On the Calculation of the Bernstein-Szegö Factor for Multivariate Polynomials

Abstract

Let IR^d be the Euclidean space with the usual norm |.|₂, ${\mathcal P}_n^d$ be the set of all polynomials over IR^d of degree n, and K ⊂ IR^d be a convex body. An algorithm for calculation of the Bernstein-Szegö factor:

$$ BS(K):= \sup_{{\bf x\in{\rm int}(K)}\atop {P\in {\cal P}_n^d, n\in{\rm I\!N}}} \Bigg\{{|\rm grad P(\bf x)|_2w(K)\sqrt{1-\alpha^2(K,\bf x)} \over n \sqrt{||P||^2_{C(K)} - P^2(\bf x)} }\Bigg\} $$

is considered, where w(K) is the width of K and α(K,x) is the generalized Minkowsky functional. It is known that $BS(K)\in [2,2\sqrt2]$. On the basis of computer experiments, we show that the existing in the literature hypothesis, that BS(K) = 2 for any convex body K ⊂ IR^d, fails to hold.