A Sparse Version of Reznick’s Positivstellensatz

Abstract

If f is a positive definite form, Reznick’s Positivstellensatz states that there exists [Formula: see text] such that [Formula: see text] is a sum of squares of polynomials. Assuming that f can be written as a sum of forms [Formula: see text], where each fl depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick’s Positivstellensatz. Namely, there exists [Formula: see text] such that [Formula: see text], where σl is a sum of squares of polynomials, Hl is a uniform polynomial denominator, and both polynomials [Formula: see text] involve the same variables as fl for each [Formula: see text]. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick’s certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly noncompact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate from Putinar and Vasilescu. Funding: V. Magron was supported by the Fondation Mathématique Jacques Hadamard Programme Gaspard Monge for Optimization (Exact Polynomial Optimization with Innovative Certifed Schemes project). This work has benefited from the Tremplin European Research Council Starting [Grant ANR-18-ERC2-0004-01] (Tremplin-Certified Optimization for Cyber-Physical Systems project), the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Actions [Grant 813211] (Polynomial Optimization, Efficiency through Moments and Algebra) as well as from the Artificial Intelligence Interdisciplinary Institute Artificial and Natural Intelligence Toulouse Institute funding, through the French “Investing for the Future Programme d’Investissements d’Avenir3” program [Grant n°ANR-19-PI3A-0004].