Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power

TL;DR

This paper characterizes the fixed divisor of polynomials over integers by examining their relation to overring ideals, providing a detailed description of polynomials with fixed divisors divisible by prime powers.

Contribution

It offers a novel characterization of fixed divisors through overring contractions and describes the primary decomposition of related ideals for prime power divisibility.

Findings

Complete description of polynomials with fixed divisor divisible by p^n

Characterization of fixed divisors via overring contractions

Primary decomposition of related ideals

Abstract

We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.