Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely
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arXiv
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| Format: | Preprint |
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2024
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| author | Nakato, Sarah Rissner, Roswitha |
| author_facet | Nakato, Sarah Rissner, Roswitha |
| contents | We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator, already the third power has to factor non-uniquely or the element is absolutely irreducible. Recently, it has been shown that for any $N\in\mathbb{N}$, there exists a discrete valuation domain $D$ and a polynomial $F\in\operatorname{Int}(D)$ such that the minimal $k$ for which $F^k$ factors non-uniquely is greater than $N$.
In this paper, we show that, over principal ideal domains with infinitely many maximal ideals of finite index, the minimal power for which an irreducible but not absolutely irreducible element has to factor non-uniquely depends on the $p$-adic valuations of the denominator and cannot be bounded by a constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely Nakato, Sarah Rissner, Roswitha Commutative Algebra 13A05, 11R09, 13B25, 13F20, 11C08 We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator, already the third power has to factor non-uniquely or the element is absolutely irreducible. Recently, it has been shown that for any $N\in\mathbb{N}$, there exists a discrete valuation domain $D$ and a polynomial $F\in\operatorname{Int}(D)$ such that the minimal $k$ for which $F^k$ factors non-uniquely is greater than $N$. In this paper, we show that, over principal ideal domains with infinitely many maximal ideals of finite index, the minimal power for which an irreducible but not absolutely irreducible element has to factor non-uniquely depends on the $p$-adic valuations of the denominator and cannot be bounded by a constant. |
| title | Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely |
| topic | Commutative Algebra 13A05, 11R09, 13B25, 13F20, 11C08 |
| url | https://arxiv.org/abs/2405.16131 |