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Singularities in any Characteristic

This paper provides a comprehensive overview of hypersurface singularities defined by convergent power series over arbitrary real-valued fields, unifying classical complex analytic and formal power series approaches while presenting new classification results and complete proofs for simple singularities in positive characteristic.

Original authors: Gert-Martin Greuel

Published 2026-02-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gert-Martin Greuel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but instead of fingerprints or footprints, your clues are mathematical shapes that have gone wrong. In mathematics, these "wrong" shapes are called singularities. Think of a singularity as a sharp point on a smooth curve, a cusp on a leaf, or a knot in a piece of string where the smoothness breaks down.

This paper is a comprehensive guidebook written by mathematician Gert-Martin Greuel, dedicated to his late friend Wolfgang Ebeling. It's a "field manual" for understanding these broken shapes, but with a twist: it works in any universe of numbers, not just the familiar ones we use every day (like real or complex numbers).

Here is a breakdown of the paper's journey, explained through simple analogies.

1. The Setting: A New Kind of Number World

Usually, when mathematicians study these shapes, they use the complex numbers (the standard "high-tech" numbers used in physics and engineering). This paper says, "Let's try this in any number system."

The Analogy: Imagine you are a carpenter. Usually, you only build with oak wood (the complex numbers). This paper asks: "What if we build with bamboo, plastic, or even alien wood?"
The Challenge: In these "alien" number systems (called fields of positive characteristic), the rules of arithmetic change. For example, in a world where the number 2 equals 0, adding a number to itself makes it vanish. This breaks many of the standard tools mathematicians use, forcing them to invent new ones.

2. The Tools: Measuring the "Brokenness"

To classify these shapes, the authors need a way to measure how "bad" the break is. They introduce two main rulers:

The Milnor Number (The "Right" Ruler): This measures how many separate pieces the shape would break into if you nudged it slightly. It's like counting how many cracks appear in a dropped vase.
The Tjurina Number (The "Contact" Ruler): This is a stricter ruler. It counts the cracks plus how the shape is glued together.

The Big Discovery: In our normal world (characteristic 0), these two rulers usually agree. But in the "alien" worlds (positive characteristic), they often disagree. A shape might look like it has 5 cracks to one ruler but 10 to the other. The paper provides the new formulas to calculate these numbers correctly in these weird worlds.

3. The "Finite Determinacy" Rule: The Snapshot Principle

One of the most important ideas in the paper is Finite Determinacy.

The Analogy: Imagine you are looking at a very long, complicated poem. You want to know if it's a specific type of poem. Do you need to read the whole thing?
- The Paper says: No! You only need to read the first few lines (the "k-jet"). If the first few lines match a known pattern, the rest of the poem doesn't matter; it's that type of poem.
Why it matters: This allows mathematicians to ignore the infinite, messy tail of the equation and focus only on the "head" of the shape. The paper proves exactly how many lines you need to read to be sure, even in those tricky alien number worlds.

4. The Splitting Lemma: Untangling the Knot

Sometimes a shape is a mess of tangled variables. The Splitting Lemma is a magic trick that says: "You can separate this mess into two parts."

The Analogy: Imagine a tangled ball of yarn. The lemma says you can pull out a simple, clean loop (a quadratic part, like a perfect circle) and leave the rest of the tangle (the "residual part") separate.
The Result: Once you pull out the simple loop, you only have to study the messy tangle. The paper proves this works even when the math rules are weird (like when $2=0$ ).

5. The Grand Classification: The Periodic Table of Shapes

The climax of the paper is the Classification of "Simple" Singularities.

What is "Simple"? In this context, "simple" doesn't mean "easy." It means "stable." If you wiggle a simple shape, it doesn't turn into a completely different, chaotic monster. It stays in its family.
The Old World (Characteristic 0): In our normal world, there is a famous list of simple shapes called ADE (named after diagrams like $A_1, D_4, E_8$ ). It's like a periodic table with a few elements.
The New World (Positive Characteristic): The paper reveals that in these alien number worlds, the list changes!
- In Characteristic 2 (where $2=0$ ): The list shrinks dramatically. For example, if you have an even number of dimensions, you might only have one simple shape. If you have an odd number, you might have none at all!
- The Surprise: In our normal world, you can have infinitely many simple shapes (like $A_1, A_2, A_3...$ forever). But in these alien worlds, the list is finite. There is a hard limit. Once you go past a certain complexity, the shape is no longer "simple"; it becomes chaotic and unpredictable.

