Effective Hilbert's Irreducibility Theorem for global fields

TL;DR

This paper establishes an effective version of Hilbert's irreducibility theorem for polynomials over global fields, providing explicit bounds on specializations that preserve irreducibility and Galois groups.

Contribution

It introduces explicit bounds for specializations in global fields, unifying the function field and number field cases with optimal bounds based on polynomial parameters.

Findings

Provides explicit bounds depending on polynomial height and degree

Unifies treatment of function field and number field cases

Bounds are optimal relative to the size of the specialization parameter

Abstract

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field $K$. More precisely, we give effective bounds for the number of specializations $t\in \mathcal{O}_K$ that do not preserve the irreducibility or the Galois group of a given irreducible polynomial $F(T,Y)\in K[T,Y]$. The bounds are explicit in the height and degree of the polynomial $F(T,Y)$, and are optimal in terms of the size of the parameter $t\in \mathcal{O}_K$. Our proofs deal with the function field and number field cases in a unified way.