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I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.

I found a paper

Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a Coefficients dans un Corps Fini, Proceedings of the Japan Academy 30, Issue 10 (1954) p. 930–933, doi:10.3792/pja/1195525873, Project Euclid.

In the paper, the author uses some theorem of Weil on an asymptotic formula of the cardinality of the zero set of an absolutely irreducible polynomial $f^{\ast}(u,v)=\frac{f(u)-f(v)}{u-v}$.

So I want to study 'quickly' Weil's theorem on this occasion. But the 1948 book of Weil 'Sur les courbes algébriques et les variétés qui s'en déduisent' (WorldCat) seems to be out of print and might not be modern.

I have a little familiarity with the first 3 chapters of Hartshorne.

I hope someone can recommend a reference. Thank you.

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    $\begingroup$ Hartshorne Exercise 1.10 of Chapter V, Section 1. $\endgroup$ Mar 23 '20 at 5:18
  • $\begingroup$ @FelipeVoloch Thank you very much for your comment. I think the exercise gives a roadmap. May I ask one more reference that might be more comprehensive please? $\endgroup$
    – gualterio
    Mar 23 '20 at 9:32
  • $\begingroup$ You may find a sketch, with some details, of the proof of Hasse-Weil bound for the number of ${\mathbb F}_q$-rational points on a smooth projective curve defined over the finite field ${\mathbb F}_q$ in the notes posted on the following link: ocf.berkeley.edu/~rohanjoshi/2019/11/25/… $\endgroup$
    – F Zaldivar
    Mar 24 '20 at 3:11

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