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

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 $F\in \mathrm{Int}(D)$ 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.

References in corpus (6)

• A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
• Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields
• Absolute irreducibility of the binomial polynomials
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• Non-absolutely irreducible elements in the ring of Integer-valued polynomials
• Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains