Finite Generation in Polynomial Semirings

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.