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Existence and maximal corank of simple $Z_p$ -invariant germs

This paper improves the upper bound on the corank of equivariantly stable singularities for groups of prime order and demonstrates that the maximal corank of simple $\mathbb{Z}_p$ -invariant germs grows indefinitely as the prime $p$ increases.

Original authors: Ivan Proskurnin

Published 2026-04-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ivan Proskurnin

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 an architect trying to build the simplest possible structures out of a special, rigid material. In the world of mathematics, these "structures" are called singularities—points where a smooth surface suddenly folds, twists, or breaks.

This paper is about finding the limits of how complex these "simple" structures can get when they are built under strict rules of symmetry. Specifically, the author, Ivan Proskurnin, is looking at shapes that must look the same after being rotated by a specific number of steps (like a clock hand moving $p$ times to return to the start, where $p$ is a prime number).

Here is the breakdown of the paper's discoveries using everyday analogies:

1. The Previous Rulebook (The "Bad News")

Before this paper, mathematicians had a rulebook (Theorem 1.1) that tried to limit how "messy" these simple shapes could be. The rule said:

"If you want a simple shape that respects these rotation rules, the number of 'messy' dimensions (called corank) cannot be too big. It's capped by a specific formula involving the number of rotation steps ( $p$ )."

The author felt this rule was too pessimistic. It implied that no matter how many rotation steps you had, there was a hard ceiling on how complex a "simple" shape could be. The author suspected this ceiling might not actually exist.

2. The New Discovery: The Ceiling is Gone

The paper proves two main things that change the game:

A. The Ceiling is Real, But Only for Certain Rules (Theorem 1.2)
The author refined the rulebook. He showed that the "messiness" limit only applies if the rotation rules are "real" (a specific mathematical property). If the rules are "imaginary" (in a technical sense), the old limit doesn't apply in the same way.

Analogy: Imagine you are building a tower with blocks. The old rule said, "You can never stack more than 10 blocks high." The new rule says, "Actually, you can only stack 10 blocks high if you are using red blocks. If you use blue blocks, the rule is different."

B. The "Messiness" Can Be Infinite (Theorem 1.3)
This is the big headline. The author proves that for any number $N$ you can think of (10, 1,000, 1,000,000), there exists a prime number $p$ (a specific rotation rule) where you can build a "simple" shape that is messier than $N$ .

Analogy: Think of the "corank" as the number of loose threads on a sweater. The old belief was that a "simple" sweater could only have a few loose threads. Proskurnin proves that if you pick the right pattern (the right prime number $p$ ), you can make a "simple" sweater with 100 loose threads, or 1,000, or a million. There is no maximum limit. As the rotation steps ( $p$ ) get larger, the potential for complexity grows without bound.

3. How Did He Prove It? (The Magic Recipe)

To prove that you can get infinitely complex "simple" shapes, the author didn't just guess; he built a specific recipe.

The Ingredients: He chose a large prime number $p$ .
The Construction: He broke the number $p-1$ down into many smaller factors (like breaking a large pizza into many slices).
The Shape: He used these factors to create a specific mathematical formula (a polynomial) that acts like a loop.
The Result: This formula creates a shape that is "stable" (it doesn't fall apart easily) and "simple" (it doesn't have unnecessary complications), yet it has a huge number of dimensions (high corank).

He used a famous math theorem by Erdős (about how prime numbers are made of many smaller factors) to guarantee that for large enough $p$ , you can always find enough "slices" to build a massive, complex structure.

Summary

The Problem: Mathematicians wanted to know if there was a limit to how complex a "simple" symmetrical shape could be.
The Old Answer: Yes, there is a limit.
The New Answer: No, there isn't a limit. As long as you pick the right type of symmetry (a large enough prime number), you can create "simple" shapes that are arbitrarily complex.
The Takeaway: The universe of these mathematical shapes is much wilder and less restricted than previously thought. The "simplest" shapes can actually be incredibly intricate, provided you have the right number of rotation steps to work with.

1. Problem Statement

The paper addresses the classification of equivariantly simple singularities under the action of a cyclic group of prime order, $\mathbb{Z}_p$ , acting on the complex space $(\mathbb{C}^n, 0)$ .

Context: Previous work (referenced as [1]) established an upper bound on the corank (defined as $n - \text{rk}(\tau)$ , where $\text{rk}(\tau)$ is the maximal rank of a $\tau$ -invariant quadratic form) for such singularities. The previous bound suggested that for a fixed prime $p$ , the corank is bounded by logarithmic functions of $p$ (specifically $\log_2(p+1)$ or $\log_2(2p-1)$ ).
The Conflict: A "tame" classification of all simple singularities generally requires a constant upper bound on the corank independent of the specific group parameters. The author investigates whether such a bound exists or if the corank can grow arbitrarily large as $p$ increases.
Core Question: Can the corank of a simple $\mathbb{Z}_p$ -invariant germ be arbitrarily large as $p \to \infty$ ? Furthermore, what are the precise constraints on the group action (specifically the determinant of the representation) for such singularities to exist?

2. Methodology

The author employs techniques from singularity theory, representation theory, and number theory.

