Finite Generation in Polynomial Semirings

Abstract

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