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.