Certificates of non-negativity on semialgebraic sets contained in cylinders

GABRIELA JERONIMO

Workshop; Workshop on Algebraic Real Geometry and Optimization (ARGO 2022); 2022

A certicate of non-negativity (resp. positivity) for a polynomial f in R[x] =R[x1,..., xn] on a semialgebraic set S in Rn is an algebraic identity that makesevident the fact that f(x) >= 0 (resp. f(x) > 0) for every x in S. These certificates go back to the classical Positivstellenstaz proved by Krivine in 1964, and they have been widely studied and applied since then.If S is the closed basic semialgebraic set defined by polynomials g1,...,gs in R[x], everypolynomial in the quadratic module M generated by g1,...,gs in R[x] is non-negative on S. Putinar's Positivstellensatz states that, if M is Archimedean (which implies that S is compact), every f positive on S lies in M. Furthermore, Nie and Schweighofer proved an upper bound for the degrees of the polynomials in a representation of f as an element of M.Putinar and Vasilescu gave a generalization of the previous result to non-compact sets, under certain assumptions on f, g1,....,gs. Recently, Escorcielo and Perrucci proved other certicates of non-negativity, extending Putinar's original result to certain non-compact situations.In this talk, we will present a new certicate of non-negativity for polynomials that are positive on a non-compact semialgebraic set included in a cylinder.Under certain assumptions on the positive polynomial f and the polynomialsg1,..., gs defining the set, we will show the existence of the proposed certificate of non-negativity of f and explain how we obtain an upper bound for the degrees of the polynomials appearing in the representation. This is joint work with Daniel Perrucci.