V,W\subset\mathbf{\mathbb{C}}^{n}$ are algebraic sets, then

[TABLE]

From Fact 4.1 and the definition of total degree of algebraic sets we have the following two facts (cf. [25]).

Fact 4.2**.**

$\delta(V)\leq\deg f_{1}\cdots\deg f_{r}$. In particular, for any irreducible component $V_{j}$ of $V$ we have

[TABLE]

Fact 4.3**.**

Let $L:\mathbf{\mathbb{C}}^{n}\to\mathbf{\mathbb{C}}^{k}$ be a linear mapping. Then

[TABLE]

We will need the following lemma (see [17, Lemma 3.20]).

Lemma 4.4**.**

Let $V_{j}$ be an irreducible component of the set $V$, and suppose $\dim V_{j}\geq 1$. Then for a generic linear mapping $L=(L_{1},\ldots,L_{n-1}):\mathbf{\mathbb{C}}^{r}\to\mathbf{\mathbb{C}}^{n-1}$ the set $V_{j}$ is an irreducible component of the set of common zeros of the system of equations

[TABLE]

In particular,

[TABLE]

Moreover, we can take $L_{1}(y_{1},\ldots,y_{r})=y_{1}$.

5. Proofs of Theorems 2.1 and 2.2

The idea of the proofs is similar to that in [17, proof of Theorem 1.2].

Without loss of generality, we may assume that $a=0$. Let $f:U\to\mathbf{\mathbb{R}}$ be a nonzero Nash function defined in an open neighbourhood $U\subset\mathbf{\mathbb{R}}^{n}$ of the origin such that $f(0)=0$ and $\nabla f(0)=0$. Let $P\in\mathbf{\mathbb{R}}[x,y]$ be the unique irreducible polynomial satisfying (2.1) and let $d=\deg P$.

Since the set of critical values of a differentiable semialgebraic function is finite, we have

Fact 5.1**.**

There exists $\varepsilon>0$ such that $f$ has no critical values in the interval $(-\varepsilon,\varepsilon)$ except [math].

Let $\varepsilon>0$ be as in Fact 5.1. Take $r>0$. Denote by $\Omega$ the closed ball

[TABLE]

and by $\partial\Omega$ the sphere $\{x\in\mathbf{\mathbb{R}}^{n}:|x|=r\}$. Suppose that $\Omega\subset U$. Define a semialgebraic set $\Gamma\subset\Omega$ by

[TABLE]

Then by the definition of $\Gamma$ we have

Fact 5.2**.**

Let $\varrho\in\mathbf{\mathbb{R}}$ and let $C>0$. If $|\nabla f(x)|\geq C|f(x)|^{\varrho}$ for $x\in\Gamma$ such that $|f(x)|<\varepsilon$, then $|\nabla f(x)|\geq C|f(x)|^{\varrho}$ for $x\in\Omega$, $|f(x)|<\varepsilon$.

Let $\varrho_{0}=\varrho_{0}(f)$. Then, decreasing $r$ if necessary, we can assume that

[TABLE]

Let us fix such an $r$.

Consider the case $n=1$. Denote by $\operatorname{ord}_{0}f$ the order of $f$ at zero. Then $f$ has an isolated zero and singularity at zero, $\operatorname{ord}_{0}f>0$ and the inequality (2.3) holds with

[TABLE]

Let the polynomial $P$ be of the form $P(x_{1},y)=p_{0}(x_{1})y^{d}+p_{1}(x_{1})y^{d-1}+\cdots+p_{d}(x_{1})$, where $p_{0},\ldots,p_{d}\in\mathbf{\mathbb{R}}[x_{1}]$. As $P$ is irreducible, $p_{d}\neq 0$ and $\operatorname{ord}_{0}p_{d}\leq d$. Since

[TABLE]

we have $\operatorname{ord}_{0}f\leq\operatorname{ord}_{0}p_{d}\leq d$. Together with (5.2) this gives (2.3) with $\varrho_{0}(f)=1-\frac{1}{d}$ and the assertions of Theorems 2.1 and 2.2 in the case $n=1$.

In the remainder of this article we will assume that $n>1$.

By (5.1) and the Curve Selection Lemma, there exists an analytic curve $\varphi:[0,1)\to\Omega$ for which $f(\varphi(0))=0$, $f(\varphi(\xi))\neq 0$ for $\xi\in(0,1)$ and for some constant $C_{1}>0$,

[TABLE]

(cf. [39]). By Fact 5.2 we may assume that $\varphi([0,1))\subset\Gamma$. Then we have two cases:

I. \varphi\big{(}(0,1)\big{)}\subset\operatorname{Int}\Omega,

II. \varphi\big{(}[0,1)\big{)}\subset\partial\Omega.

We will use the Lagrange multipliers theorem to describe the relation between the values $y=f(x)$ and $u=|\nabla f(x)|^{2}$ for $x\in\Gamma$, so we put

[TABLE]

[TABLE]

To fulfill the assumptions of the Lagrange theorem we will need

Lemma 5.3**.**

There exists $\varepsilon>0$ such that for every $x\in\partial\Omega$ and every $y\in\mathbf{\mathbb{R}}$ such that $0<|y|<\varepsilon$ and $y=f(x)$, the vectors \nabla\big{(}|x|^{2}-r^{2}\big{)} and $\nabla f(x)$ (that is, $2x$ and $\nabla f(x)$) are linearly independent.

