[Une nouvelle approche du théorème de Hilbert sur les quartiques ternaires.]
Hilbert a démontré qu'une forme réelle non négative 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 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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@article{CRMATH_2004__339_9_617_0,
author = {Powers, Victoria and Reznick, Bruce and Scheiderer, Claus and Sottile, Frank},
title = {A new approach to {Hilbert's} theorem on ternary quartics},
journal = {Comptes Rendus. Math\'ematique},
pages = {617--620},
publisher = {Elsevier},
volume = {339},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.09.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.09.014/}
}
TY - JOUR AU - Powers, Victoria AU - Reznick, Bruce AU - Scheiderer, Claus AU - Sottile, Frank TI - A new approach to Hilbert's theorem on ternary quartics JO - Comptes Rendus. Mathématique PY - 2004 SP - 617 EP - 620 VL - 339 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.09.014/ DO - 10.1016/j.crma.2004.09.014 LA - en ID - CRMATH_2004__339_9_617_0 ER -
%0 Journal Article %A Powers, Victoria %A Reznick, Bruce %A Scheiderer, Claus %A Sottile, Frank %T A new approach to Hilbert's theorem on ternary quartics %J Comptes Rendus. Mathématique %D 2004 %P 617-620 %V 339 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.09.014/ %R 10.1016/j.crma.2004.09.014 %G en %F CRMATH_2004__339_9_617_0
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. http://www.numdam.org/articles/10.1016/j.crma.2004.09.014/
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