On the polynomial-like behaviour of certain algebraic functions

Feffermann, Charles, and Narasimhan, Raghavan. "On the polynomial-like behaviour of certain algebraic functions." Annales de l'institut Fourier 44.4 (1994): 1091-1179. <http://eudml.org/doc/75091>.

@article{Feffermann1994,
abstract = {Given integers $D&gt;0,\, n&gt;1,\, 0&lt; r&lt; n$ and a constant $C&gt;0$, consider the space of $r$-tuples $\vec\{P\}=(P_1\ldots P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\{\rm det\}\left(\{\partial P_i\over \partial x_i\}(0)\right)_\{1\le i, j\le r\}=1$. We study the family $\lbrace f\vert V\rbrace $ of algebraic functions, where $f$ is a polynomial, and $V=\lbrace \vert x\vert \le \delta , \vec\{P\}(x)=0\rbrace ,\; \delta &gt;0$ 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.},
author = {Feffermann, Charles, Narasimhan, Raghavan},
journal = {Annales de l'institut Fourier},
keywords = {real polynomials; ideals of polynomials; semi-algebraic sets; Bernstein inequality},
language = {eng},
number = {4},
pages = {1091-1179},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the polynomial-like behaviour of certain algebraic functions},
url = {http://eudml.org/doc/75091},
volume = {44},
year = {1994},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 1091
EP - 1179
AB - Given integers $D&gt;0,\, n&gt;1,\, 0&lt; r&lt; n$ and a constant $C&gt;0$, consider the space of $r$-tuples $\vec{P}=(P_1\ldots P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying ${\rm det}\left({\partial P_i\over \partial x_i}(0)\right)_{1\le i, j\le r}=1$. We study the family $\lbrace f\vert V\rbrace $ of algebraic functions, where $f$ is a polynomial, and $V=\lbrace \vert x\vert \le \delta , \vec{P}(x)=0\rbrace ,\; \delta &gt;0$ 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.
LA - eng
KW - real polynomials; ideals of polynomials; semi-algebraic sets; Bernstein inequality
UR - http://eudml.org/doc/75091
ER -