Using hyperelliptic curves to find posi - tive polynomials that are not sum of three squares in

This article deals with a quantitative aspect of Hilbert’s seventeenth problem : producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahé, a given monic squarefree positive polynomial in two variables x and y of degree in y divisible by 4 is a sum of three squares of rational fractions if and only if the jabobian variety of some hyperelliptic curve (associated to P) has an ”antineutral” point. Using this criterium, we follow a method developped by Cassels, Ellison and Pfister to solve our problem : at first we show the Mordell-Weil rank of the jacobian variety J associated to some polynomial is zero (this step is done by doing a 2-descent), and then we check that the jacobian variety J has no antineutral torsion point. Acknowledgements: This work is part of my PhD thesis at the université de Rennes 1 and was financed by the french governement. The editing of this article was supported by EPSRC grant EP/E012590/1. I want to thank warmly my thesis advisors D. Lubicz and L. Mahé. I also thank G. Everest and P. Satgé for their suggestions and comments.