Finite Generation in Polynomial Semirings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914622631575552 |
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| author | Asal, Mohammad El Mahboub, Wael |
| author_facet | Asal, Mohammad El Mahboub, Wael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_11569 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite Generation in Polynomial Semirings Asal, Mohammad El Mahboub, Wael Commutative Algebra 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. |
| title | Finite Generation in Polynomial Semirings |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2604.11569 |