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

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