Given integers and a constant , consider the space of -tuples of real polynomials in variables of degree , whose coefficients are in absolute value, and satisfying . We study the family of algebraic functions, where is a polynomial, and being a constant depending only on . The main result is a quantitative extension theorem for these functions which is uniform in . This is used to prove Bernstein-type inequalities which are again uniform with respect to .
The proof is based on some quantitative results on ideals of polynomials and on the theory of semi-algebraic sets.
Étant donné des entiers et une constante , on considère l’espace des -uples de polynômes réels à variables, de degré , à coefficients en valeur absolue, et satisfaisant à . On étudie la famille des fonctions algébriques, où est un polynôme et ne dépendant que de . Le résultat principal est un théorème quantitatif d’extension de ces fonctions qui est uniforme par rapport à . Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à , 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.
@article{AIF_1994__44_4_1091_0, 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}, zbl = {0811.14046}, mrnumber = {95k:32011}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1428/} }
TY - JOUR AU - Feffermann, Charles AU - Narasimhan, Raghavan TI - On the polynomial-like behaviour of certain algebraic functions JO - Annales de l'Institut Fourier PY - 1994 SP - 1091 EP - 1179 VL - 44 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1428/ DO - 10.5802/aif.1428 LA - en ID - AIF_1994__44_4_1091_0 ER -
%0 Journal Article %A Feffermann, Charles %A Narasimhan, Raghavan %T On the polynomial-like behaviour of certain algebraic functions %J Annales de l'Institut Fourier %D 1994 %P 1091-1179 %V 44 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1428/ %R 10.5802/aif.1428 %G en %F AIF_1994__44_4_1091_0
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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