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arXiv · 2507.17078

The Splitting Lemma in any Characteristic

Gert-Martin Greuel·Gerhard Pfister
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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.

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Gert-Martin Greuel, Gerhard Pfister. 2025-07-22. The Splitting Lemma in any Characteristic. https://doi.org/10.1016/j.jalgebra.2025.10.023

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