Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations

TL;DR

This paper constructs integer-valued polynomials over valuation rings of global fields with prescribed lengths of factorizations, solving an open problem in the structure of factorizations in these rings.

Contribution

It demonstrates the existence of polynomials with exactly specified factorization lengths in valuation rings of global fields, extending previous results and solving an open problem.

Findings

Existence of polynomials with prescribed factorization lengths

Applicable to valuation rings with finite residue fields

Solves an open problem in factorization theory

Abstract

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}$, which has precisely $k$ essentially different factorizations into irreducible elements of $\text{Int}(V)$ whose lengths are exactly $n_1,\ldots,n_k$. In fact, we show more, namely that the same result holds true for every discrete valuation domain $V$ with finite residue field such that the quotient field of $V$ admits a valuation ring independent of $V$ whose maximal ideal is principal or whose residue field is finite. If the quotient field of $V$ is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.