On continuous selections of polynomial functions

TL;DR

This paper studies continuous selections of polynomial functions, proving their finiteness, semi-algebraic nature, and analyzing their critical points and values, with implications for inequalities and error bounds.

Contribution

It establishes the finiteness and semi-algebraic property of continuous polynomial selections and analyzes their critical points and values under generic conditions.

Findings

Finitely many continuous selections exist for a given polynomial set.

Critical points of these selections are finite and have distinct critical values.

Some classical inequalities extend to all continuous selections of polynomial sets.

Abstract

A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that there are only finitely many continuous selections of it and each one is semi-algebraic. Then, we establish some generic properties regarding the critical points, defined by the Clarke subdifferential, of these continuous selections. In particular, given a set of finitely many polynomials with generic coefficients, we show that the critical points of all continuous selections of it are finite and the critical values are all different, and we also derive the coercivity of those continuous selections which are bounded from below. We point out that some existing results about {\L}ojasiewicz's inequality and error bounds for the maximum function of some finitely many polynomials are also valid for all the continuous selections of them.