An elementary recursive bound for effective Positivstellensatz and   Hilbert 17-th problem
  We prove elementary recursive bounds in the degrees for Positivstellensatz and Hilbert 17-th problem, which is the expression of a nonnegative polynomial as a sum of squares of rational functions. We obtain a tower of five exponentials. A precise bound in terms of the number and degree of the polynomials and their number of variables is provided in the paper. 
Henri Lombardi, Daniel Perrucci, Marie-Fran\c{c}oise Roy
math.AG