On the polynomial-like behaviour of certain algebraic functions

Feffermann, Charles; Narasimhan, Raghavan

Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 1091-1179.

Étant donné des entiers $D>0,n>1,0<r<n$ et une constante $C>0$, on considère l’espace des $r$-uples $\overrightarrow{P}=(P_1...P_r)$ de polynômes réels à $n$ variables, de degré $\le D$, à coefficients $\le C$ en valeur absolue, et satisfaisant à $\det {\left(\frac{\partial P_i}{\partial x_i}\left(0\right)\right)}_{1\le i,j\le r}=1$. On étudie la famille $\left\{f\right|V\}$ des fonctions algébriques, où $f$ est un polynôme et $V=\left\{\right|x|\le \delta ,\overrightarrow{P}(x)=0\},\delta >0$ 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 $\overrightarrow{P}$. Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à $\overrightarrow{P}$, 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.

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

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

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