On exact Reznick, Hilbert-Artin and Putinar's representations

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