Finite Generation in Polynomial Semirings

TL;DR

This paper characterizes finite generation in polynomial semirings over algebraic numbers, linking it to properties of minimal polynomials and weak Perron numbers, with applications to cubic polynomials.

Contribution

It provides a complete characterization of finite generation for certain polynomial semirings and shows that such generation implies the algebraic number is a weak Perron number.

Findings

Finite generation depends on divisibility of minimal polynomials.

Finite generation implies the algebraic number is a weak Perron number.

Partial classification of rank-3 monoids based on generation and factorization.

Abstract

We study the semiring $\mathbb{N}_0[\alpha]$ as an additive monoid where $\alpha$ is a positive real algebraic number. In the atomic case, the atoms of $\mathbb{N}_0[\alpha]$ are precisely the powers $\alpha^n$ up to a certain nonnegative integer $n$, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form $\mathfrak{m}_\alpha(X)=p_\alpha(X)-c$ with $c\in\mathbb{N}$. Our second main result shows that finite generation forces $\alpha$ to be a weak Perron number. As an application, we analyze cubic minimal polynomials and obtain a partial classification of rank-$3$ monoids $\mathbb{N}_0[\alpha]$ by generation and factorization type, including coefficient constraints, non--length-factoriality results for a large family, and examples with prescribed numbers of atoms.