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Preprints, Working Papers, ... Year : 2022

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 semialgebraic 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 ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g = (g 1 ,. .. , g r) defining S. They are polynomial in c and ε −1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalties g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.
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Dates and versions

hal-03899373 , version 1 (14-12-2022)

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Lorenzo Baldi, Bernard Mourrain, Adam Parusinski. On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz. 2022. ⟨hal-03899373⟩
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