On irreducible factors of polynomials over integers

TL;DR

This paper develops new methods for analyzing the irreducibility of integer polynomials by examining prime factorizations and derivatives at large integers, providing explicit lower degree bounds for factors.

Contribution

It introduces novel factorization results for specific polynomial classes using prime factorization and derivatives, with explicit bounds derived via Newton polygons.

Findings

New factorization results for integer polynomials

Explicit lower degree bounds for factors

Application of Newton polygons for bounds

Abstract

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their higher order formal derivatives at sufficiently large integer arguments. If a lower bound for the minimum possible degree of a factor of such a polynomial is known a priori, then the integer argument becomes significantly smaller, which makes the underlying factorization result easier to apply. A result on explicit lower degree factor bound for the classes of polynomials considered in this paper is also proved via Newton polygons.