On the probability that given polynomials have a specified highest common factor

The polynomial functions f, f,…, f are found to have highest common factor h for a set of values of the variables x, x,…,x whose asymptotic density is 1/h∑d∣hμ(d)Π ϱ(f, dh)/dΠp∣h(1−Πϱ(f, p)/p) For the special case f(x) = f(x) = … = f(x) = x and h = 1 the above formula reduces to Π(1 − 1/p) = 1/ζ(m), the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f, f,…, f for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density