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

Abstract : 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.
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Contributor : Valery Mahe <>
Submitted on : Monday, May 10, 2010 - 2:14:52 PM
Last modification on : Thursday, November 15, 2018 - 11:56:35 AM

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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⟩

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