Gaps of Binary Numerical Semigroups and of Binary Inclusion-Exclusion Polynomials

TL;DR

This paper analyzes the gaps in binary inclusion-exclusion polynomials and numerical semigroups, providing complete descriptions of their structures and properties.

Contribution

It offers a detailed analysis of linear permutations of residue systems and applies this to characterize gaps in binary inclusion-exclusion polynomials and distances in numerical semigroups.

Findings

Complete description of gapsets of binary inclusion-exclusion polynomials

Characterization of all possible distances between consecutive elements in numerical semigroups

Analysis of dominant pairs in linear permutations of residue systems

Abstract

Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that a pair of integers $(a,b)$ is a dominant pair of this permutation if either the inequality $\max(\langle ua\rangle_p,\langle ub\rangle_p)<\min_{a<n<b}\langle un\rangle_p$, or the inequality $\min(\langle ua\rangle_p,\langle ub\rangle_p)>\max_{a<n<b}\langle un\rangle_p$ hold. The main technical part of this work gives analysis of this property of linear permutations of residue systems. We then apply this analysis to the problems that motivated it, and give (i) complete description of the gapsets of binary inclusion-exclusion polynomials $Q_{\{p,q\}}$ (which include binary cyclotomic polynomials $\Phi_{pq}$ as its principal special case), and (ii) complete description of all possible distances between consecutive elements of a numerical semigroup $\langle p,q\rangle$.