Nonnegative Morse polynomial functions and polynomial optimization

TL;DR

This paper investigates the representation of nonnegative Morse polynomial functions on compact semi-algebraic sets, introducing non-degenerate classes and applying findings to polynomial optimization problems.

Contribution

It introduces two classes of non-degenerate polynomials ensuring compactness and studies their role in representing nonnegative Morse polynomials for optimization.

Findings

Defined non-degenerate polynomial classes ensuring compact algebraic sets

Characterized the representation of nonnegative Morse polynomials on these sets

Applied results to polynomial optimization problems involving Morse functions

Abstract

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $\mathbb R^n$, and having only finitely many zeros in this set. Following C. Bivi\`{a}-Ausina, we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.