Families of polynomials and their specializations

TL;DR

This paper explores how families of multivariable polynomials generally remain irreducible when parameters vary, revisiting classical results by Grothendieck and Gao from this perspective.

Contribution

It provides a unified framework for understanding irreducibility of parameterized polynomials, extending and recasting previous classical results.

Findings

Most parameter choices yield irreducible polynomials.

Reinterpretation of Grothendieck and Gao's results.

Framework applicable to various polynomial families.

Abstract

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results of Grothendieck and of Gao.