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

(18.1) |

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 ...

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