Comptes Rendus
Algebraic Geometry
A new approach to Hilbert's theorem on ternary quartics
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 617-620.

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.

Received:
Accepted:
Published online:
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/

[1] M.D. Choi; T.Y. Lam; B. Reznick Sums of squares of real polynomials, (Santa Barbara, 1992) (Proc. Symp. Pure Math.), Volume vol. 58, American Mathematical Society, Providence, RI (1995), pp. 103-126

[2] A.B. Coble Algebraic Geometry and Theta Functions, Amer. Math. Soc. Colloq. Publ., vol. 10, American Mathematical Society, 1929

[3] W.-D. Geyer Ein algebraischer Beweis des Satzes von Weichold über reelle algebraische Funktionenkörper (H. Hasse; P. Roquette, eds.), Algebraische Zahlentheorie (Oberwolfach, 1964), Mannheim, 1966, pp. 83-98

[4] D. Hilbert Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., Volume 32 (1888), pp. 342-350

[5] A. Pfister On Hilbert's theorem about ternary quartics, Algebraic and Arithmetic Theory of Quadratic Forms, Contemp. Math., vol. 344, American Mathematical Society, Providence, RI, 2004

[6] V. Powers; B. Reznick Notes towards a constructive proof of Hilbert's theorem on ternary quartics, (Dublin, 1999) (Contemp. Math.), Volume vol. 272, American Mathematical Society, Providence, RI (2000), pp. 209-227

[7] W. Rudin Sums of squares of polynomials, Amer. Math. Monthly, Volume 107 (2000), pp. 813-821

[8] R.G. Swan Hilbert's theorem on positive ternary quartics, (Dublin, 1999) (Contemp. Math.), Volume vol. 272, American Mathematical Society, Providence, RI (2000), pp. 287-292

[9] C.T.C. Wall Is every quartic a conic of conics?, Math. Proc. Cambridge Philos. Soc., Volume 109 (1991), pp. 419-424

[10] G. Weichold Über symmetrische Riemannsche Flächen und die Periodizitätsmoduln der zugehörigen Abelschen Normalintegrale erster Gattung, Z. Math. Phys., Volume 28 (1883), pp. 321-351

[11] E. Witt Zerlegung reeller algebraischer Funktionen in Quadrate, Schiefkörper über reellem Funktionenkörper, J. Reine Angew. Math., Volume 171 (1934), pp. 4-11

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