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.
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.
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Victoria Powers 1; Bruce Reznick 2; Claus Scheiderer 3; Frank Sottile 4
@article{CRMATH_2004__339_9_617_0, author = {Victoria Powers and Bruce Reznick and Claus Scheiderer and Frank Sottile}, 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}, }
TY - JOUR AU - Victoria Powers AU - Bruce Reznick AU - Claus Scheiderer AU - Frank Sottile 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 DO - 10.1016/j.crma.2004.09.014 LA - en ID - CRMATH_2004__339_9_617_0 ER -
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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