Proof.

If $f|_{\partial\Omega}$ is a constant function then the assertion is obvious. Assume that $f$ is not constant on $\partial\Omega$. Then, by Fact 5.1, there exists $\varepsilon>0$ such that $\nabla f(x)\neq 0$ for $x\in\partial\Omega$, $0<|f(x)|<\varepsilon$.

Suppose to the contrary that for any $\varepsilon>0$ there exist $x\in\partial\Omega$ and $y_{\varepsilon}\in\mathbf{\mathbb{R}}$ with $0<|y_{\varepsilon}|<\varepsilon$ such that $y_{\varepsilon}=f(x)$ and $\nabla f(x)=\xi\cdot 2x$ for some $\xi\in\mathbf{\mathbb{R}}\setminus\{0\}$. Then by the Curve Selection Lemma there exist analytic curves $\gamma:[0,1)\to\partial\Omega$ with $\gamma((0,1))\subset\Omega\setminus f^{-1}(0)$ and $f(\gamma(0))=0$, and $\alpha:[0,1)\to\mathbf{\mathbb{R}}$, such that for $t\in(0,1)$,

[TABLE]

Then

[TABLE]

and consequently $f\circ\gamma$ is a constant function equal to [math]. This contradicts the choice of $\gamma$ and ends the proof. ∎

By the Lagrange multipliers theorem, Fact 5.1 and Lemma 5.3 we obtain

Fact 5.4**.**

Let $\varepsilon>0$ fulfill Fact 5.1 and Lemma 5.3. Take a point $x_{0}\in\Omega$ such that $0<|f(x_{0})|<\varepsilon$.

(a)* If $x_{0}\in\Gamma\cap\operatorname{Int}\Omega$ then $x_{0}$ is a lower critical point of the function $\Omega\ni x\mapsto|\nabla f(x)|^{2}\in\mathbf{\mathbb{R}}$ on the set $f^{-1}(f(x_{0}))\cap\Omega$. In particular, $\Gamma\cap\operatorname{Int}\Omega\subset\Gamma_{I}$.*

(b)* If $n\geq 3$, $x_{0}\in\Gamma\cap\partial\Omega$ then $x_{0}$ is a lower critical point of the function $\partial\Omega\ni x\mapsto|\nabla f(x)|^{2}\in\mathbf{\mathbb{R}}$ on the set $f^{-1}(f(x_{0}))\cap\partial\Omega$. In particular, $\Gamma\cap\partial\Omega\subset\Gamma_{II}$.*

Let $\mathbf{\mathbb{M}}=\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}^{n}$, and let $\mathbf{\mathbb{X}}\subset\mathbf{\mathbb{M}}$ be the Zariski closure of the set

[TABLE]

We will determine polynomials describing a certain algebraic set $\mathbf{\mathbb{Y}}\subset\mathbf{\mathbb{M}}$ containing $\mathbf{\mathbb{X}}$ as an irreducible component. Let $G\in\mathbf{\mathbb{C}}[x,y,u]$, where $u$ is a variable, be the polynomial defined by

[TABLE]

It is easy to observe that $G(x,f(x),|\nabla f(x)|^{2})=0$ for $x\in\Omega$. In particular, the polynomial $G$ vanishes on $\mathbf{\mathbb{X}}$.

Take systems of variables $t=(t_{1},\ldots,t_{n})$, $z=(z_{1},\ldots,z_{n})$, and let $G_{1},G_{2,i},G_{3,i}\in\mathbf{\mathbb{C}}[x,y,u,t,z]$ be defined by

[TABLE]

Let $\mathbf{\mathbb{Y}}\subset\mathbf{\mathbb{M}}$ be the closure of the constructible set

[TABLE]

Obviously $\mathbf{\mathbb{X}}\subset\mathbf{\mathbb{Y}}$, and locally $\mathbf{\mathbb{Y}}^{0}$ is the graph of a complex Nash mapping (i.e., a holomorphic mapping with semialgebraic graph). Moreover, we have