6. The "Modality" Mystery

The paper ends with a fascinating conjecture (a guess based on strong evidence).

The Idea: As the "brokenness" (Milnor number) of a shape gets bigger and bigger, does it become more chaotic?
The Guess: Yes. In normal worlds, you can have simple shapes with huge numbers. But in these alien worlds, as the number gets huge, the shape loses its simplicity and becomes infinitely complex. The paper suggests that in positive characteristic, there is a "speed limit" on how complex a simple shape can be.

Summary

This paper is a bridge. It takes the beautiful, well-understood theory of shapes from our familiar world and successfully extends it to strange, exotic mathematical universes. It shows us that while the basic rules of geometry hold up, the "flavor" of the shapes changes drastically depending on the number system you use.

In our world: Infinite variety, familiar rules.
In the "alien" worlds: Finite variety, surprising rules, and a need for new tools to measure the cracks.

It's a testament to the power of mathematics to adapt and find order, even in the most bizarre number systems imaginable.

Technical Summary: Hypersurface Singularities in any Characteristic

1. Problem Statement and Context
The paper addresses the theory of hypersurface singularities defined by convergent power series over a field $K$ of arbitrary characteristic ( $p \geq 0$ ). While the theory of complex analytic singularities ( $K=\mathbb{C}$ ) is well-established, the study of singularities over fields of positive characteristic (and general real-valued fields) has remained less explored, particularly regarding the convergence of power series rather than just formal ones.

The central problem is to generalize fundamental definitions, invariants, and classification results from the complex analytic case to the setting of convergent power series $K\{x\}$ over an arbitrary real-valued field. A specific challenge lies in the fact that many results valid in characteristic 0 (or large characteristic) fail or require significantly different proofs in small positive characteristic due to issues with derivations, the Jacobian ideal, and the behavior of the Milnor and Tjurina numbers.

2. Methodology
The author employs a unified approach that bridges the gap between formal power series ( $K[[x]]$ ) and convergent power series ( $K\{x\}$ ) over real-valued fields. The core methodological pillars include:

Real-Valued Fields and Convergence: The work defines $K\{x\}$ as the ring of power series convergent with respect to a real valuation $|\cdot|$ on $K$ . This generalizes the classical cases of $\mathbb{R}$ and $\mathbb{C}$ while accommodating non-Archimedean fields (e.g., $p$ -adic numbers).
Grauert's Approximation Theorem: A critical tool used throughout is Grauert's approximation theorem (adapted for real-valued fields). This theorem allows the lifting of formal solutions to analytic (convergent) solutions. It is used to prove the existence of semiuniversal deformations and to establish finite determinacy bounds for convergent series.
The Lefschetz Principle: For fields of characteristic 0, the author utilizes the Lefschetz principle to transfer results known for $\mathbb{C}$ to arbitrary real-valued fields of characteristic 0, provided the field is algebraically closed or can be embedded into $\mathbb{C}$ .
Splitting Lemma: The paper provides a new, simplified proof of the Splitting Lemma for any characteristic. This lemma states that a singularity can be decomposed into a non-degenerate quadratic part and a "residual" part of higher order, with the residual part being unique up to coordinate change.
Algebraic Geometry and Deformation Theory: The classification relies heavily on the theory of deformations, specifically the existence of semiuniversal unfoldings and the concept of "modality" (the dimension of the family of orbits).

