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An upper bound on the number of rational points of arbitrary projective varieties over finite fields


Author: Alain Couvreur
Journal: Proc. Amer. Math. Soc. 144 (2016), 3671-3685
MSC (2010): Primary 11G25, 14J20
DOI: https://doi.org/10.1090/proc/13015
Published electronically: February 12, 2016
MathSciNet review: 3513530
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Abstract: We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field $ \mathbf {F}_q$. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.


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Additional Information

Alain Couvreur
Affiliation: INRIA & LIX, UMR 7161, École Polytechnique, 91128 Palaiseau Cedex, France
Email: alain.couvreur@lix.polytechnique.fr

DOI: https://doi.org/10.1090/proc/13015
Received by editor(s): September 26, 2014
Received by editor(s) in revised form: October 27, 2015
Published electronically: February 12, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society

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