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Hilbert's Seventeenth Problem and Hyperelliptic Curves

Published online by Cambridge University Press:  01 February 2010

Valéry Mahé
Valéry Mahé
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom, [email protected]

Abstract

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This article deals with a constructive aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables, of degree 8 in one variable, which are positive but are not sums of three squares of rational fractions.

To do this we use a reformulation of this problem in terms of hyperelliptic curves due to Huisman and Mahé and we follow a method of Cassels, Ellison and Pfister which involves the computation of a Mordell–Weil rank over ℝ(x).

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 298 - 325
Copyright
Copyright © London Mathematical Society 2008

References

1.Artin, E., ‘Über die Zerlegung definiter Funktionen in Quadrate’, Hamb. Abh 5 (1927) 100–115.CrossRefGoogle Scholar
2.Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 36 (Springer, Berlin, 1998). Translated from the 1987 French original, revised by the authors.CrossRefGoogle Scholar
3.Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 21 (Springer, Berlin, 1990).Google Scholar
4.BOST, J.-B. and Mestre, J.-F., ‘Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2’, Gaz. Math. 38 (1988) 36–64.Google Scholar
5.Cantor, D. G., ‘Computing in the Jacobian of a hyperelliptic curve’, Math. Gornp. 48 (1987) 95–101.Google Scholar
6.Cassels, J. W. S., ‘The Mordell-Weil group of curves of genus 2’, Arithmetic and geometry, Vol. I, Progr. Math. 35 (Birkhäuser, Boston, MA, 1983) 27–60.Google Scholar
7.Cassels, J. W. S., Ellison, W. J. and Pfister, A., ‘On sums of squares and on elliptic curves over function fields’, J. Number Theory 3 (1971) 125–149.Google Scholar
8.Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series 230 (Cambridge University Press, Cambridge, 1996).Google Scholar
9.Christie, M. R., ‘Positive definite rational functions of two variables which are not the sum of three squares’, J. Number Theory 8 (1976) 224–232.CrossRefGoogle Scholar
10.Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K. and Vercauteren, F. (eds), Handbook of elliptic and hyperelliptic curve cryptography, Discrete Mathematics and its Applications (Boca Raton) (Chapman & Hall/CRC, Boca Raton, FL, 2006).Google Scholar
11.Colliot-Thélène, J.-L., ‘The Noether-Lefschetz theorem and sums of 4-squares in the rational function field ℝ(x,y)’, Compositio Math. 86 (1993) 235–243.Google Scholar
12.Gaudry, P., ‘Algorithmique des courbes hyperelliptiques et applications à la cryptologie’, PhD thesis, École Polytechnique, 2000.Google Scholar
13.Hilbert, D., ‘Über die Darstellung definiter Formen als Summen von Formen-Quadraten’, Math. Ann 32 (1888) 342–350.CrossRefGoogle Scholar
14.Hindry, M. and Silverman, J. H., Diophantine geometry, Graduate Texts in Mathematics 201 (Springer, New York, 2000).CrossRefGoogle Scholar
15.Huisman, J. and Mahé, L., ‘Geometrical aspects of the level of curves’, J. Algebra 239 (2001) 647–674.CrossRefGoogle Scholar
16.Lam, T. Y., The algebraic theory of quadratic forms, Mathematics Lecture Note Series (W. A. Benjamin, Reading, MA, 1973).Google Scholar
17.Lang, S., Survey of Diophantine Geometry (Springer, 1997).Google Scholar
18.Macé, O., ‘Sommes de trois carres en deux variables et représentation de bas degré pour le niveau des courbes réelles’, PhD thesis, Université de Rennes 1, 2000, http://tel.ccsd.cnrs.fr/documents/archives0/00/00/62/39/indexir.html.Google Scholar
19.Macé, O. and Mahé, L., ‘Sommes de trois carrés de fractions en deux variables’, Manuscripta Math. 116 (2005) 421–447.Google Scholar
20.Mahé, L., ‘Level and Pythagoras number of some geometric rings’, Math. Z. 204 (1990) 615–629; erratum Math. Z. 209 (1992) 481–483.CrossRefGoogle Scholar
21.Mahé, V. A., ‘Calculs dans les jacobiennes de courbes algébriques, applications en géométrie algébrique réelle’, PhD thesis, Université de Rennes 1, 2006, http://tel.archives-ouvertes.fr/tel-00124040/Google Scholar
22.Mahé, V. A., ‘Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in ℝ(x, y)’, Preprint, 2007, http://arxiv.org/abs/math/0703722v2.Google Scholar
23.Mumford, D., Tata lectures on theta II, Progress in Mathematics 43 (Birkhäuser, Boston, MA, 1984). Jacobian theta functions and differential equations. With the collaboration of Musili, C., Nori, M., Previato, E., Still-man, M. and Umemura, H..Google Scholar
24.Oort, F. and Ueno, K., ‘Principally polarized abelian varieties of dimension two or three are Jacobian varieties’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973) 377–381.Google Scholar
25.Pfister, A., ‘Zur Darstellung definiter Funktionen als Summe von Quadraten’, Invent. Math. 4 (1967) 229–237.CrossRefGoogle Scholar
26.Schaefer, E. F., ‘2-descent on the Jacobians of hyperelliptic curves’, J. Number Theory 51 (1995) 219–232.Google Scholar
27.Silverman, J. H. and Tate, J., Rational points on elliptic curves, Undergraduate Texts in Mathematics (Springer, New York, 1992).Google Scholar
28.Stichtenoth, H.Algebraic function fields and codes, Universitext (Springer, Berlin, 1993).Google Scholar
29.Stoll, M., ‘Implementing 2-descent for s of hyperelliptic curves’, Acta Arith. 98 (2001) 245–277.Google Scholar
30.Zariski, O. and Samuel, P., Commutative algebra, Vol. I, The University Series in Higher Mathematics (Van Nostrand, Princeton, NJ, 1958). With the cooperation of I. S. Cohen.Google Scholar