3. Key Contributions and Results

A. General Theory and Invariants (Section 1)

Generalized Definitions: The paper rigorously defines the Milnor number ( $\mu$ ) and Tjurina number ( $\tau$ ) for $f \in K\{x\}$ .
Finite Determinacy: It establishes bounds for finite determinacy (the degree up to which a power series determines its equivalence class).
- In characteristic 0 or large characteristic ( $p \geq k+2-\text{ord}(f)$ ), finite determinacy is equivalent to the singularity being isolated ( $\mu < \infty$ or $\tau < \infty$ ).
- In small positive characteristic, the standard bounds fail. The author provides new, sharper bounds using Grauert's theorem: if $m^{k+2} \subseteq m^2 \cdot j(f)$ , then $f$ is right $(2k - \text{ord}(f) + 2)$ -determined.
Mather-Yau Theorem: The paper generalizes the Mather-Yau theorem, which states that the isomorphism class of the Milnor/Tjurina algebra determines the equivalence class of the singularity.
- In characteristic 0, the standard algebras suffice.
- In positive characteristic, the author proves that higher Tjurina and Milnor algebras ( $T_k(f)$ and $M_k(f)$ ) are required to distinguish singularities, providing explicit bounds for $k$ .

B. Versal Deformations and Splitting (Section 2)

Semiuniversal Deformations: The existence of a semiuniversal unfolding for isolated singularities in $K\{x\}$ is proven for any real-valued field, extending previous results from formal series.
Splitting Lemma: A complete proof is provided for the splitting lemma in any characteristic.
- For $char(K) \neq 2$ , $f \sim x_1^2 + \dots + x_k^2 + g$ .
- For $char(K) = 2$, the quadratic part involves terms like $x_i x_{i+1}$ and $x_i^2$ , reflecting the classification of quadratic forms in characteristic 2. The uniqueness of the residual part $g$ is rigorously established.

C. Classification of Simple Singularities (Sections 3 & 4)
The paper classifies "simple" singularities (those with 0 modality) in positive characteristic, distinguishing between Contact Equivalence and Right Equivalence.

Contact Simple Singularities (Section 3):
- The classification is largely similar to the characteristic 0 case (ADE classification: $A_k, D_k, E_6, E_7, E_8$ ).
- However, in small characteristics (specifically $p=2, 3, 5$ ), new normal forms appear (e.g., $E_6^1, E_7^1$ ).
- The paper also characterizes non-isolated singularities of type $A_\infty$ and $D_\infty$ as the limits of simple singularities.
Right Simple Singularities (Section 4):
- Fundamental Difference: Unlike the contact case, the classification of right simple singularities in positive characteristic is drastically different from characteristic 0.
- Finiteness: For a fixed prime $p$ , there are only finitely many right simple singularities.
- Specific Results:
  - If $p=2$ and $n$ is even, the only right simple singularity is $A_1$ (sum of hyperbolic terms). If $n$ is odd, no right simple singularities exist.
  - If $p > 2$ , the list of right simple singularities is a truncated version of the ADE list, where the indices are bounded by $p$ (e.g., $A_k$ exists only for $k \leq p-2$ ).
- Modality: The paper proves that for univariate series, the right modality is $\lfloor \mu/p \rfloor$ . Consequently, a singularity is right simple if and only if $\mu < p$ .

4. Significance

Unification: The paper successfully unifies the theory of singularities over complex numbers, real numbers, and arbitrary fields (including $p$ -adic and non-Archimedean fields) under the framework of convergent power series.
Novelty in Positive Characteristic: It resolves long-standing gaps in the classification of singularities in positive characteristic, particularly the distinction between contact and right equivalence, which coincides in characteristic 0 but diverges significantly in $p > 0$ .
New Proofs: The use of Grauert's approximation theorem provides a robust mechanism to move from formal to analytic settings, offering complete proofs for results that were previously only known for formal series or complex analytic spaces.
Conjecture: The paper concludes with a significant conjecture: as the Milnor number $\mu$ tends to infinity in positive characteristic, the right modality also tends to infinity, implying that the set of right simple singularities is strictly finite for any fixed $p$ .

In summary, this work provides a comprehensive, rigorous, and generalized framework for hypersurface singularities, correcting and extending classical results to the realm of arbitrary characteristic and convergent power series.