Effective Hilbert's irreducibility theorem for global fields (Q6594750)

The text under review proves an effective version of Hilbert's Irreducibility Theorem for arbitrary global fields. The theorem states that for an irreducible polynomial \(F(T,Y)\) in two variables with coefficients in a global field \(K\), the set of values of \(Y\) of height at most \(B\) for which \(F(T,Y)\) becomes reducible has natural density zero in the set of all possible \(Y\). The text under review gives an effective bound for this number which is polynomial in the degree of \(F\) and in the logarithmic height of \(F\).\N\NThe authors also prove similar bounds for the set of values of \(Y\) for which the Galois group of \(F(T,Y)\) over \(K\) is a proper subgroup of the Galois group over \(K(Y)\), and for the set of \(Y\) for which \(F(T,Y)\) acquires a root in \(K\).\N\NIn the theorem, there is also a constant that depends only on the field \(K\). This constant is made completely explicit in the case of a function field \(K\) of positive characteristic, but may be hard to compute in other cases such as number fields.