Finiteness Theorems for Binary Forms with Given Discriminant
Finiteness Theorems for Binary Forms with Given Discriminant
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给定判别式的二元形式的有限定理
DOI：
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
J. Merriman
中科院分区：
文献类型：
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作者：
B. Birch;J. Merriman
1. 经典不变量理论研究线性群在代数形式空间上的作用以及这种作用下的代数不变量；本文讨论了这类空间中最简单的一种，即给定次的二元形式空间，以及最简单的不变量之一，即二元形式的判别式。我们特别证明了如果Do是一个给定的整数，那么只有有限个二元形式的GL2(Z)-轨道/具有给定阶数n且判别D(f) = Do。在我们陈述主要定理之前，我们先建立记号并回顾一些标准定义。首先，假设f(x, y) =£？- 0aix ~y是n次的二进制形式；如果f（x,y）因子为H^=i{j~Pjy)>，则/的判别式为
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