Bounds for Hilbert's irreducibility theorem
Hilbert's irreducibility theorem asserts that for every irreducible polynomial \(f(T,X)\) in the ring \(\mathbb{Q}[T,X]\) with \(\deg_X(f)\geq 1\) there exists an irreducible specialization in \(\mathbb{N}\), i.e., a positive integer \(t\) such that \(f(t,X)\) is irreducible in \(\mathbb{Q}[X]\). It is important for applications to find effective versions of Hilbert's irreducibility theorem. For example a bound on the minimal irreducible specialization in \(\mathbb{N}\) that is polynomial in the degree of \(f\) and in \(\log(H)\) (the logarithmic height of \(f\)) would provide a deterministic algorithm for the factorization of bivariate polynomials in polynomial time. \textit{Y. Walkowiak} [Acta Arith.\ 116, No.~4, 343--362 (2005; Zbl 1071.12002)] established a bound that is polynomial in \(\deg_T(f)\) and \(\log(H)\) but exponential in \(\deg_X(f)\). Moreover, the bound becomes fully polynomial if \(f(T,X)\) is Galois over \(\mathbb{Q}(T)\) (i.e., \(f(T,X)\) splits over \(\mathbb{Q}(T)[X]/(f)\)). The paper under review introduces the notion of \textit{Hilbert index} of a degree \(n\) transitive group \(G\), denoted \({\text{Hi}}(G,n)\),that comes from the maximal subgroup lattice of \(G\). The Hilbert index \({\text{Hi}}(f)\) of an irreducible polynomial \(f(T,X)\in \mathbb{Q}[T,X]\) is then defined to be the Hilbert index of the Galois group of \(f\) over \(\mathbb{Q}(T)\) (w.r.t.\ the Galois action on the roots of \(f(T,X)\) in an algebraic closure of \(\mathbb{Q}(T)\)). The main result of the paper asserts that the bound on the least irreducible specialization in \(\mathbb{N}\) for \(f\) is polynomial in \(\deg_T(f)\), \(\log(H)\), and \(\deg_X(f)^{{\text{Hi}}(f)}\). In several cases the authors absolutely bound the Hilbert index and hence achieve the desired bound on the least irreducible specialization. Most notably the Hilbert index is absolutely bounded whenever \(|G|\leq n^\alpha\), for some absolute constant \(\alpha\). Note that if \(f(T,X)\) is Galois over \(\mathbb{Q}(T)\), then \(|G|=n\), hence the main result generalizes the earlier work of Walkowiak mentioned above. The authors also note that in the generic case, i.e.\ when \(G\cong S_n\), one has \({\text{Hi}}(S_n,n)\gg n/\log n\), hence the method fails to achieve the desired bound. The proof is based on a combinatorial refinement of some reduction step in the classical theory of Hilbert's irreducibility theorem together with \textit{Y. Walkowiak}'s application [loc. cit.] of \textit{D. R. Heath-Brown}'s result on the density of rational points [Ann. Math. (2) 155, No. 2, 553--598 (2002; Zbl 1039.11044)]. The paper also discusses bounds that do not depends on \(\log (H)\), and connects those to Lang's conjecture using \textit{L. Caporaso, J. Harris} and \textit{B. Mazur} [J. Am. Math. Soc. 10, No. 1, 1--35 (1997; Zbl 0872.14017)].
scientific article; zbMATH DE number 5380355
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