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Sharp Degree Bounds for Putinar Certificates in Terms of the Positivity Margin via Analytic Sums of Squares

2026-10-18 23:10:30
Yuxiang Zhao, Ting Gan
arXiv_CL
   
Abstract

We establish sharp degree bounds for Putinar sum-of-squares (SOS) certificates in terms of the positivity margin $\delta$. For any fixed Archimedean quadratic module with prescribed generators and any fixed bound on the degree of the objective, we prove a uniform certificate-degree upper bound of $O(\delta^{-1/2}\log^2(2/\delta))$ for normalized polynomial objectives. We show that this bound is sharp by constructing a matching lower-bound example in three variables, thereby establishing that the dependence on $\delta$ is optimal. These matching bounds also yield sharp worst-case convergence rates for the Lasserre hierarchy.

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URL

https://arxiv.org/abs/2610.26976

PDF

https://arxiv.org/pdf/2610.26976.pdf

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