Finite Generation in Polynomial Semirings

Abstract

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