# Stability of associated forms

## TL;DR

This paper proves that the associated form of certain Artinian complete intersections is polystable and applies this to develop an invariant-theoretic version of the Mather-Yau theorem for homogeneous hypersurface singularities.

## Contribution

It establishes the polystability of associated forms for Artinian complete intersections and introduces an invariant-theoretic approach to the Mather-Yau theorem.

## Key findings

- Associated forms are polystable for Artinian complete intersections of type (d,...,d).
- Provides an invariant-theoretic variant of the Mather-Yau theorem.
- Connects stability properties with singularity classification.

## Abstract

We show that the associated form, or equivalently a Macaulay inverse system, of an Artinian complete intersection of type $(d,\dots, d)$ is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.