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In (Davis and Papp, 2022), the authors introduced the concept of dualcertificates of (weighted) sum-of-squares polynomials, which are vectors fromthe dual cone of weighted sums of squares (WSOS) polynomials that can beinterpreted as nonnegativity certificates. This initial theoretical work showedthat for every polynomial in the interior of a WSOS cone, there exists arational dual certificate proving that the polynomial is WSOS. In this article,we analyze the complexity of rational dual certificates of WSOS polynomials bybounding the bit sizes of integer dual certificates as a function of parameterssuch as the degree and the number of variables of the polynomials, or theirdistance from the boundary of the cone. After providing a general bound, weexplore several special cases, such as univariate polynomials nonnegative overthe real line or a bounded interval, represented in different commonly usedbases. We also provide an algorithm which runs in rational arithmetic andcomputes a rational certificate with boundable bit size for a WSOS lower boundof the input polynomial.

Rational Dual Certificates for Weighted Sums-of-Squares Polynomials with Boundable Bit Size