The Splitting Lemma in any Characteristic

TL;DR

This paper provides a new, simple proof of the splitting lemma in singularity theory applicable to any characteristic, including characteristic two, and extends results to algebraic and analytic power series.

Contribution

It introduces a novel proof of the splitting lemma valid in all characteristics and extends the lemma's applicability to algebraic and analytic series with non-isolated singularities.

Findings

Proof of the splitting lemma in arbitrary characteristic

Uniqueness of the residual part in characteristic two

Extensions to algebraic and convergent power series

Abstract

We give a simple proof of the splitting lemma in singularity theory, also known as generalized Morse lemma, for formal power series over arbitrary fields. Our proof for the uniqueness of the residual part in any characteristic is new and was previously unknown in characteristic two. Beyond the formal case, we give proofs for algebraic power series and for convergent real and complex analytic power series, which are new for non-isolated singularities.