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On numerical semigroups with embedding dimension four

This paper introduces a geometric procedure for determining the Apéry set of numerical semigroups with embedding dimension four, which is then applied to compute key invariants such as Frobenius numbers and Betti elements for semigroups generated by four consecutive squares and four consecutive triangular numbers.

Original authors: Kazimierz Chomicz

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kazimierz Chomicz

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 have a magical vending machine that only accepts specific types of coins. Let's say you only have coins worth 4, 6, and 9 cents. You can buy items that cost 4, 6, 8 (4+4), 9, 10 (4+6), 12, 13 (4+9), and so on. But no matter how you combine your coins, you can never make exactly 1, 2, 3, 5, or 7 cents.

In the world of mathematics, this collection of "buyable" amounts is called a Numerical Semigroup. The "missing" amounts are the gaps. The Frobenius number is simply the price of the most expensive item you cannot buy. In our example, that would be 7 cents.

This paper, written by a high school student named Kazimierz Chomicz, tackles a very specific and tricky version of this problem: What happens when your vending machine accepts four specific types of coins, and those coins follow a very neat pattern? Specifically, the author looks at two patterns:

Four consecutive squares: Like 1, 4, 9, 16 (or 100, 121, 144, 169).
Four consecutive triangular numbers: Like 1, 3, 6, 10 (or 15, 21, 28, 36).

For a long time, mathematicians knew how to solve this "missing price" problem if you had only two or three types of coins. But once you hit four, it gets incredibly messy, and for a long time, no one had a general recipe to find the answer.

The Main Tool: The "3D Lego Castle"

To solve this, the author invents a visual method. Imagine you are building a castle out of 3D Lego blocks in a corner of a room.

Each block represents a way to combine your coins.
The author builds a giant, infinite castle.
Then, he starts "demolishing" specific sections of the castle based on mathematical rules. He cuts away huge chunks of the structure that represent combinations that are too big or redundant.

After all the demolition, what's left is a specific, oddly shaped structure that looks a bit like the letter L (or a set of stairs). The author proves that the "height" of the highest block in this remaining L-shaped castle tells you exactly what the Frobenius number (the most expensive unbuyable item) is.

This "L-shape" is the key. It's like a map that tells you exactly which numbers you can and cannot make. If you can count the blocks in this L-shape correctly, you can calculate not just the most expensive unbuyable item, but also:

The Genus: The total number of "missing prices" (how many items you can't buy at all).
The Catenary Degree: A measure of how "confusing" the ways to make a price are. If you can make a price of 100 cents in many different ways (e.g., 25+25+25+25 or 40+60), this number tells you how hard it is to switch from one combination to another without getting stuck.
Minimal Presentations: The absolute shortest list of "rules" needed to describe how all these coins interact.

The Results: Finding the Patterns

The author didn't just build the castle; he calculated the exact dimensions for every possible starting number. He found that the answer depends on what the starting number is when you divide it by 12 (for squares) or 6 (for triangular numbers).

Think of it like a weather forecast. If you start with a number that leaves a remainder of 0 when divided by 12, the "Frobenius number" follows one specific formula. If it leaves a remainder of 1, it follows a slightly different formula.

The paper provides these exact formulas for:

Frobenius Numbers: The exact price of the most expensive unbuyable item for any set of four consecutive squares or triangular numbers.
Genus: The exact count of all the unbuyable prices.
Catenary Degree: How complex the combinations are.
Minimal Presentations: The exact number of rules needed to describe the system.

Why This Matters (According to the Paper)

The author notes that while we can solve this for four coins, trying to do it for five or six coins (an infinite sequence) is much harder. He proves that for an infinite sequence of squares, the "missing price" grows too fast to be described by a simple quadratic formula (like $n^2$ ). It grows faster, like $n^{2.5}$ or higher.

The "High School" Twist

Perhaps the most surprising part of the paper is the author's bio. Kazimierz Chomicz was a high school student when he wrote this. He used computer tools (like Mathematica and GAP) to help verify his complex 3D geometric calculations, but the core logic and the geometric "demolition" method were his own.

In summary: The paper takes a notoriously difficult math puzzle (finding the "missing price" for four specific types of coins) and solves it by turning the problem into a 3D geometry game. By carving out a specific "L-shaped" castle from a block of infinite possibilities, the author provides a complete recipe to calculate every important number associated with these specific coin sets.

1. Problem Statement

The paper addresses the Frobenius problem for numerical semigroups with embedding dimension four ( $e(S)=4$ ).

Context: For a numerical semigroup $S = \langle d_0, d_1, d_2, d_3 \rangle$ , the Frobenius number $F(S)$ is the largest integer not in $S$ . While explicit formulas exist for $e(S)=2$ (Sylvester's formula) and various special cases for $e(S)=3$ , no general polynomial formula exists for $e(S) \ge 3$ (proven to be NP-hard).
Specific Goal: The author aims to compute the Frobenius number, genus, Betti elements, minimal presentations, and catenary degrees for two specific families of semigroups:
1. Semigroups generated by four consecutive squares: $S = \langle n^2, (n+1)^2, (n+2)^2, (n+3)^2 \rangle$ .
2. Semigroups generated by four consecutive triangular numbers: $S = \langle \binom{n+1}{2}, \binom{n+2}{2}, \binom{n+3}{2}, \binom{n+4}{2} \rangle$ .

