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

Authors : Asaph Keikara MUHUMUZA, Rebecca NALULE MUHUMUZA, Santosh KUMAR

Pages : 355-367

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Publication Date : 2022-10-31

Article Type : Research Paper

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 insert ignore into journalissuearticles values(or determinant); when optimized over a given p-norm surface expressed by univariate polynomialinsert ignore into journalissuearticles values(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