Comptes Rendus
Algebraic Geometry
A new approach to Hilbert's theorem on ternary quartics
[Une nouvelle approche du théorème de Hilbert sur les quartiques ternaires.]
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 617-620.

Hilbert a démontré qu'une forme réelle non négative f(x,y,z) de degré 4 est la somme de trois carrés de formes quadratiques. Nous donnons une nouvelle démonstration qui montre que si la courbe plane Q definie par f est non singulière, alors f a exactement 8 telles représentations, à equivalence près. Elles correspondent aux points de 2- torsion du jacobien de Q qui ne sont pas représentés par un diviseur de Q invariant par conjugaison.

Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q.

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DOI : 10.1016/j.crma.2004.09.014

Victoria Powers 1 ; Bruce Reznick 2 ; Claus Scheiderer 3 ; Frank Sottile 4

1 Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA
2 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
3 Institut für Mathematik, Fakultät 4, Universität Duisburg, 47048 Duisburg, Germany
4 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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Victoria Powers; Bruce Reznick; Claus Scheiderer; Frank Sottile. A new approach to Hilbert's theorem on ternary quartics. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 617-620. doi : 10.1016/j.crma.2004.09.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.014/

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