Lemma 5.5**.**

The set $\mathbf{\mathbb{X}}$ is an irreducible component of $\,\mathbf{\mathbb{Y}}$. Moreover, $\mathbf{\mathbb{Y}}^{0}$ is a Zariski open and dense subset of $\mathbf{\mathbb{Y}}$, and any point $w=(x_{0},y_{0},u_{0},t_{0},z_{0})\in\mathbf{\mathbb{Y}}^{0}$ has a neighbourhood $B\subset\mathbf{\mathbb{M}}$ such that $\mathbf{\mathbb{Y}}\cap B=\mathbf{\mathbb{Y}}^{0}\cap B$ and

[TABLE]

for some holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset\mathbf{\mathbb{C}}^{n}$ is a neighbourhood of $x_{0}$, and $h(x)=\left(\frac{\partial g}{\partial x_{1}}(x)\right)^{2}+\cdots+\left(\frac{\partial g}{\partial x_{n}}(x)\right)^{2}$.

Proof.

Since $P$ is an irreducible polynomial, $\frac{\partial P}{\partial y}$ does not vanish on $\mathbf{\mathbb{X}}$. So, by the Implicit Function Theorem, $\{w=(x,y,u,t,z)\in\mathbf{\mathbb{X}}:\frac{\partial P}{\partial y}(x,y)\neq 0\}$ is an open and dense subset of $\mathbf{\mathbb{X}}$, and moreover it is a smooth and connected submanifold of $\mathbf{\mathbb{Y}}^{0}$. Consequently, $\mathbf{\mathbb{X}}$ is an irreducible component of $\mathbf{\mathbb{Y}}$. The “moreover” part of the assertion follows immediately from the Implicit Function Theorem. ∎

Define $G_{0},G_{4,i,j},G_{4,i,j,k}\in\mathbf{\mathbb{C}}[x,y,u,t,z]$ by

[TABLE]

where the polynomials $G_{4,i,j,k}$ are defined if $n\geq 3$. Put

[TABLE]

where the sets $\mathbf{\mathbb{X}}_{II}$, $\mathbf{\mathbb{L}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$ are defined for $n\geq 3$.

Obviously $\mathbf{\mathbb{X}}_{I}\subset\mathbf{\mathbb{Y}}_{I}$ and $\mathbf{\mathbb{X}}_{II}\subset\mathbf{\mathbb{Y}}_{II}$. Moreover, any irreducible component of $\mathbf{\mathbb{X}}_{I}$ is an irreducible component of $\mathbf{\mathbb{Y}}_{I}$. The same holds for $\mathbf{\mathbb{X}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$. Additionally, by the Lagrange multipliers theorem and Facts 5.1, 5.4 we immediately obtain

Fact 5.6**.**

(a)* Let*

[TABLE]

If $\varphi((0,1))\subset\operatorname{Int}\Omega$ then $\mathscr{F}\subset{\mathbf{\mathcal{Z}}}_{I}\subset A_{I}\subset\mathbf{\mathbb{X}}_{I}\subset\mathbf{\mathbb{Y}}_{I}$ and there exists an irreducible component $\mathbf{\mathbb{X}}_{I,*}$ of $\overline{A_{I}}$ which contains $\mathscr{F}$ and is an irreducible component of $\mathbf{\mathbb{X}}_{I}$.

(b)* Let*

[TABLE]

If $\varphi((0,1))\subset\partial\Omega$ then $\mathscr{F}\subset{\mathbf{\mathcal{Z}}}_{II}\subset A_{II}\subset\mathbf{\mathbb{X}}_{II}\subset\mathbf{\mathbb{Y}}_{II}$ and there exists an irreducible component $\mathbf{\mathbb{X}}_{II,*}$ of $\overline{A_{II}}$ which contains $\mathscr{F}$ and is an irreducible component of $\mathbf{\mathbb{X}}_{II}$.

Proof.

From Fact 5.4(a) we have $\mathscr{F}\subset\{(x,y,u,t,z)\in\mathbf{\mathbb{X}}:x\in\Gamma_{I}\}\subset A_{I}$. Since all the polynomials $G_{4,i,j}$ vanish on $\mathbf{\mathbb{X}}_{I}$, the vectors $t,\,z$ are linearly dependent provided $(x,y,u,t,z)\in\mathbf{\mathbb{X}}_{I}$ for some $x,y,u$. So $\mathbf{\mathbb{X}}_{I}=\mathcal{X}_{I}\cup A_{I}$, where

[TABLE]

Obviously, the set $\mathcal{X}_{I}$ is contained in the hyperplane $H$ defined by $t=0$, and by Fact 5.1 we have $\mathscr{F}\setminus H\neq\emptyset$, so $\overline{A_{I}}$ has an irreducible component containing $\mathscr{F}$ which is an irreducible component of $\mathbf{\mathbb{X}}_{I}$. This gives assertion (a).

