ON THE NUMBER OF RATIONAL-POINTS ON AN ALGEBRAIC CURVE OVER A FINITE-FIELD

A new bound for the number of rational points on an algebraic curve ov er a finite field is obtained in Theorem 1.3. It is derived from previ ous work on the upper bounds for the size of a complete are in a finit e projective plane. In the terminology of plane curves, the main resul t is Theorem 1.4, and considers an absolutely irreducible, plane curve C of degree d defined over F-q,F- q = p(h) with p prime and p 1 3. An upper bound is obtained for the number of branches of C that are cent red at F-q-rational points. To do this, two types of branches are dist inguished: (a) branches of order and class equal to r; (b) branches of order r and class different from r. The main theorem counts twice the number of branches of type (a) plus the number of branches of type (b ). As a corollary, this theorem gives an upper bound for the number of Fg-rational points of C, since simple non-inflexion points are branch es of order 1 and class 1, while inflexions points are branches of ord er 1 and class greater than 1.