An Overwiev of the Number of Points of Algebraic Sets over Finite Fields

We determine an upper bound on the number of rational points of an affine or projective algebraic set defined over an algebraic closure of a finite field by a system of polynomial equations. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We determine in the affine case some algebraic sets having the maximum number of rational points.