On Exact Reznick, Hilbert-Artin and Putinar's Representations

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 singly exponential in the cardinality of the Newton polytope (or doubly 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.