Analogously, from Fact 5.4(b) we obtain $\mathscr{F}\subset A_{II}$. Moreover, the vectors $x,t,z$ are linearly dependent provided $(x,y,u,t,z)\in\mathbf{\mathbb{X}}_{II}$ for some $y,u$, so $\mathbf{\mathbb{X}}_{II}=\mathcal{X}_{II}\cup A_{II}$, where

[TABLE]

Obviously, $\mathcal{X}_{II}$ is contained in the set $W$ defined by $G_{4,i,j}(x,t)=0$, $1\leq i<j\leq n$. By Lemma 5.3 we have $\mathscr{F}\setminus W\neq\emptyset$, so as above, the set $A_{II}$ has an irreducible component satisfying (b). ∎

From Fact 5.6 and Lemmas 4.4 and 5.5 and the definition of $\mathbf{\mathbb{Y}}$ we have

Fact 5.7**.**

$\delta(\mathbf{\mathbb{X}}_{I,*})\leq\delta(\mathbf{\mathbb{Y}}_{I})\leq 2(2d-1)^{3n+1}$* and $\delta(\mathbf{\mathbb{X}}_{II,*})\leq\delta(\mathbf{\mathbb{Y}}_{II})\leq 2(2d-1)^{3n+1}$.*

The proofs of Theorems 2.1 and 2.2 consist in showing that the projections of the sets $\mathbf{\mathbb{X}}_{I,*}$ and $\mathbf{\mathbb{X}}_{II,*}$ onto the space of $(y,u)\in\mathbf{\mathbb{C}}^{2}$ are proper algebraic subsets of $\mathbf{\mathbb{C}}^{2}$, since we have

Lemma 5.8**.**

If $Q\in\mathbf{\mathbb{C}}[y,u]$ is a nonzero polynomial of degree $D$ such that

[TABLE]

where $\varphi$ is the curve fulfilling (5.3), then

(a)* $\varrho_{0}(f)\leq 1-\frac{1}{D}$ if $D$ is even,*

(b)* $\varrho_{0}(f)\leq 1-\frac{1}{D+1}$ if $D$ is odd.*

Proof.

Let $\operatorname{ord}_{0}(f\circ\varphi)=M$ and $\operatorname{ord}_{0}|\nabla f\circ\varphi|^{2}=K$. Then $M,K>0$ and

[TABLE]

i.e., $|f\circ\varphi|^{\frac{K}{2M}}\sim|\nabla f\circ\varphi|$ near zero111That is, there are $C_{1},C_{2}>0$ such that $C_{1}|f\circ\varphi|^{\frac{K}{2M}}\leq|\nabla f\circ\varphi|\leq C_{2}|f\circ\varphi|^{\frac{K}{2M}}$ near zero., so by (5.3) we have

[TABLE]

Then, by definitions of $M$ and $K$ there exists a pair of different monomials $\alpha u^{N}y^{S}$ and $\beta u^{N_{1}}y^{S_{1}}$ of the polynomial $Q$ such that

[TABLE]

and

[TABLE]

Hence $N-N_{1}\neq 0$, $S_{1}-S\neq 0$, and

[TABLE]

Since $M>0$, we have $\operatorname{ord}_{0}\nabla f\circ\varphi\leq M-1$, and so $K\leq 2M-2$, and $\frac{K}{2M}<1$. On the other hand, $|S_{1}-S|,|N-N_{1}|\in\{1,\ldots,D\}$, so by (5.5), $\varrho_{0}(f)$ is estimated from above by the maximal possible rational number less than $1$ with numerator from the set $\{1,\ldots,D\}$ and denominator from $\{2,4,\ldots,2D\}$. Consequently, we obtain the assertion. ∎

5.1. Proof of Theorem 2.1 in case I when $\varphi((0,1))\subset\operatorname{Int}\Omega$

By the assumption (2.2), in the definition of $\mathbf{\mathbb{Y}}$ one can take the polynomials

[TABLE]

instead of $G_{3,i}$, $1\leq i\leq n$; also in the definitions of $\mathbf{\mathbb{X}}_{I}$ and $\mathbf{\mathbb{Y}}_{I}$ one can take

[TABLE]

instead of $G_{4,i,j}$, $1\leq i<j\leq n$.

From the above and Fact 5.6 we obtain the following fact.

Fact 5.9**.**

For $x\in\Gamma_{I}$ and $v=(x,y,u)=(x,f(x),|\nabla f(x)|^{2})$ we have

[TABLE]

Let $\mathbf{Y}_{I,0}\subset\bf{M}$, where ${\bf M}=\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$, be an algebraic set defined by the system of equations (5.7)–(5.9), and let

[TABLE]

We have the following fact (cf. [17, Fact 2.11]).

Fact 5.10**.**

The mapping

[TABLE]

is a bijection.

Proof.

