Reduction of Binary Cubic and Quartic Formsj

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coef-cients. The resulting coeecient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number elds, and 2-descent on elliptic curves deened over Q. Remarks are given concerning the extension of these results to forms deened over number elds. 1. Introduction Reduction theory for polynomials has a long history and numerous applications, some of which have grown considerably in importance in recent years with the growth of algorithmic and computational methods in mathematics. It is therefore quite surprising to nd that even for the case of binary forms of degree three and four with integral coeecients, the results in the existing literature, which are widely used, can be improved. The two basic problems which we will address for forms