Geodesic
Algebraic Geometry
A new approach to Hilbert's theorem on ternary quartics
[Une nouvelle approche du théorème de Hilbert sur les quartiques ternaires.]
Présenté par : Michel Raynaud
Powers, Victoria1 ; Reznick, Bruce2 ; Scheiderer, Claus3 ; Sottile, Frank4
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
Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 617-620 Cet article a éte moissonné depuis la source Numdam

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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.

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.

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DOI : 10.1016/j.crma.2004.09.014

Powers, Victoria 1 ; Reznick, Bruce 2 ; Scheiderer, Claus 3 ; Sottile, Frank 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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     title = {A new approach to {Hilbert's} theorem on ternary quartics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {617--620},
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     language = {en},
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Powers, Victoria; Reznick, Bruce; Scheiderer, Claus; Sottile, Frank. A new approach to Hilbert's theorem on ternary quartics. Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 617-620. doi: 10.1016/j.crma.2004.09.014

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