Taking any $(x,y,u,t,z)\in\mathbf{\mathbb{Y}}^{0}_{I}$ (respectively $(x,y,u)\in\mathbf{Y}^{0}_{I}$), by the Implicit Function Theorem there are a neighbourhood $\Delta\subset\mathbf{\mathbb{C}}^{n}$ of $x$, a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$ and neighbourhoods $U_{1}\subset\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}^{n}$ and $U_{2}\subset\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$ of $(y,u,t,z)$ and $(y,u)$ respectively such that

[TABLE]

where $h(\zeta)=\left(\frac{\partial g}{\partial x_{1}}(\zeta)\right)^{2}+\cdots+\left(\frac{\partial g}{\partial x_{n}}(\zeta)\right)^{2}$, and

[TABLE]

In particular, $g(x)=y$, $u=h(x)$, $t=\nabla g(x)$ and $z=\nabla h(x)$. Thus, we obtain the assertion. ∎

Let $\mathbf{L}_{I}\subset{\bf M}\times\mathbf{\mathbb{C}}$ be the Zariski closure of the set

[TABLE]

From Fact 5.6(a) we obtain

Fact 5.11**.**

There exists an irreducible component $\mathbf{L}_{I,*}$ of $\mathbf{L}_{I}$ which contains a Zariski open and dense subset $\mathcal{U}$ such that for any $(x,y,u,\lambda)\in\mathcal{U}$ there exist $t,z\in\mathbf{\mathbb{C}}^{n}$ such that $(x,y,u,t,z)\in\mathbf{\mathbb{X}}_{I,*}$ and in particular $z=\lambda t$.

Proof.

The set $\mathbf{L}_{I}$ is the projection of the union of some irreducible components of $\mathbf{\mathbb{L}}_{I}$ onto $(x,y,u,\lambda)\in{\bf M}\times\mathbf{\mathbb{C}}$. So by Fact 5.6(a) we obtain the assertion. ∎

Let

[TABLE]

let $\mathbf{L}_{I,*}$ be an irreducible component of $\mathbf{L}_{I}$ as in Fact 5.11 and let

[TABLE]

Lemma 5.12**.**

The set ${\mathbf{X}}_{I}$ is an irreducible component of the algebraic set $\mathbf{Y}_{I}$. Moreover, ${\mathbf{X}}_{I}$ contains a Zariski open and dense subset $\mathcal{U}_{I}$ such that $\mathcal{U}_{I}\subset\mathbf{Y}_{I}^{0}\cap\pi(\mathbf{L}_{I,*})$, and any point $(x_{0},y_{0},u_{0})\in\mathcal{U}_{I}$ has a neighbourhood $B\subset{\bf M}$ such that $\mathbf{Y}_{I}\cap B=\mathcal{U}_{I}\cap B$ and

[TABLE]

for some analytic set $V\subset\Delta$ with $x_{0}\in V$ and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset\mathbf{\mathbb{C}}^{n}$ is a neighbourhood of $x_{0}$.

Proof.

By Facts 5.6, 5.10 and 5.11 we have $\pi(\mathbf{L}_{I,0})\subset\mathbf{Y}_{I}$, so ${\mathbf{X}}_{I}\subset\mathbf{Y}_{I}$ and ${\mathbf{X}}_{I}$ is an algebraic subset of $\mathbf{Y}_{I}$. Since any irreducible component of $\mathbf{\mathbb{X}}_{I}$ is an irreducible component of $\mathbf{\mathbb{Y}}_{I}$, the same holds for $\pi(\mathbf{L}_{I})$ and $\mathbf{Y}_{I}$, because these sets are projections onto the space ${\bf M}$ of some collections of irreducible components of $\mathbf{\mathbb{X}}_{I}$ and $\mathbf{\mathbb{Y}}_{I}$, respectively. In particular, this holds for ${\mathbf{X}}_{I}$ and $\mathbf{Y}_{I}$. This gives the first part of the assertion. We prove the “moreover” part analogously to Fact 5.10. ∎

Let

[TABLE]

[TABLE]

We have the following lemma (cf. [17, Lemmas 2.12, 2.14]).

Lemma 5.13**.**

For generic $y_{0}\in\mathbf{\mathbb{C}}$, i.e., for any $y_{0}\in\mathbf{\mathbb{C}}$ off a finite set, the function $\pi_{u}$ is constant on each connected component of $(\pi_{y})^{-1}(y_{0})$.

Proof.

If $\dim{\mathbf{X}}_{I}=0$ or $\dim(\pi_{y})^{-1}(y)\leq 0$ for generic $y\in\mathbf{\mathbb{C}}$, then the assertion holds. Assume that $\dim{\mathbf{X}}_{I}>0$ and $\dim(\pi_{y})^{-1}(y)>0$ for generic $y\in\mathbf{\mathbb{C}}$. Then by Lemma 5.12, and under the notations of this lemma, we have $\overline{\pi_{y}(\mathcal{U}_{I})}=\overline{\pi_{y}({\mathbf{X}}_{I})}=\mathbf{\mathbb{C}}$ and $(\pi_{y})^{-1}(y)\cap\mathcal{U}_{I}\neq\emptyset$ for generic $y\in\mathbf{\mathbb{C}}$.

