An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem

Henri Lombardi (1) , Daniel Perrucci (2) , Marie-Françoise Roy (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Departamento de Matemática [Buenos Aires]
3 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domaines: Géométrie algébrique [math.AG]

Henri Lombardi, Daniel Perrucci, Marie-Françoise Roy. An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem. Memoirs of the American Mathematical Society, 2020, 263 (1277), ⟨10.1090/memo/1277⟩. ⟨hal-00980282⟩