Curves with abnormally many integral points

Abstract

i. It is well known that if f(x) is a polynomial with integer coefficients and if for each integer z, f(z) is an m-th power, then f(x) = hm(x), where h(x) is a polynomial with integer coefficients. There are several ways one could hope to generalize this result. The most obvious is to ask what relation must exist between polynomials f, g with integer coefficients, if f(~) c g(Z). (Here we let denote the ring of rational integers, ~ the field of rational numbers, ~p the field of p-adic integers, ~p the ring of p-adic integers, ~ the complex field and f(x) the set [f(~) I ~ ranges over X].) In particular, is f(x) = g(h(x)) for some polynomial h(x)? The answer is no if we require h to have integer coefficients as is demonstrated by the example: f = x(x + i), g = 2x. However, the answer is yes provided we allow h to have rational coefficients. In that case h is an integer valued function at integer points. Actually, the hypothesis can be relaxed considerably and the same conclusion obtained.