Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains

Let [Formula: see text] be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal [Formula: see text] of [Formula: see text], the residue field [Formula: see text] is finite. Let [Formula: see text] be the quotient field of [Formula: see text]. We investigate sets of lengths in the ring of integer-valued polynomials on [Formula: see text], [Formula: see text]. For every multiset of integers [Formula: see text], we explicitly construct an element of [Formula: see text] with exactly [Formula: see text] essentially different factorizations into irreducible elements of [Formula: see text] whose lengths are [Formula: see text]. Furthermore, we show that [Formula: see text] is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings [Formula: see text], where [Formula: see text] is an infinite subset of [Formula: see text].