Integer-valued polynomials on 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 $1<n_1\le \cdots \le n_k$ there exists an integer-valued polynomial on V, that is, an element of $\text{Int}\left(V\right)=\{f\in K[X]\mid f(V)\subseteq V\}$, which has precisely k essentially different factorizations into irreducible elements of $\text{Int}\left(V\right)$ 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.