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Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming

In this article, we propose a geometric programming method in order to compute lower bounds for real polynomials. We provide new sufficient conditions for polynomials to be nonnegative as well as to have a sum of binomial squares representation. These criteria rely on the coefficients and the support of a polynomial and generalize all previous ones by Lasserre, Ghasemi, Marshall, Fidalgo and Kovacec to polynomials with arbitrary simplex Newton polytopes. This generalization yields a geometric programming approach for computing lower bounds for polynomials that significantly extends the geometric programming method proposed by Ghasemi and Marshall. Furthermore, it shows that geometric programming is strongly related to nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the authors.