Take any $y_{0}\in\mathbf{\mathbb{C}}$ such that $(\pi_{y})^{-1}(y_{0})\cap\mathcal{U}_{I}\neq\emptyset$. Take any $x_{0}\in\mathbf{\mathbb{C}}^{n}$ and $u_{0}\in\mathbf{\mathbb{C}}$ such that $(x_{0},y_{0},u_{0})\in\mathcal{U}_{I}$. By Lemma 5.12 there exist a neighbourhood $B\subset{\bf M}$ of $(x_{0},y_{0},u_{0})$ and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset\mathbf{\mathbb{C}}^{n}$ is a neighbourhood of $x_{0}$, such that (5.10) holds for some analytic set $V\subset\Delta$.

Take any smooth curve $\gamma:[0,1]\to\Delta\cap V$ such that $g(\gamma(t))=y_{0}$ for $t\in[0,1]$. Let $h(x)=\left(\frac{\partial g}{\partial x_{1}}(x)\right)^{2}+\cdots+\left(\frac{\partial g}{\partial x_{n}}(x)\right)^{2}$ for $x\in\Delta$ and take a function $u:[0,1]\to\mathbf{\mathbb{C}}$ defined by

[TABLE]

Observe that the function $u$ is constant. Indeed, by definition of $\mathcal{U}_{I}$ we see that for any $x\in\Delta\cap V$ there exists $\lambda_{x}\in\mathbf{\mathbb{C}}$ such that

[TABLE]

So,

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in $\mathbf{\mathbb{C}}^{n}$. Since $g(\gamma(t))=y_{0}$ for $t\in[0,1]$, we have $\langle\nabla g(\gamma(t)),\overline{\gamma^{\prime}(t)}\rangle=0$, and consequently $u^{\prime}(t)=0$ for $t\in[0,1]$ and $u$ is constant. Summing up, the function $\pi_{u}$ is constant on each connected component of $(\pi_{y})^{-1}(y_{0})\cap\mathcal{U}_{I}$.

Since $\mathcal{U}_{I}$ is a Zariski open and dense subset of ${\mathbf{X}}_{I}$, any irreducible component of ${\mathbf{X}}_{I}\setminus\mathcal{U}_{I}$ has dimension smaller than the dimension of ${\mathbf{X}}_{I}$, and for generic $y\in\mathbf{\mathbb{C}}$ any irreducible component $A$ of the fibre $\pi_{y}^{-1}(y)$ has a dense subset of the form $A\cap\mathcal{U}_{I}$ (see [28, Chapter 3]). Then by the above we obtain the assertion. ∎

Since $\Gamma$ is an infinite set, it follows that $\dim\mathbf{L}_{I,0}\geq 1$, so by Fact 5.10, $\dim{\mathbf{L}_{I}}\geq 1$, and since $d=\deg P\geq 2$, Lemma 4.4 and the definition of $\mathbf{Y}_{I}$ yield $\delta({\mathbf{X}}_{I})\leq d(3d-2)^{n}$, where $\delta({\mathbf{X}}_{I})$ is the total degree of ${\mathbf{X}}_{I}$. So, from Lemma 5.13, the closure of the projection of ${\mathbf{X}}_{I}$, $W=\overline{\{(y,u)\in\mathbf{\mathbb{C}}^{2}:\exists_{x\in\mathbf{\mathbb{C}}^{n}}\;(x,y,u)\in{\mathbf{X}}_{I}\}}$, is a proper algebraic subset of $\mathbf{\mathbb{C}}^{2}$ and by Fact 4.3, $\delta(W)\leq\delta({\mathbf{X}}_{I})$. Then there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ such that

[TABLE]

and $Q(y,u)=0$ for $(x,y,u)\in{\mathbf{X}}_{I}$. In particular, $Q(f(\varphi(t)),|\nabla f(\varphi(t))|^{2})=0$ for $t\in[0,1)$. Since $D=d(3d-2)^{n}$ may be odd, by Lemma 5.8(b) we obtain the assertion of Theorem 2.1 in case I.

5.2. Proof of Theorem 2.1 in case II when \varphi\big{(}[0,1)\big{)}\subset\partial\Omega

For any $x\in\partial\Omega\setminus f^{-1}(0)$ sufficiently close to $f^{-1}(0)$ the tangent spaces to $\partial\Omega$ and $f^{-1}(f(x))$ are transversal, as shown in Lemma 5.3.

We will prove Theorem 2.1 in two dimensions and in the multidimensional case separately.

