On the number of rational points on an algebraic curve over a finite field
James W. P. Hirschfeld, Gábor Korchmáros · Bulletin of the Belgian Mathematical Society - Simon Stevin · 1998
A new bound for the number of rational points on an algebraic curve over a finite field is obtained in Theorem 1.3.It is derived from previous work on the upper bounds for the size of a complete arc in a finite projective plane.In the terminology of plane curves, the main result is Theorem 1.4, and considers an absolutely irreducible, plane curve C of degree d defined over F q , q = p h with p prime and p ≥ 3.An upper bound is obtained for the number of branches of C that are centred at F q -rational points.To do this, two types of branches are distinguished: (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 F q -rational points of C, since simple non-inflexion points are branches of order 1 and class 1, while inflexions points are branches of order 1 and class greater than 1.