Extreme-Fixed Points of the Vandermonde Polynomial when Optimized over p-Normed Surfaces Defined by Univariate Polynomials

Abstract

The concept of the extreme-fixed point is coined from the facts of zeros of polynomials, extreme/optimal points on optimization of a given function and fixed points of a given continuous function. In this article, we establish the close relations between zeros, extreme, fixed points, and also what we define as extreme-fixed points. We illustrate this result with the Vandermonde polynomial (or determinant) when optimized over a given p-norm surface expressed by univariate polynomial(s). It is further, established that indeed the coordinates of the extreme-fixed points on such a surface like a p-sphere are given as roots of some classical orthogonal polynomials.