Associated Forms and Hypersurface Singularities: The Binary Case

TL;DR

This paper proves a conjecture relating invariants of binary forms to associated forms, enabling the extraction of complete invariants of plane curve singularities from their Milnor algebras, advancing classical invariant theory and singularity reconstruction.

Contribution

It establishes the conjecture for binary forms of any degree, providing an explicit method to derive invariants of plane curve singularities from Milnor algebras.

Findings

Confirmed the conjecture for binary forms of arbitrary degree.

Provided an explicit method to extract invariants of plane curve singularities.

Enabled reconstruction of singularity invariants from Milnor algebras.

Abstract

In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of degree $n(d-2)$ by means of assigning every form with nonvanishing discriminant the so-called associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the well-known Mather-Yau theorem. The conjecture was confirmed by Eastwood and Isaev for binary forms of degree $d \le 6$ as well as ternary cubics. Furthermore, a weaker version of it was settled by Alper and Isaev for arbitrary $n$ and $d$. In the present paper, we focus on the case $n=2$ and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree. This result allows one to extract a complete system of biholomorphic invariants of homogeneous plane curve singularities from their Milnor algebras.