Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)

Valéry Mahé (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 Mahe, 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.

Mots clés: jacobian varieties; Hilbert's seventeenth problem; hyperelliptic curves; 2-descent; Mordell-Weil's rank; sums of squares; real algebraic geometry

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Valéry Mahé. Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y). The London Mathematical Society Journal of Computations and Mathematics, 2008, 11, pp.298 -- 325. ⟨hal-00482405⟩