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,n>1,0<r<n and a constant C>0, consider the space of r-tuples P =(P 1 ...P r ) of real polynomials in n variables of degree D, whose coefficients are C in absolute value, and satisfying det P i x i (0) 1i,jr =1. We study the family {f|V} of algebraic functions, where f is a polynomial, and V={|x|δ,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 P . This is used to prove Bernstein-type inequalities which are again uniform with respect 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<r<n et une constante C>0, on considère l’espace des r-uples P =(P 1 ...P r ) de polynômes réels à n variables, de degré D, à coefficients C en valeur absolue, et satisfaisant à det P i x i (0) 1i,jr =1. On étudie la famille {f|V} des fonctions algébriques, où f est un polynôme et V={|x|δ,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 P . Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à 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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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. http://archive.numdam.org/articles/10.5802/aif.1428/

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