On the Generation of Positivstellensatz Witnesses in Degenerate Cases

David Monniaux (1) , Pierre Corbineau (1)

1 VERIMAG - IMAG - VERIMAG

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.

Mots clés: semidefinite programming; Positivstellensatz; polynomials; witness; Coq; LLL

Domaines: Logique [math.LO]; Géométrie algébrique [math.AG]; Recherche opérationnelle [math.OC]; Calcul formel [cs.SC]

David Monniaux, Pierre Corbineau. On the Generation of Positivstellensatz Witnesses in Degenerate Cases. Interactive Theorem Proving, Aug 2011, Nijmegen, Netherlands. pp.249-264. ⟨hal-00594761⟩