Weighted Homogeneous Polynomials with Isomorphic Milnor Algebras

TL;DR

This paper demonstrates that the Milnor algebra uniquely classifies weighted homogeneous polynomials under right-equivalence, providing a complete invariant for their classification.

Contribution

It introduces the use of Milnor algebras as complete invariants for weighted homogeneous polynomials under right-equivalence.

Findings

Milnor algebra is a complete invariant for classification

Weighted homogeneous functions and their filtrations are analyzed

Analogues for diffeomorphisms and vector fields are introduced

Abstract

We recall first some basic facts on weighted homogeneous functions and filtrations in the ring $A$ of formal power series. We introduce next their analogues for weighted homogeneous diffeomorphisms and vector fields. We show that the Milnor algebra is a complete invariant for the classification of weighted homogeneous polynomials with respect to right-equivalence, i.e. change of coordinates in the source and target by diffeomorphism.