[Paper Review] Numerical semigroups, cyclotomic polynomials and Bernoulli numbers

Pieter Moree|arXiv (Cornell University)|Aug 19, 2013

Analytic Number Theory Research10 references3 citations

TL;DR

This paper establishes a deep connection between numerical semigroups, binary cyclotomic polynomials, and Bernoulli numbers by proving that the semigroup polynomial of a numerical semigroup generated by two coprime integers $ p $ and $ q $ equals the cyclotomic polynomial $ \Phi_{pq}(x) $. It provides a conceptual proof of the gap structure in $ \Phi_{pq}(x) $, showing that the maximum gap between non-zero monomial exponents is $ \min\{p,q\} - 1 $, and generalizes this to arbitrary numerical semigroups using inclusion-exclusion polynomials and the LLL-diagram framework.

ABSTRACT

We give two proofs of a folkore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences. In particular, we give a more conceptual reproof of a result of Hong et al. (2012) on gaps between the exponents of non-zero monomials in a binary cyclotomic polynomial. The intent of the author with this expositional paper is to better unify the various results within the cyclotomic polynomial and numerical semigroup communities.

Motivation & Objective

• To unify results across numerical semigroups and cyclotomic polynomials by proving a folklore result on binary cyclotomic polynomials.
• To provide a conceptual proof of the gap structure in $ \Phi_{pq}(x) $, specifically that $ g(\Phi_{pq}) = p-1 $ for odd primes $ p < q $.
• To generalize the gap structure result to arbitrary numerical semigroups using inclusion-exclusion polynomials and the LLL-diagram.
• To explore connections between cyclotomic polynomial coefficients and Bernoulli numbers through recurrence relations.
• To demonstrate how the semigroup polynomial $ P_S(x) $ encodes Frobenius number and gap information via $ g(P_S(x)) = m(S) - 1 $.

Proposed method

• Uses the semigroup polynomial $ P_S(x) = (1 - x)H_S(x) $, where $ H_S(x) $ is the Hilbert series of a numerical semigroup $ S $.
• Applies the inclusion-exclusion construction to define generalized cyclotomic-like polynomials $ Q_\rho(x) $, with $ Q_{\{p,q\}}(x) = \frac{(x^{pq}-1)(x-1)}{(x^p-1)(x^q-1)} $.
• Establishes that $ P_{S(p,q)}(x) = \Phi_{pq}(x) $ when $ p $ and $ q $ are distinct primes, linking numerical semigroups to cyclotomic polynomials.
• Uses the LLL-diagram to visualize and analyze the distribution of gaps and elements in $ S(p,q) $, particularly in the range $ 0 $ to $ pq-1 $.
• Applies coefficient-level analysis of $ P_S(x) $ to show that non-zero coefficients alternate between $ 1 $ and $ -1 $, leading to gap block structure.
• Derives a recurrence for Bernoulli numbers using the structure of $ S(4,7) $, linking semigroup gaps to number-theoretic identities.

Experimental results

Research questions

• RQ1 What is the precise relationship between the semigroup polynomial of a numerical semigroup $ S(p,q) $ and the cyclotomic polynomial $ \Phi_{pq}(x) $?
• RQ2 Why does the maximum gap between non-zero monomial exponents in $ \Phi_{pq}(x) $ equal $ \min\{p,q\} - 1 $?
• RQ3 How can the LLL-diagram be used to systematically analyze the structure of $ S(p,q) $ and its associated polynomial?
• RQ4 Can the coefficient structure of $ P_S(x) $ be used to derive new identities for Bernoulli numbers?
• RQ5 What is the generalization of the gap structure in $ \Phi_n(x) $ to arbitrary numerical semigroups?

Key findings

• The semigroup polynomial $ P_{S(p,q)}(x) $ equals $ \Phi_{pq}(x) $ when $ p $ and $ q $ are distinct primes, establishing a direct link between numerical semigroups and cyclotomic polynomials.
• The maximum gap between non-zero monomial exponents in $ \Phi_{pq}(x) $ is $ \min\{p,q\} - 1 $, confirming a result of Hong et al. with a conceptual proof via numerical semigroups.
• The gap structure of $ P_S(x) $ satisfies $ g(P_S(x)) = m(S) - 1 $, where $ m(S) $ is the multiplicity of the semigroup $ S $.
• The non-zero coefficients of $ Q_{\{p,q\}}(x) $ alternate between $ 1 $ and $ -1 $, reflecting the alternating block structure of elements and gaps in the LLL-diagram.
• The number of element blocks and gap blocks in $ S(p,q) $ is $ \rho\sigma - 1 $, where $ \rho $ and $ \sigma $ are the number of gaps and elements in the fundamental domain.
• A recurrence for Bernoulli numbers is derived using the semigroup $ S(4,7) $, with coefficients corresponding to integers not in $ S(4,7) $, linking semigroup gaps to number-theoretic identities.