Factorization of integer-valued polynomials with square-free denominator

TL;DR

This paper presents an algorithm for factoring integer-valued polynomials with square-free denominators by transforming the problem into a combinatorial one, enabling the computation of essentially different factorizations.

Contribution

It introduces a novel algorithm that reduces the factorization of such polynomials to a combinatorial problem, assuming known factorizations of numerator and denominator.

Findings

Algorithm effectively computes different factorizations.

Transforms algebraic problem into combinatorial framework.

Applicable to primitive integer-valued polynomials with square-free denominators.

Abstract

We describe an algorithm to compute the essentially different factorizations of a given image primitive integer-valued polynomial $f(X)=g(X)/d\in\Q[X]$, where $g\in\Z[X]$ and $d\in\N$ is square-free, assuming that the factorization of $g(X)$ in $\Z[X]$ and $d$ in $\Z$ is known. We translate this problem into a combinatorial one.