Abstract
This article deals with a constructive 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 sums of three squares of rational fractions.To do this we use a reformulation of this problem in terms of hyperelliptic curves due to Huisman and Mahé and we follow a method of Cassels, Ellison and Pfister which involves the computation of a Mordell–Weil rank over ℝ(x).
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