# ANNALES DE L'INSTITUT FOURIER

On the polynomial-like behaviour of certain algebraic functions
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1091-1179.

Given integers $D>0,\phantom{\rule{0.166667em}{0ex}}n>1,\phantom{\rule{0.166667em}{0ex}}0 and a constant $C>0$, consider the space of $r$-tuples $\stackrel{\to }{P}=\left({P}_{1}...{P}_{r}\right)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\mathrm{det}{\left(\frac{\partial {P}_{i}}{\partial {x}_{i}}\left(0\right)\right)}_{1\le i,j\le r}=1$. We study the family $\left\{f|V\right\}$ of algebraic functions, where $f$ is a polynomial, and $V=\left\{|x|\le \delta ,\stackrel{\to }{P}\left(x\right)=0\right\},\phantom{\rule{0.277778em}{0ex}}\delta >0$ being a constant depending only on $n,\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{0.166667em}{0ex}}C$. The main result is a quantitative extension theorem for these functions which is uniform in $\stackrel{\to }{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\stackrel{\to }{P}$.

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

Étant donné des entiers $D>0,n>1,0 et une constante $C>0$, on considère l’espace des $r$-uples $\stackrel{\to }{P}=\left({P}_{1}...{P}_{r}\right)$ de polynômes réels à $n$ variables, de degré $\le D$, à coefficients $\le C$ en valeur absolue, et satisfaisant à $\mathrm{det}{\left(\frac{\partial {P}_{i}}{\partial {x}_{i}}\left(0\right)\right)}_{1\le i,\phantom{\rule{0.166667em}{0ex}}j\le r}=1$. On étudie la famille $\left\{f|V\right\}$ des fonctions algébriques, où $f$ est un polynôme et $V=\left\{|x|\le \delta ,\stackrel{\to }{P}\left(x\right)=0\right\},\phantom{\rule{0.277778em}{0ex}}\delta >0$ ne dépendant que de $n,\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{0.166667em}{0ex}}C\phantom{\rule{0.166667em}{0ex}}$. Le résultat principal est un théorème quantitatif d’extension de ces fonctions qui est uniforme par rapport à $\stackrel{\to }{P}$. Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à $\stackrel{\to }{P}$, 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.

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author = {Feffermann, Charles and Narasimhan, Raghavan},
title = {On the polynomial-like behaviour of certain algebraic functions},
journal = {Annales de l'Institut Fourier},
pages = {1091--1179},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {44},
number = {4},
year = {1994},
doi = {10.5802/aif.1428},
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Feffermann, Charles; Narasimhan, Raghavan. On the polynomial-like behaviour of certain algebraic functions. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1091-1179. doi : 10.5802/aif.1428. https://aif.centre-mersenne.org/articles/10.5802/aif.1428/

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