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

Abstract

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1<n_1\le \dots \le n_k$ 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 $n_1,\dots ,n_k$. 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.