In this short interlude we consider some results midway between the Pólya Theorem of Chapter 9 and the Schneider–Lang Theorem of Chapter 19. Some higher-dimensional versions have recently become very useful in the study of unlikely intersections in semiabelian schemes and Shimura varieties.
In Chapter 9 we considered integral values of
f (1), f (2), f (3), . . . .
Here we treat rational values of
the values at first assumed to lie in Z/n. How many of these n + 1 values can there be?
As in Pólya’s Theorem, polynomials in Z[z] play a special role: if the degree is at most 1 then all n + 1 values are in ...