# Deformation of Milnor Algebras

## TL;DR

This paper studies how Milnor algebras of smooth homogeneous polynomials deform and shows that such polynomials are uniquely determined by certain components of their Jacobian ideals, extending previous reconstruction results.

## Contribution

It proves that smooth degree d homogeneous polynomials not of Sebastiani-Thom type are uniquely determined by specific homogeneous components of their Jacobian ideals.

## Key findings

- Polynomials are determined by Jacobian ideal components within a certain degree range.
- Generalization of polynomial reconstruction from Jacobian ideals.
- Extension of previous results on polynomial reconstruction.

## Abstract

We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree $k$ homogeneous component of its Jacobian ideal for any $d-1\leq k\leq (n+1)(d-2)$. Our results generalize the previous result on the reconstruction of a homogeneous polynomial from its Jacobian ideal.