Associated Forms in Classical Invariant Theory

Abstract

It was conjectured in a recent article by M. Eastwood and the second author that all absolute classical invariants of forms of degree $m\ge 3$ on ${\mathbb{C}}^n$ can be extracted, in a canonical way, from those of forms of degree $n(m-2)$ by means of assigning every form with non-vanishing discriminant the so-called associated form. In that paper, this surprising conjecture was confirmed for binary forms of degree $m\le 6$ and ternary cubics. In the present article, we settle the conjecture in full generality. In addition, we propose a stronger version of this statement and obtain evidence supporting it.