A new approach to Hilbert's theorem on ternary quartics

Victoria Powers; Bruce Reznick; Claus Scheiderer; Frank Sottile

doi:10.1016/j.crma.2004.09.014

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

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 617-620.

Résumé
Abstract

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.

Reçu le : 2004-06-07
Accepté le : 2004-09-13
Publié le : 2004-10-18

DOI : 10.1016/j.crma.2004.09.014

Affiliations des auteurs :

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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     title = {A new approach to {Hilbert's} theorem on ternary quartics},
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     pages = {617--620},
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     volume = {339},
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     year = {2004},
     doi = {10.1016/j.crma.2004.09.014},
     language = {en},
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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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