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

Abstract

AbstractLet V be a valuation ring of a global field K. We show that for all positive integers k and $$1 < n_1 \le \cdots \le n_k$$ 1 < n 1 ≤ ⋯ ≤ n k there exists an integer-valued polynomial on V, that is, an element of $${{\,\textrm{Int}\,}}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}$$ Int ( V ) = { f ∈ K [ X ] ∣ f ( V ) ⊆ V } , which has precisely k essentially different factorizations into irreducible elements of $${{\,\textrm{Int}\,}}(V)$$ Int ( V ) whose lengths are exactly $$n_1,\ldots ,n_k$$ n 1 , … , 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.