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Reduction of Binary Cubic and Quartic Forms

Published online by Cambridge University Press:  01 February 2010

J. E. Cremona
J. E. Cremona
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD

Abstract

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A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over the set of rational numbers. Remarks are given concerning the extension of these results to forms defined over number fields.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 2 , 1999 , pp. 62 - 92
Copyright
Copyright © London Mathematical Society 1999

References

1.Belabas, K., ’Computing cubic fields in quasi-linear time’, [5] 17–25.CrossRefGoogle Scholar
2.Belabas, K., ‘A fast algorithm to compute cubic fields’, Math. Comp. 66 (1997) 1213–1237.CrossRefGoogle Scholar
3.Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘Notes on elliptic curves I’, J. Reine Angew. Math. 212 (1963) 7–25.CrossRefGoogle Scholar
4.Cassels, J. W. S., An introduction to the geometry of numbers, Classics in Mathematics (Springer, 1997).Google Scholar
5.Cohen, H. (ed.), Algorithmic number theory, Lecture Notes in Computer Science 1122 (Springer-Verlag, 1996).CrossRefGoogle Scholar
6.Cohen, H., A course in computational algebraic number theory, 3rd corrected printing, Graduate Texts in Mathematics 138 (Springer-Verlag, 1996).Google Scholar
7.Cremona, J. E., ‘Classical invariants and 2-descent on elliptic curves’, J. Symbolic Comput., to appear.Google Scholar
8.Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, 1997).Google Scholar
9.Cremona, J. E. and Serf, P., ‘Computing the rank of elliptic curves over real quadratic fields of class number 1’, Math. Comp. 68 (1999) 1187–1200.CrossRefGoogle Scholar
10.Elliott, E. B., An introduction to the algebra of quantics, 2nd edn (Oxford University Press, 1913).Google Scholar
11.Hilbert, D., Theory of algebraic invariants (Cambridge University Press, 1993).Google Scholar
12.Julia, G., ‘Étude sur les formes binaires non quadratiques à indéterminées réelles ou complexes’, Mem. Acad. Sci. I'Inst. France 55 (1917) 1–293.Google Scholar
13.Mathews, G.-B., ‘On the reduction and classification of binary cubics which have a negative discriminant’, Proc. London Math. Soc. (3) 10 (1912) 128–138.CrossRefGoogle Scholar
14.Serf, P., ’The rank of elliptic curves over real quadratic number fields of class number 1’, Ph.D. thesis, Universität des Saarlandes, 1995.Google Scholar