On exact Reznick, Hilbert-Artin and Putinar's representations. (November 2021)

• Record Type: Journal Article
• Authors: Magron, Victor; Safey El Din, Mohab
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 221
• Page End: 250
• Publication Date: 2021-11
• Subjects: Real algebraic geometry -- Semidefinite programming -- Sums of squares decomposition -- Reznick's representation -- Putinar's representation -- Hybrid numeric-symbolic algorithm; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.03.005
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and singly exponential in the number of variables. Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.