Proof of Theorem 2.1 in case II for $n=2$. Take a polynomial $G\in\mathbf{\mathbb{C}}[x,y,u]$, where $x=(x_{1},x_{2})$ and $y$, $u$ are single variables, defined by (5.4), i.e., $G(x,y,u)=\sum_{i=1}^{2}\left(\frac{\partial P}{\partial x_{i}}(x,y)\right)^{2}-\left(\frac{\partial P}{\partial y}(x,y)\right)^{2}\cdot u$. Let

[TABLE]

Then for any $x\in\Gamma\cap\partial\Omega$ we have $(x,f(x),|\nabla f(x)|^{2})\in\mathbf{Y}_{II}$. Consequently,

[TABLE]

In particular, $\dim\mathbf{Y}_{II}\geq 1$ and by Fact 4.2 we have $\delta(\mathbf{Y}_{II})\leq 2d(2d-1)$.

Since $P$ is an irreducible polynomial of positive degree with respect to $y$, for any $y\in\mathbf{\mathbb{C}}\setminus\{0\}$ sufficiently close to [math] the set $\{x\in\mathbf{\mathbb{C}}^{2}:P(x,y)=0,\,G_{0}(x)=0\}$ is finite, so the set $\{(x,u)\in\mathbf{\mathbb{C}}^{2}\times\mathbf{\mathbb{C}}:(x,y,u)\in\mathbf{Y}_{II}\}$ is also finite. Then the projection

[TABLE]

is contained in a proper algebraic subset of $\mathbf{\mathbb{C}}^{2}$. By Fact 4.3,

[TABLE]

Then there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ of degree $\deg Q\leq\delta(\overline{W})\leq\mathcal{R}(n,d)$ which vanishes on $W$. Since $2d(2d-1)$ is even, by Lemma 5.8(a) we obtain the assertion of Theorem 2.1 in case II for $n=2$. $\square$

Let us consider the case $n\geq 3$. Let $\varepsilon>0$ be as in Lemma 5.3.

By the assumption (2.2), in the definition of the set $\mathbf{\mathbb{Y}}$ one can take the polynomials $K_{3,i}$ of the form (5.6) instead of $G_{3,i}$; also, in the definitions of $\mathbf{\mathbb{X}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$, one can take the polynomials

[TABLE]

instead of $G_{4,i,j,k}$ for $1\leq i<j<k\leq n$, where $G$ is defined in (5.4). Then

[TABLE]

Let $\mathbf{Y}_{II,0}\subset{\bf M}$, where ${\bf M}=\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$, be the algebraic set defined by

[TABLE]

and let

[TABLE]

By an analogous argument to the proof of Fact 5.10 we obtain

Fact 5.14**.**

The mapping

[TABLE]

is a bijection.

Let $\mathbf{L}_{II}\subset{\bf M}\times\mathbf{\mathbb{C}}^{2}$ be the Zariski closure of the set

[TABLE]

By a similar argument to the proof of Fact 5.11, from Fact 5.6(b) we obtain

Fact 5.15**.**

There exists an irreducible component $\mathbf{L}_{II,*}$ of $\mathbf{L}_{II}$ which contains a Zariski open, dense subset $\mathcal{U}$ such that for any $(x,y,u,\lambda_{1},\lambda_{2})\in\mathcal{U}$ there exist $t,z\in\mathbf{\mathbb{C}}^{n}$ such that $(x,y,u,t,z)\in\mathbf{\mathbb{X}}_{II,*}$ and in particular $z=\lambda_{1}t+\lambda_{2}x$.

Let

[TABLE]

and let

[TABLE]

By an analogous argument to the proof of Lemma 5.12 we obtain

Lemma 5.16**.**

The set ${\mathbf{X}}_{II}$ is an irreducible component of the algebraic set $\mathbf{Y}_{II}$. Moreover, ${\mathbf{X}}_{II}$ contains a Zariski open and dense subset $\mathcal{U}_{II}$ such that $\mathcal{U}_{II}\subset\mathbf{Y}^{0}_{II}\cap\pi^{\prime}(\mathbf{L}_{II,*})$ and any point $(x_{0},y_{0},u_{0})\in\mathcal{U}_{II}$ has a neighbourhood $B\subset{\bf M}$ such that $\mathbf{Y}_{II}\cap B=\mathcal{U}_{II}\cap B$ and

[TABLE]

for some analytic set $V\subset\Delta$, where $x_{0}\in V$ and $G_{0}$ vanishes on $V$, and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset\mathbf{\mathbb{C}}^{n}$ is a neighbourhood of $x_{0}$.

Let

[TABLE]

We have the following lemma (cf. Lemma 5.13 and [17, Lemmas 2.12, 2.14]).

Lemma 5.17**.**

For generic $y_{0}\in\mathbf{\mathbb{C}}$ the function $\pi_{u}$ is constant on each connected component of $(\pi_{y})^{-1}(y_{0})$.

