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Auxiliary Polynomials in Number Theory by David Masser

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18

Counting – Bombieri–Pila

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

image (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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