Calculs dans les jacobiennes de courbes algébriques, applications en géométrie algébrique réelle.

Abstract : This thesis 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 Mahé, 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 jacobian variety of some hyperelliptic curve (associated to P) has an ”antineutral” point.

Using this criterium, we follow a method developed by Cassels, Ellison and Pfister to solve our problem : at first we show that the Mordell-Weil's 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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Valéry Mahé. Calculs dans les jacobiennes de courbes algébriques, applications en géométrie algébrique réelle.. Mathématiques [math]. Université Rennes 1, 2006. Français. ⟨tel-00124040⟩

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