A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

TL;DR

This paper presents a method to construct integer-valued polynomials with any prescribed finite set of factorization lengths, advancing the understanding of factorization properties in integer-valued polynomial rings.

Contribution

It introduces a construction technique for integer-valued polynomials with arbitrary finite sets of factorization lengths, filling a gap in the factorization theory of these polynomials.

Findings

Constructed polynomials with prescribed length sets

Demonstrated the flexibility of factorization lengths in Int(Z)

Extended the understanding of factorization structures in integer-valued polynomials

Abstract

For an arbitrary finite set S of natural numbers greater 1, we construct an integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of lengths of f is the set of all natural numbers n, such that f has a factorization as a product of n irreducibles in Int(Z)={g in Q[x] | g(Z) contained in Z}.