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. The main question considered in the paper [DZ] was the following: given a polynomial f ∈ Q[x, y], say absolutely irreducible, and given a sequence {θj} satisfying f(j, θj) = 0, estimate the degree over the rationals of the fields Q(θ1, . . . , θn) as n→∞. It was shown (Theorem 2(a)) that, provided d = degy f > 1, a lower bound of type c logn holds for large n, where c > 1 depends only on f (the simple examples f(x, y) = y − x show that the bound is not far from best possible). Also, a sufficient condition for an estimate of type c was given (Theorem 2(b)). That condition, which looked somewhat unnatural, was not made more general and complete, due among other things to the lack of certain estimates connected with power free values of polynomials. In that context Professor A. Schinzel formulated an elegant conjecture giving the exact condition for an exponential lower bound to hold: this is stated below (but already appeared in [DZ]). Let us first introduce some notation. Let Σ denote the splitting field of f over Q(x) and, for a positive integer m, define Σ(m) as the splitting field of the polynomial f(m, y). Let G be the Galois group of kΣ over k(x), k being some number field containing the algebraic closure of Q in Σ. The function field kΣ corresponds to a nonsingular curve C over k. Let S(x) be a squarefree polynomial whose roots are precisely the finite ramification points of x (viewed as a function on C). It is well known that, in view of our assumptions, S is nonconstant (see Lemma 1.2 in [DZ]), and moreover, since f has rational coefficients, S may also be assumed to have rational integral coefficients. Let D(n) be the degree over Q of the composite of the fields Σ(j) for 1 ≤ j ≤ n. By means of a quantitative version of Hilbert’s Irreducibility Theorem it was shown in [DZ] (see Remark 1, p. 21 of the old preprint or the end of the proof of Theorem 2(a) in the published version) that D(n) and the degree of Q(θ1, . . . , θn) have logarithms of the same order of magnitude (their quotient is bounded above and below by positive con-

Fields containing values of algebraic functions II (On a conjecture of Schinzel)