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 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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