Explicit Hilbert's Irreducibility Theorem in Function Fields

2019/12/11 by Bary-Soroker, Lior, Entin, Alexei
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.05162

Abstract

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If f(T₁,…, T_n,X) is an irreducible polynomial over the field of rational functions over a finite field \mathbbF_q of characteristic p, then the proportion of n-tuples (t₁,…, t_n) of monic polynomials of degree d for which f(t₁,…, t_n,X) is reducible out of all n-tuples of degree d monic polynomials is O(dq^-d/2).

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