Density results for Hilbert subsets. (English) Zbl 0923.12001

A classical tool for studying Hilbert’s irreducibility theorem is Siegel’s finiteness theorem for \(S\)-integral points on algebraic curves. In a previous paper [P. Dèbes, Manuscr. Math. 89, No. 1, 107-137 (1996; Zbl 0853.12001)], the author presented a different approach, based on \(s\)-integral points. Given an integer \(s>0\), the \(s\)-integral points of a field \(K\) with the product formula are those points \(t\) for which the set of places \(v\in M_K\) such that \(| t| _v>1\) is of cardinality \(\leq s\). Now the author gives new applications of this method, which include the possibility that \(K\) is of characteristic \(p>0\). Under the assumptions that the polynomials involved are separable and tamely ramified over \(\infty\), the applications essentially say that, except for trivial cases, the Hilbert subsets of \(K\) contain infinitely many powers of a given element \(b\in K\) of height \(h(b)>0\). For a field \(K\) with the product formula, which is either of characteristic 0 or of characteristic \(p\) and imperfect, the author proves also the following density property: Hilbert subsets of \(K\) are dense in \(K\) for the strong approximation topology, provided that \(0\) is not isolated in \(K\).


12E25 Hilbertian fields; Hilbert’s irreducibility theorem
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G20 Curves over finite and local fields


Zbl 0853.12001