A Wandermonde-like Determinant

Abstract

We present a determinant which arises from try- ing to find the extreme points of the unit ball in the space of n-homogeneous polynomials defined on the Euclidean plane. The purpose of this note is to present a Wandermonde-like determi- nant which arises from trying to find the extreme points of the unit ball in the space of n-homogeneous polynomials on the Euclidean plane. In order to explain this, let us start with some background information on homogeneous polynomial (see (1) for more details). However, this not needed by the reader merely interested in the de- terminant announced in the title. We say that P is an n-homogeneous polynomial on a (real or com- plex) normed space X if there exists an n-linear form B on the prod- uct X n such that P(x) = B (x;:::;x) for every x in X. We denote by P ( n X) the space of all continuous n-homogeneous polynomials on X endowed with the natural norm kPk = supfjP(x)j : kxk = 1g and by Ls( n X) the space of continuous symmetric n-linear forms on X with the supremum norm. According to the polarization for- mula (1) for each P there exists a unique A in Ls( n X) such that