ON THE NUMBER OF RATIONAL-POINTS ON CODIMENSION-1 ALGEBRAIC-SETS IN P-N(F-Q)

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.