Hilbert's seventeenth problem and hyperelliptic curves
Valéry Mahé
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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