Proof.

As in the proof of Lemma 5.13, we may assume that $\dim{\mathbf{X}}_{II}>0$ and $\dim\,(\pi_{y})^{-1}(y)>0$ for generic $y\in\mathbf{\mathbb{C}}$. Then by Lemma 5.16, and under the notations of that lemma, $\overline{\pi_{y}(\mathcal{U}_{II})}=\overline{\pi_{y}({\mathbf{X}}_{II})}=\mathbf{\mathbb{C}}$ and $(\pi_{y})^{-1}(y)\cap\mathcal{U}_{II}\neq\emptyset$ for generic $y\in\mathbf{\mathbb{C}}$.

Take any $y_{0}\in\mathbf{\mathbb{C}}$ such that $(\pi_{y})^{-1}(y_{0})\cap\mathcal{U}_{II}\neq\emptyset$. Take any $x_{0}\in\mathbf{\mathbb{C}}^{n}$ and $u_{0}\in\mathbf{\mathbb{C}}$ such that $(x_{0},y_{0},u_{0})\in\mathcal{U}_{II}$. By Lemma 5.16 there exist a neighbourhood $B\subset\mathbf{\mathbb{C}}^{n}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$ of $(x_{0},y_{0},u_{0})$ and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset\mathbf{\mathbb{C}}^{n}$ is a neighbourhood of $x_{0}$, such that (5.11) holds for some analytic set $V\subset\Delta$ such that $G_{0}$ vanishes on $V$.

Take a smooth curve $\gamma=(\gamma_{1},\ldots,\gamma_{n}):[0,1]\to\Delta\cap V$ such that $g(\gamma(t))=y_{0}$. Then

[TABLE]

Let $h(x)=\left(\frac{\partial g}{\partial x_{1}}(x)\right)^{2}+\cdots+\left(\frac{\partial g}{\partial x_{n}}(x)\right)^{2}$, $x\in\Delta$. Take a function $u:[0,1]\to\mathbf{\mathbb{C}}$ defined by

[TABLE]

Observe that the function $u$ is constant. Indeed, by definition of $\mathcal{U}_{II}$, for any $x\in\Delta\cap V$ there exist $\lambda_{1,x},\lambda_{2,x}\in\mathbf{\mathbb{C}}$ such that

[TABLE]

So

[TABLE]

Since $g(\gamma(t))=y_{0}$, we have $\langle\nabla g(\gamma(t)),\overline{\gamma^{\prime}(t)}\rangle=0$ for $t\in[0,1]$. Moreover, by (5.12) we have $\langle\gamma(t),\overline{\gamma^{\prime}(t)}\rangle=0$ for $t\in[0,1]$. Consequently, $u^{\prime}(t)=0$ for $t\in[0,1]$ and $u$ is constant. Summing up, the function $\pi_{u}$ is constant on each connected component of $(\pi_{y})^{-1}(y_{0})\cap\mathcal{U}_{II}$. Since $\mathcal{U}_{II}$ is a dense subset of ${\mathbf{X}}_{II}$, we obtain the assertion. ∎

Since $\Gamma$ is an infinite set, we have $\dim\mathbf{L}_{II,0}\geq 1$, so by Fact 5.14, $\dim{\mathbf{L}_{II}}\geq 1$, and since $d=\deg P\geq 2$, Lemma 4.4 and the definition of $\mathbf{Y}_{II}$ yield $\delta({\mathbf{X}}_{II})\leq d(3d-2)^{n}$. So, from Lemma 5.17, the closure of the projection of ${\mathbf{X}}_{II}$,

[TABLE]

is a proper algebraic subset of $\mathbf{\mathbb{C}}^{2}$ and $\delta({W})\leq\delta({\mathbf{X}}_{II})$. Then there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ such that $\deg Q\leq 2(3d-2)^{n}\leq\mathcal{R}(n,d)-1$ and $Q(y,u)=0$ for $(x,y,u)\in{\mathbf{X}}_{II}$. Since $D=2(3d-2)^{n}$ is an even number, by Lemma 5.8(a) we obtain the assertion of Theorem 2.1 in case II.

5.3. Proof of Theorem 2.2

Analogously to the proof of Lemma 5.13, we prove that the set

[TABLE]

is a proper algebraic subset of $\mathbf{\mathbb{C}}^{2}$. Moreover, by Fact 5.7 we have $\delta(W)\leq\delta(\mathbf{\mathbb{Y}}_{I})\leq 2d(2d-1)$ if $n=1$ and $\delta(W)\leq\delta(\mathbf{\mathbb{Y}}_{I})\leq 2(2d-1)^{3n+1}$ for $n\geq 2$. Then by Lemma 5.8(a) we obtain the assertion of Theorem 2.2 in case I. An analogous argument gives the assertion in case II.