Linearization and Diagonalization: Since any analytic action of a finite group on $(\mathbb{C}^n, 0)$ is linearizable, and representations of finite abelian groups are diagonalizable, the study is reduced to linear actions of the form $g \cdot (x_1, \dots, x_n) = (\chi_1(g)x_1, \dots, \chi_n(g)x_n)$ , where $\chi_i$ are characters of $\mathbb{Z}_p$ .
Equivariant Jacobian Algebra: The paper utilizes the Jacobian algebra $Q_f = \mathbb{C}\{x_1, \dots, x_n\} / J_f$ (where $J_f$ is the Jacobian ideal). The group $G$ acts on $Q_f$ , and the equivariant Milnor number $\mu_G(f)$ is defined as the representation of $G$ on this algebra.
Stability Criteria: The author relies on the characterization that a germ is equivariantly stable if and only if the dimension of the space of invariant elements in $Q_f$ (denoted $\nu(f)$ ) is equal to 1.
Splitting Lemma: The analysis uses the splitting lemma to decompose a stable germ $f$ into a non-degenerate quadratic part (of rank $\text{rk}(\tau)$ ) and a residual part $h$ with $j^2(h)=0$ . This allows the problem to be reduced to analyzing the corank of $h$ .
Number Theoretic Construction: To prove the existence of high-corank examples, the author constructs specific polynomials (loop-type invertible polynomials) and utilizes Erdős's theorem on the distribution of prime divisors of $p-1$ to show that for large primes, $p-1$ can be factored into a large number of integers.

3. Key Contributions and Theorems

A. Refinement of Existence Conditions (Theorem 1.2)

The paper improves the conditions under which equivariantly simple germs can exist.

Previous Result: Simple germs could exist if $\det(\tau) \neq 1$ (with bound $\log_2(p+1)$ ) or if $\det(\tau) = 1$ (with bound $\log_2(2p-1)$ ).
New Result: The author proves that if $\det(\tau) = 1$ , simple germs cannot exist unless the action is a real group action (admitting an invariant quadratic form of full rank $n$ ).
Implication: This significantly narrows the range of possible group actions. For complex actions with $\det(\tau)=1$ , simple singularities are impossible.

B. Proof of Unbounded Corank (Theorem 1.3)

This is the paper's primary contribution.

Statement: For any integer $N$ , there exists a prime $p$ and a $\mathbb{Z}_p$ -invariant simple germ $f: (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$ with $j^2(f)=0$ and corank $n > N$ .
Significance: This disproves the hope for a constant upper bound on the corank of simple singularities. The maximal corank tends to infinity as $p$ increases.

C. Classification of Equivariant Milnor Numbers (Theorem 3.3 & 4.1)

The author classifies the possible isomorphism classes of the equivariant Milnor number $\mu_{\mathbb{Z}_p}(f)$ for stable germs.

It is shown that the class $\tilde{\mathbb{C}} \oplus 2W$ (where $W$ is the restriction of the regular representation) cannot occur for stable germs.
This elimination is crucial for the proof of Theorem 1.2, as it restricts the possible dimensions of the Jacobian algebra, forcing the corank to be bounded in specific cases but unbounded in others.

4. Results and Construction

Construction of High-Corank Examples:
To prove Theorem 1.3, the author constructs a specific family of polynomials:
$f = x_1^{d_1}x_2 + x_2^{d_2}x_3 + \dots + x_{n-1}^{d_{n-1}}x_n + x_n^{d_n}x_1$

Group Action: A generator $g$ of $\mathbb{Z}_p$ (where $p = d_1 \dots d_n + 1$ ) acts on coordinates with specific weights involving a primitive root of unity.
Properties: This polynomial is an invertible polynomial of "loop type," known to have an isolated critical point. It is proven to be equivariantly stable.
Corank: The corank is determined by the number of variables $n$ .
Number Theory Link: The author invokes Erdős's Theorem, which states that for "almost all" primes $p$ , the number of distinct prime divisors of $p-1$ grows as $\log(\log(p))$ . By choosing $p$ such that $p-1$ has many factors, one can construct a decomposition $p-1 = d_1 \dots d_n$ with arbitrarily large $n$ .
Asymptotic Behavior: The maximal corank for a prime $p$ is at least $O(\log(\log(p)))$ .

5. Significance

Resolution of Classification Limits: The paper definitively answers that a "tame" classification of all $\mathbb{Z}_p$ -invariant simple singularities (with a uniform bound on corank) is impossible. The complexity of the classification grows with the prime order of the group.
Refinement of Stability Theory: By proving that $\det(\tau)=1$ generally precludes simple singularities (except for real actions), the paper clarifies the geometric constraints required for stability in the equivariant setting.
Interdisciplinary Connection: The work successfully bridges singularity theory with analytic number theory, using the distribution of prime factors to construct geometric counterexamples to boundedness conjectures.
Foundation for Future Work: By narrowing the existence conditions (Theorem 1.2) and establishing the growth rate of corank (Theorem 1.3), the paper sets the stage for a more nuanced classification that accounts for the dependence of the corank on the prime $p$ .

In summary, Proskurnin demonstrates that while simple singularities are rare for small primes or specific determinant conditions, they become increasingly complex (higher corank) as the prime $p$ grows, fundamentally altering the expectations for the classification of equivariant singularities.