Sylvester versus Gundelfinger

TL;DR

This paper determines the exact number of generators needed for the algebra of polynomial invariants under SL_2 action on a specific module, resolving a long-standing mathematical question from 143 years ago.

Contribution

It establishes that 63 generators are necessary for the algebra of invariants of a particular SL_2-module, settling a question that has remained open for over a century.

Findings

The algebra of invariants is generated by exactly 63 elements.

The result resolves a 143-year-old open problem in invariant theory.

Provides a complete description of the minimal generating set for the algebra.

Abstract

Let $V_n$ be the ${\rm SL}_2$-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 $R = \mathbb{C}[V]^{{\rm SL}_2}$ of polynomial functions on $V$ invariant under the action of ${\rm SL}_2$ equals 63. This settles a 143-year old question.