Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely

TL;DR

This paper investigates the minimal power at which irreducible integer-valued polynomials, not absolutely irreducible, must factor non-uniquely, revealing dependence on p-adic valuations and showing no universal bound exists.

Contribution

It demonstrates that over certain principal ideal domains, the minimal power for non-unique factorization depends on p-adic valuations and cannot be universally bounded.

Findings

The minimal power depends on p-adic valuations of the denominator.

No universal bound exists for the minimal power in certain domains.

Factorization non-uniqueness occurs at arbitrarily high powers.

Abstract

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.