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

Year 2022, Volume: 10 Issue: 2, 355 - 367, 31.10.2022

Asaph Keikara Muhumuza , Rebecca Nalule Muhumuza , Santosh Kumar

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.

Keywords

Vandermonde Determinant, Orthogonal Polynomials, Extreme Points, Fixed Points, $p$-Norm.

Supporting Institution

None

Project Number

NIL

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