An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem

We prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely we express a nonnegative polynomial as a sum of squares of rational functions, and we obtain as degree estimates for the numerators and denominators the following tower of five exponentials 2(22d4k) where d is the degree and k is the number of variables of the input polynomial. Our method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely we give an algebraic certificate of the emptyness of the realization of a system of sign conditions and we obtain as degree bounds for this certificate a tower of five exponentials, namely 2(2)((2max {2, d}4k +) (s2k) (max {2, d}16k bit(d))) where d is a bound on the degrees, s is the number of polynomials and k is the number of variables of the input polynomials.