Bounds for Hilbert's Irreducibility Theorem

Abstract

In the context of Hilbert’s irreducibility theorem, it is an open question whether there exists a bound for the least hilbertian specialization in N that is polynomial in the degree d and the logarithmic height log(H) of the polynomial P (T,Y ) in question. A positive answer would be useful, notably for algorithmic applications. We obtain a polynomial bound in log(H) and dHi(P) where Hi(P) — the Hilbert index of P — is a pure group-theoretical invariant we define and which we show to be absolutely bounded for many classes of polynomials. We also discuss further questions related to effectiveness in Hilbert’s irreducibility theorem. 2000 MSC. Primary 12E25, 14G05 ; Secondary 11C08, 12E05