Sylvester versus Gundelfinger

Abstract

Let $V_n$ be the ${\mathrm{SL}}_2$-module of binary forms of degree $n$ and let $V=V_1\oplus V_3\oplus V_4$. We show that the minimum number of generators of the algebra $R=\mathbb{C}[V{]}^{{\mathrm{SL}}_2}$ of polynomial functions on $V$ invariant under the action of ${\mathrm{SL}}_2$ equals 63. This settles a 143-year old question.