Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains

TL;DR

This paper studies the structure of factorizations in rings of integer-valued polynomials over prime ideals of principal ideal domains, explicitly constructing elements with prescribed factorization lengths and analyzing their algebraic properties.

Contribution

It provides explicit constructions of elements with specified sets of factorization lengths in rings of integer-valued polynomials on prime ideals of PIDs, and shows these rings are not transfer Krull domains.

Findings

Constructed elements with arbitrary sets of factorization lengths.

Demonstrated non-transfer Krull domain property.

Extended study to rings of integer-valued polynomials on infinite subsets.

Abstract

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 $D$.