A new preprint by Minglang Li and Yizhi Zhang settles an open problem in the theory of numerical semigroups by completely characterizing which pairs of positive integers can appear as the embedding dimension and multiplicity of a complete intersection numerical semigroup.
The Problem
A numerical semigroup is a cofinite additive submonoid of the non-negative integers containing 0. Two fundamental invariants are:
- The embedding dimension e(Γ): the number of elements in the minimal generating set
- The multiplicity m(Γ): the smallest positive element in the semigroup
A numerical semigroup is called a complete intersection if its semigroup ring k[[Γ]] is a complete intersection ring — equivalently, if the number of relations in a minimal presentation equals e(Γ) − 1. Complete intersections are built recursively through a construction called gluing.
Moscariello and Sammartano posed the problem of characterizing all realizable pairs (e, m): for which positive integers e and m does there exist a complete intersection numerical semigroup with embedding dimension e and multiplicity m?
The Answer
Li and Zhang prove a remarkably clean characterization:
A pair (e, m) is realizable if and only if (e, m) = (1, 1) or e ≥ 2 and m ≥ 2^(e−1).
This means the set of realizable pairs is exactly:
{(1, 1)} ∪ {(e, m) ∈ ℤ² : e ≥ 2, m ≥ 2^(e−1)}
The lower bound m ≥ 2^(e−1) was known to be sharp — it is attained by certain semigroups — but the contribution of this paper is showing that every multiplicity at least this large is achievable.
Proof Strategy
The proof has two parts:
- Necessity: The bound m(Γ) ≥ 2^(e(Γ)−1) for complete intersections follows from classical results.
- Sufficiency: The authors construct a family of "covering" semigroups Cₑ(x) that serve as building blocks. For each e ≥ 2 and x ≥ 2ᵉ, they produce a complete intersection of embedding dimension e and multiplicity 2^(e−1) where x is not a minimal generator. Using gluing, they then build semigroups with any desired multiplicity m ≥ 2^(e−1).
Comparison with Symmetric Semigroups
The result contrasts with the more complex situation for symmetric semigroups. As noted in the paper, symmetric semigroups with given (e, m) exist if and only if 2 ≤ e ≤ m − 1 or (e, m) ∈ {(1, 1), (2, 2)} — a characterization that involves a linear constraint rather than an exponential one.
The simplicity of the complete intersection answer — once m is at least 2^(e−1), all larger multiplicities are realizable — reflects the rigid structure imposed by the complete intersection property.
Publication Details
The paper "A Complete Characterization of Realizable (Embedding Dimension, Multiplicity) Pairs for Complete Intersection Numerical Semigroups" was posted to arXiv on August 21, 2026. It builds on the theory of gluings developed by Delorme and others, and answers Question 7.5 from a 2024 survey on numerical semigroup invariants.
