2007/03/24 by Valéry Mahé, Mahé, Valéry
Computer Science · Mathematics · #14G05 #14H05 #14H40 #14P99 #14Q05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:14G05 #msc:14H05 #msc:14H40 #msc:14P99 #msc:14Q05
paper · pdf · doi:10.48550/arxiv.math/0703722
63 pages, a proposition has been added (proposition 2.8)
openalex publication_date 2007/03/24 · arxiv created 2007/09/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.