On the polynomial-like behaviour of certain algebraic functions

Feffermann, Charles; Narasimhan, Raghavan

doi:10.5802/aif.1428

Feffermann, Charles ; Narasimhan, Raghavan

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

Abstract (VO)
Résumé

Given integers $D > 0, n > 1, 0 < r < n$ and a constant $C > 0$ , consider the space of $r$ -tuples $\vec{P} = (P_{1} ... P_{r})$ of real polynomials in $n$ variables of degree $\leq D$ , whose coefficients are $\leq C$ in absolute value, and satisfying $\det {(\frac{\partial P_{i}}{\partial x_{i}} (0))}_{1 \leq i, j \leq r} = 1$ . We study the family ${f | V}$ of algebraic functions, where $f$ is a polynomial, and $V = {| x | \leq δ, \vec{P} (x) = 0}, δ > 0$ 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.

Étant donné des entiers $D > 0, n > 1, 0 < r < n$ et une constante $C > 0$ , on considère l’espace des $r$ -uples $\vec{P} = (P_{1} ... P_{r})$ de polynômes réels à $n$ variables, de degré $\leq D$ , à coefficients $\leq C$ en valeur absolue, et satisfaisant à $\det {(\frac{\partial P_{i}}{\partial x_{i}} (0))}_{1 \leq i, j \leq r} = 1$ . On étudie la famille ${f | V}$ des fonctions algébriques, où $f$ est un polynôme et $V = {| x | \leq δ, \vec{P} (x) = 0}, δ > 0$ 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.

EuDML MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1428

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     title = {On the polynomial-like behaviour of certain algebraic functions},
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     pages = {1091--1179},
     year = {1994},
     publisher = {Association des Annales de l'Institut Fourier},
     volume = {44},
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Feffermann, Charles; Narasimhan, Raghavan. On the polynomial-like behaviour of certain algebraic functions. Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 1091-1179. doi: 10.5802/aif.1428

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