On {\L}ojasiewicz Inequalities and the Effective Putinar's Positivstellensatz

TL;DR

This paper provides new bounds and a simplified proof for the effective Putinar's Positivstellensatz, linking the degree bounds to Łojasiewicz inequalities and analyzing their behavior under certain regularity conditions.

Contribution

The paper introduces improved degree bounds for positive polynomial representations on semi-algebraic sets, utilizing Łojasiewicz inequalities and analyzing their properties under constraint qualification.

Findings

New polynomial bounds involving Łojasiewicz parameters.

Simplified proof of the effective Putinar's Positivstellensatz.

Analysis of Łojasiewicz inequality when constraints satisfy CQ.

Abstract

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a {\L}ojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $\epsilon^{-1}$ with an exponent depending only on $L$. We analyse in details the {\L}ojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the {\L}ojasiewicz exponent $L$ is $1$ and we relate the {\L}ojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems.