[Paper Review] Improved bounds on the number of numerical semigroups of a given genus

Sergi Elizalde|ArXiv.org|May 4, 2009

Commutative Algebra and Its Applications5 references3 citations

TL;DR

This paper improves the best-known lower and upper bounds for the number of numerical semigroups of genus $ g $, using a generating tree structure and functional equations derived from succession rules. It establishes tighter bounds involving Fibonacci numbers and generating functions, and provides an asymptotic upper bound of $ oxed{ ext{limsup } m_g^{1/g} o ext{sqrt}(2)} $, with a conjectured tighter bound of $ ext{sqrt}( ext{phi}) ext{ if } n_g ext{ grows like Fibonacci numbers}.

ABSTRACT

We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use them to construct approximations of T by generating trees whose nodes are labeled by certain parameters of the semigroups. We then translate the succession rules of these trees into functional equations for the generating functions that enumerate their nodes, and solve these equations to obtain the bounds. Some of our bounds involve the Fibonacci numbers, and the others are expressed as generating functions. We also give upper bounds on the number of numerical semigroups having an infinite number of descendants in T.

Motivation & Objective

• To improve the previously known lower and upper bounds for the number $ n_g $ of numerical semigroups of genus $ g $, particularly for large $ g $.
• To analyze the generating tree $ frac{\mathcal{T}}{} $ of numerical semigroups by studying its structural properties and succession rules.
• To derive functional equations for generating functions enumerating nodes in approximated subtrees of $ \mathcal{T} $, enabling precise bounds on $ n_g $.
• To provide upper bounds on the number of semigroups with infinitely many descendants in $ \mathcal{T} $, using gcd-based characterization from prior work.
• To refine asymptotic estimates for $ m_g $, the number of semigroups with infinite chains, under the conjecture that $ n_g $ grows like Fibonacci numbers.

Proposed method

• Constructs approximations of the generating tree $ \mathcal{T} $ by labeling nodes with semigroup parameters (e.g., multiplicity, effective generators) and deriving succession rules.
• Translates the succession rules of these approximated trees into functional equations for generating functions that enumerate their nodes.
• Solves the resulting functional equations using algebraic manipulation, including kernel method techniques and variable substitutions (e.g., $ w = uv^2 $).
• Uses the characterization from [3] that a semigroup has infinitely many descendants iff $ \gcd(\lambda_0, \dots, \lambda_{f-g}) \neq 1 $, to bound the number of such semigroups.
• Applies injective mapping from infinite-descendant semigroups to smaller semigroups and divisors to derive cardinality bounds.
• Combines bounds from generating functions and gcd-based counting to derive asymptotic limits for $ m_g^{1/g} $.

Experimental results

Research questions

• RQ1 What are the improved lower and upper bounds for $ n_g $, the number of numerical semigroups of genus $ g $, especially for large $ g $?
• RQ2 Can the generating tree structure of numerical semigroups be leveraged to derive functional equations that yield tighter bounds on $ n_g $?
• RQ3 What is the asymptotic growth rate of $ m_g $, the number of numerical semigroups with infinitely many descendants in the tree $ \mathcal{T} $?
• RQ4 How does the conjectured Fibonacci-like growth of $ n_g $ affect the asymptotic upper bound on $ m_g $?
• RQ5 Can the number of semigroups with infinite chains be bounded using the gcd condition on small elements below the Frobenius number?

Key findings

• The paper improves the lower bound for $ n_g $ to $ 2F_g $, where $ F_g $ is the $ g $-th Fibonacci number, and provides a new upper bound expressed as a generating function.
• A generating function for the number of semigroups with infinite chains is derived, leading to the bound $ m_g eq 1 + (g-1) imes \sum_{i=0}^{\lfloor(g-1)/2\rfloor} n_i $, which is used to analyze asymptotic growth.
• The asymptotic upper bound $ \limsup_{g \to \infty} m_g^{1/g} \leq \sqrt{2} $ is established, improving upon previous estimates.
• Under the conjecture that $ \lim_{g \to \infty} n_{g+1}/n_g = \phi $, the bound tightens to $ \limsup_{g \to \infty} m_g^{1/g} \leq \sqrt{\phi} \approx 1.27201965 $.
• The functional equation for the generating function of the tree nodes is solved via kernel method, yielding $ \sum_{g \geq 1} d_g t^g = t + t^3 + \sum_{g \geq 1} 2F_{g-1} t^g $, which gives exact counts for small $ g $.
• For $ 1 \leq g \leq 4 $, the derived bound $ d_g = g $ matches the true number of semigroups $ m_g $, validating the method on small cases.