On Exact Polya and Putinar's Representations

TL;DR

This paper introduces a hybrid numeric-symbolic algorithm for computing exact sums of squares decompositions of non-negative polynomials, leveraging SDP solvers and perturbation techniques, with applications to Polya and Putinar's representations.

Contribution

It presents a novel hybrid algorithm that computes exact rational SOS decompositions for polynomials in the interior of the SOS cone, with polynomial bit complexity in degree and exponential in variables.

Findings

Algorithm successfully computes exact SOS decompositions.

Compared favorably with existing algebraic methods.

Provides complexity estimates for the algorithm.

Abstract

We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.