The LMS JCM, (2) 62-92. Published 10 Jun 1999. First received 29 Jun 1998.


Reduction of binary cubic and quartic forms

J. E. Cremona



Abstract: 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.

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"Reduction of binary cubic and quartic forms" has been subsequently referenced by the following articles :

  • Computing the rank of elliptic curves over number fields (17 Apr 2002)
  • Corrigendum: Reduction of binary cubic and quartic forms (24 May 2001)
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