  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.
