On the number of rational points on codimension-1 algebraic sets in Pn( Fq)

Abstract

An upper bound on the number of F q -rational points on a pure ( n - 1)-dimensional algebraic set of low degree defined over F q in P n ( F q ) is given, using simple counting arguments, and the result is generalized to all degrees using results from coding theory. The bound depends on n, q, d, where d is the degree of the algebraic set. A number of corollaries are deduced and applications to coding theory are mentioned.