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

Ce livre est une introduction a la theorie de la complexite algebrique basee sur un panorama des methodes algorithmiques en algebre lineaire exacte. Il donne en particulier les principaux algorithmes pour le calcul du polynome caracteristique. Ce livre se remarque par l'etendue des sujets traites tout en restant tres lisible.