Singularities in any Characteristic

TL;DR

The work develops a unified theory of hypersurface singularities defined by convergent power series over real-valued fields of arbitrary characteristic. It introduces generalized invariants $\mu(f)$ and $\tau(f)$ via Milnor and Tjurina algebras $M_f$ and $T_f$, proves right/ contact-invariance properties, and establishes finite determinacy bounds through Grauert's approximation, along with Mather–Yau-type reconstruction in characteristic zero. A splitting lemma and semiuniversal deformation theory are developed in full generality, enabling structured deformation analysis across variable blocks. The paper then delivers a comprehensive classification of simple singularities in positive characteristic, distinguishing between contact and right equivalence and providing ADE-type normal forms with characteristic-dependent refinements, including finite right-simple classes and distinctive 2-characteristic phenomena. Together, these results extend classical complex singularity theory to broad arithmetic settings and furnish complete proofs in this general framework.

Abstract

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of analytic power series over the complex numbers with formal power series over arbitrary fields, but on the other hand, it goes significantly beyond that. Besides general definitions and basic results, we report on the classification of contact simple and right simple singularities in positive characteristic. Some of the results are new in this general setting, for which we provide complete proofs.

Theorems & Definitions (53)

• Remark 1
• Definition 1.1
• Definition 1.2
• Lemma 1.3
• proof
• proof
• Theorem 1.5
• proof
• Theorem 1.6
• proof