Prime numbers and factorization of polynomials

TL;DR

This paper develops upper bounds on the number of irreducible factors of certain integer polynomials, extending irreducibility criteria using prime factorization and root location, including bivariate cases over arbitrary fields.

Contribution

It introduces new bounds on polynomial irreducibility and extends existing results to bivariate polynomials over arbitrary fields using non-Archimedean valuations.

Findings

Derived upper bounds on irreducible factors of integer polynomials.

Extended irreducibility criteria to bivariate polynomials over arbitrary fields.

Utilized prime factorization and root location techniques in proofs.

Abstract

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we use the information about prime factorization of the values taken by such polynomials at sufficiently large integer arguments along with the information about their root location in the complex plane. Further, these techniques are extended to bivariate polynomials over arbitrary fields using non-Archimedean absolute values, yielding extensions of the irreducibility results of M. Ram Murty and S. Weintraub to bivariate polynomials.