Symbolic computation of degree-three covariants for a binary form

Abstract

We use elementary linear algebra to explicitly calculate a basis for, and the dimension of, the space of degree-three covariants for a binary form of arbitrary degree. We also give an explicit basis for the subspace of covariants complementary to the space of degree-three reducible covariants. The study of invariant functions was one of the main influences on the development of modern algebra. Consider the following simple example. The group GD Z acts on R by addition: g xD gC x. We define a G-invariant function to be a real-valued function f.x/ on R such that f gD f for all g2 G. In other words, f.x/D f.gC x/ for all g2 Z, x2 R. The invariant functions are precisely the real-valued functions with period one. Hence, geometric information, such as periodicity, can be recovered by studying functions with certain algebraic properties. In Section 1, we introduce the concepts of an invariant and covariant function for a binary form Q.x; y/. The problem of determining the complete set of these functions was widely worked on during the late nineteenth century. Gordan [1868] proved that the ring of invariants (and the ring of covariants) for a degree-n binary form is finitely generated. A milestone in the history of modern algebra was Hilbert’s nonconstructive proof [1890] of the following fundamental theorem. Theorem [Hilbert 1890]. The ring of invariants (and the ring of covariants) for a degree-n homogeneous polynomial in m variables is finitely generated. Hilbert’s theorem says that all invariants (resp. covariants) for a homogeneous polynomial can be expressed as polynomials in a certain finite set of invariants (resp. covariants). Hilbert [1893] subsequently gave a constructive proof of this theorem. The minimal size of the generating set is only known for a few values of .m; n/. When mD 2, this number has been determined for n 8 [Bedratyuk 2009; Bedratyuk and Bedratyuk 2008].