On a generalization of Pólya’s and Putinar-Vasilescu’s Positivstellensätze

In this paper we provide a generalization of two well-known positivstellensätze, namely the positivstellensatz from Pólya [Georg Pólya. Über positive darstellung von polynomen vierteljschr. In Naturforsch. Ges. Zürich, 73: 141-145, 1928] and the positivestellensatz from Putinar and Vasilescu [Mihai Putinar and Florian-Horia Vasilescu. Positive polynomials on semialgebraic sets. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328(7), 1999]. We show that if a homogeneous polynomial is strictly positive over the intersection of the non-negative orthant and a given basic semialgebraic cone, then there exists a "Pólya type" certificate for non-negativity.





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