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 {{,textrm{Int},}}(V) = { f in K[X] mid f(V) subseteq V }, which has precisely k essentially different factorizations into irreducible elements of {{,textrm{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.