Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915876120297472 |
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| author | Kansiime, Zaituni Luambano, Sholastica Nakato, Sarah Nalule, Hadijah Ndayikunda, Yvette |
| author_facet | Kansiime, Zaituni Luambano, Sholastica Nakato, Sarah Nalule, Hadijah Ndayikunda, Yvette |
| contents | Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03246 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains Kansiime, Zaituni Luambano, Sholastica Nakato, Sarah Nalule, Hadijah Ndayikunda, Yvette Commutative Algebra 13A05, 13B25, 13F20, 11R09, 11C08 Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$. |
| title | Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains |
| topic | Commutative Algebra 13A05, 13B25, 13F20, 11R09, 11C08 |
| url | https://arxiv.org/abs/2601.03246 |