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Finiteness Theorems for Binary Forms with Given Discriminant
Author(s) -
Birch B. J.,
Merriman J. R.
Publication year - 1972
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-24.3.385
Subject(s) - discriminant , citation , mathematics , binary number , computer science , arithmetic , combinatorics , library science , artificial intelligence
1. Classical invariant theory is concerned with the action of linear groups on spaces of algebraic forms and the algebraic invariants under such actions; in this paper we are concerned with one of the simplest of such spaces, the space of binary forms of given degree, and one of the simplest invariants, the discriminant of the form. We prove in particular that if Do is a given integer then there are only finitely many GL2(Z)-orbits of binary forms / of given degree n with discriminant D(f) = Do. Before we state our main theorems, we establish our notation and recall some standard definitions. First, suppose that f(x, y) = £?-o aix ~y is a binary form of degree n; if f(x,y) factors as H^=i{j~Pjy)> the discriminant of/ is