On the polynomial-like behaviour of certain algebraic functions

Feffermann, Charles ; Narasimhan, Raghavan

Annales de l'Institut Fourier, Tome 44 (1994), p. 1091-1179 / Harvested from Numdam

Résumé

Étant donné des entiers $D>0, n>1, 0< r< n$ et une constante $C>0$, on considère l’espace des $r$-uples $\vec{P}=(P_1\ldots P_r)$ de polynômes réels à $n$ variables, de degré $\le D$, à coefficients $\le C$ en valeur absolue, et satisfaisant à ${\rm det}\left({\partial P_i\over \partial x_i}(0)\right)_{1\le i, \,j\le r}=1$. On étudie la famille $\lbrace f\vert V\rbrace $ des fonctions algébriques, où $f$ est un polynôme et $V=\lbrace \vert x\vert \le \delta , \vec{P}(x)=0\rbrace ,\; \delta >0$ ne dépendant que de $n,\, D,\, C\,$. Le résultat principal est un théorème quantitatif d’extension de ces fonctions qui est uniforme par rapport à $\vec{P}$. Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à $\vec{P}$, du type de celle de Bernstein.

La démonstration s’appuie sur des résultats quantitatifs concernant les idéaux de polynômes et sur la théorie des ensembles semi-algébriques.

Abstract

Given integers $D>0,\, n>1,\, 0< r< n$ and a constant $C>0$, consider the space of $r$-tuples $\vec{P}=(P_1\ldots P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying ${\rm det}\left({\partial P_i\over \partial x_i}(0)\right)_{1\le i, j\le r}=1$. We study the family $\lbrace f\vert V\rbrace $ of algebraic functions, where $f$ is a polynomial, and $V=\lbrace \vert x\vert \le \delta , \vec{P}(x)=0\rbrace ,\; \delta >0$ being a constant depending only on $n,\, D,\, C$. The main result is a quantitative extension theorem for these functions which is uniform in $\vec{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\vec{P}$.

The proof is based on some quantitative results on ideals of polynomials and on the theory of semi-algebraic sets.

Détail

Publié le : 1994-01-01

MR 95k:32011 | Zbl 0811.14046 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1428

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