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The Splitting Lemma in any Characteristic

Gert-Martin Greuel, Gerhard Pfister

TL;DR

The paper proves a simple, constructive splitting lemma for formal power series over any field, decomposing $f$ as $f^{(2)}+g$ with disjoint variable sets, where $f^{(2)}$ is the quadratic part and $g$ has order at least 3. It shows existence and uniqueness of the residual part in all characteristics; in characteristic not equal to 2 this uses the implicit function theorem after diagonalizing the 2-jet, while in characteristic 2 Arf's classification yields a canonical 2-jet form. The authors extend the formal result to algebraic power series and to convergent real/complex analytic power series via nested Artin approximation and versal unfoldings, respectively, even without isolated singularities. They provide corresponding results for the algebraic and analytic settings, with the residual part unique up to right equivalence in the appropriate subring. Overall, the work broadens applicability of the generalized Morse splitting lemma and provides constructive, characteristic-flexible proofs across formal, algebraic, and analytic contexts.

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.

The Splitting Lemma in any Characteristic

TL;DR

The paper proves a simple, constructive splitting lemma for formal power series over any field, decomposing as with disjoint variable sets, where is the quadratic part and has order at least 3. It shows existence and uniqueness of the residual part in all characteristics; in characteristic not equal to 2 this uses the implicit function theorem after diagonalizing the 2-jet, while in characteristic 2 Arf's classification yields a canonical 2-jet form. The authors extend the formal result to algebraic power series and to convergent real/complex analytic power series via nested Artin approximation and versal unfoldings, respectively, even without isolated singularities. They provide corresponding results for the algebraic and analytic settings, with the residual part unique up to right equivalence in the appropriate subring. Overall, the work broadens applicability of the generalized Morse splitting lemma and provides constructive, characteristic-flexible proofs across formal, algebraic, and analytic contexts.

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.

Paper Structure

This paper contains 3 sections, 16 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Splitting Lemma in Characteristic $\ne 2$
  3. Splitting Lemma in Characteristic 2

Key Result

Theorem 1.1

Let $K$ be a field with a real valuation. Let $x=x_1,\ldots,x_n$ and $y=y_1,\ldots,y_m$ and $f=(f_1,\ldots,f_m)\in K\langle x,y\rangle^m$ such that Then there exists $y(x)\in K\langle x\rangle^m$ such that

Theorems & Definitions (30)

  • Theorem 1.1: Implicit Function Theorem for Convergent Power Series
  • Remark 1.2
  • Theorem 1.3: Artin Approximation
  • Lemma 1.4
  • proof
  • Theorem 1.5: Nested Artin Approximation
  • Theorem 1.6
  • proof
  • Corollary 1.7
  • Theorem 2.1: Formal splitting lemma in characteristic $\ne 2$
  • ...and 20 more