On the number of rational points on an algebraic curve over a finite field (Q1281141)

The authors obtain a new upper bound for the number of rational points on a plane curve, possibly singular, absolutely irreducible, defined over a finite field. The bound holds for the degree of the plane curve in a certain range that depends only on the cardinality of the finite field. The bound is derived from Stöhr-Voloch theory of Frobenius orders of morphisms [\textit{K.-O. Stöhr} and \textit{J. F. Voloch}, Proc. Lond. Math. Soc. (3) 52, 1-19 (1986; Zbl 0593.14020)] and from previous work of the authors on arcs in a finite plane [Finite Fields Appl. 2, 274-292 (1996; Zbl 0890.05013)].