2. Methodology

The core of the paper is a geometric procedure to determine the Apéry set ( $Ap(S, d_0)$ ) of a numerical semigroup with embedding dimension four. The Apéry set is crucial because $F(S) = \max(Ap(S, d_0)) - d_0$ and the genus $G(S)$ can be derived from the sum of its elements.

A. Geometric Construction (Section 2)

Lattice Representation: The method visualizes the semigroup in $\mathbb{R}^3$ using unit cubes labeled by values $id_1 + jd_2 + kd_3$ .
Minimal Relations: The author defines "minimal relations" (e.g., $a_{00}d_0 = a_{01}d_1 + a_{02}d_2 + a_{03}d_3$ ) and constructs specific points in $\mathbb{Z}^3$ based on the coefficients of these relations.
Region Deletion: The procedure involves deleting specific regions (orthants) from the initial collection of cubes associated with these points.
- Theorem 2.2: Provides a rigorous algorithm for deleting regions associated with minimal relations and specific identities to isolate the Apéry set.
- L-shapes: The goal is to reduce the remaining collection of cubes to an L-shape (a specific geometric configuration where every element of the Apéry set labels exactly one cube).
- Proposition 2.5 & 2.8: Establish conditions under which the resulting figure is guaranteed to be an L-shape. This relies on the equality of coefficients in minimal relations ( $a_{ii} = b_{ii}$ ).
Uniqueness: The paper discusses the uniqueness of these L-shapes (Proposition 2.10), linking it to the structure of the minimal relations.

B. Factorization Theory (Section 3)

Once the Apéry set (as an L-shape) is identified, the author uses it to compute factorization invariants:

Betti Elements: Identified as elements in the Apéry set that correspond to specific "corners" or relations in the geometric figure.
Catenary Degree ( $c(S)$ ): Calculated by analyzing the distances between different factorizations of the Betti elements.
Minimal Presentations: Constructed using the sets of factorizations of the Betti elements (Theorem 3.2).

3. Key Contributions

General Geometric Algorithm: The paper proposes a modified, generalizable geometric procedure to find the Apéry set for any numerical semigroup with embedding dimension four, overcoming limitations of previous methods (like Killingbergtrø's) that relied on specific properties (e.g., telescopic sequences).
Explicit Formulas for Special Families: The author successfully applies this method to derive closed-form polynomial formulas for the invariants of semigroups generated by four consecutive squares and triangular numbers.
Connection to L-shapes: The work rigorously connects the geometric shape of the Apéry set (specifically the L-shape property) to the algebraic properties of the semigroup, such as the existence of minimal presentations and the calculation of the catenary degree.

4. Detailed Results

A. Four Consecutive Squares ( $S = \langle n^2, (n+1)^2, (n+2)^2, (n+3)^2 \rangle$ )

The results depend on $n \pmod{12}$ (and $n \pmod 4$ for specific cases).

Frobenius Number ( $F(S)$ ): Derived as a cubic polynomial in $n$ (e.g., for $n \equiv 0 \pmod{12}$ , $F(S) = \frac{11}{12}n^3 + 4n^2 + \frac{9}{2}n - 2$ ).
Genus ( $G(S)$ ): Derived as a cubic polynomial in $n$ .
Catenary Degree ( $c(S)$ ): Found to be linear in $n$ (e.g., $c(S) = \frac{11}{12}n + 2$ for $n \equiv 0 \pmod{12}$ ).
Minimal Presentation: The cardinality of the minimal presentation is linear in $n$ (e.g., $\frac{1}{3}n + 6$ ).
Betti Elements: The number of Betti elements is explicitly calculated (e.g., $4k+6$ for $n=12k$ ).

B. Four Consecutive Triangular Numbers ( $S = \langle \binom{n+1}{2}, \dots, \binom{n+4}{2} \rangle$ )

The results depend on $n \pmod 6$ .

Frobenius Number: Derived as a cubic polynomial in $n$ (e.g., for $n \equiv 0 \pmod 6$ , $F(S) = \frac{1}{3}n^3 + \frac{11}{6}n^2 + \frac{5}{2}n - 1$ ).
Genus: Derived as a cubic polynomial in $n$ .
Catenary Degree: Linear in $n$ (e.g., $c(S) = \frac{2}{3}n + 1$ for $n \equiv 0 \pmod 6$ ).
Minimal Presentation: Cardinality is linear in $n$ (e.g., $\frac{1}{3}n + 3$ ).

Note: The author used Mathematica for identity discovery and the numericalsgps GAP package for verification.

5. Significance and Future Work

Asymptotic Behavior: The paper discusses the asymptotic behavior of the Frobenius number for infinite sequences of powers. It proves that for $k$ consecutive squares, the Frobenius number grows as $O(n^{2 + 2/k})$ , meaning it cannot be a quadratic polynomial for $k \ge 3$ . This contrasts with the cubic results found for $k=4$ in this paper.
Generalization: The author highlights that the current method relies on the condition $a_{ii} = b_{ii}$ (Proposition 2.5). A significant open problem proposed is to extend the procedure to cases where $a_{ii}d_i = a_{jj}d_j$ (non-telescopic cases).
Tame Degree: The paper identifies the tame degree as a more complex invariant that remains to be systematically computed for these families using the developed geometric framework.

In summary, this paper provides a powerful geometric toolkit for analyzing numerical semigroups of embedding dimension four, successfully solving the Frobenius problem and related factorization invariants for two important families of generators, and establishing a framework for future research into more general semigroup structures.