On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz

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 Łojasiewicz 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 Łojasiewicz inequality when the defining inequalities $\mathbf{g}$ satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent $L$ is $1$ and we relate the Łojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf{g}$ to the set of singular systems.

Final version, accepted in Journal of Algebra (2024)

References in corpus (1)

• Degree bounds for Putinar's Positivstellensatz on the hypercube

Cited by in corpus (2)

• Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems
• An Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets