Elementary recursive degree bounds for Positivstellensatz and Hilbert?s 17-th problem

HENRI LOMBARDI; DANIEL PERRUCCI; MARIE-FRANCOISE ROY

Quito

Conferencia; XXII Coloquio Latinoamericano de Álgebra; 2017

Coloquio Latinoamericano de Álgebra

Hilbert?s 17th problem is to express a non-negative polynomial as a sum ofsquares of rational functions. The original proof by Artin is non-constructive and gives noinformation on degree bounds. A more general problem is to give an identity which certifiesthe unrealizability of a system of polynomial equations and inequalities. The existence ofsuch an identity is guaranteed by the Positivstellensatz, and Hilbert 17th problem as well asReal Nullstellensatz follow easily from such identity. In this talk, we present a constructiveproof which provides elementary recursive bounds for the Positivstellensatz, Hilbert?s 17thproblem, and the Real Nullstellensatz, namely a tower of five levels of exponentials.