Elementary recursive degree bounds for the Positivstellensatz, Hilbert ?s 17th problem, and the Real Nullstellensatz

HENRI LOMBARDI; DANIEL PERRUCCI; MARIE-FRANCOISE ROY

Luminy

Conferencia; Ordered Algebraic Structures and Related Topics; 2015

CIRM

Hilbert?s 17th problem is to express a non-negative polynomial as a sum of squares of rational functions. Artin?s original proof is non-constructive and gives no information on degree bounds. A more general problem is to give an identity which certifies the unrealizabilityof a system of polynomial equations and inequalities. The existence of such an identity is guaranteed by the Positivstellensatz, and Hilbert?s 17th problem as well as the Real Nullstellensatz follow easily from such identity. In this talk, we explain a new constructive proof which provides elementary recursive bounds for the Positivstellensatz, Hilbert?s 17th problem, and the Real Nullstellensatz, namely a tower of five levels of exponentials.