Algebraic Curves

Abstract

Let F(X. Y) be a polynomial of two variables over the field F _q . A point (a, b) lying in the plane is called the root of the polynomial if F(a, b) = O. All these roots can be found by enumeration and define a plane affine curve. Usually, one considers all points with coordinates in the algebraic closure of the field F _q , i.e., $a,b \in {F_{{q^m}}},m = 1,2 \cdots .$ Points of the curve such that (a, b) ∈ F _q are said to be rational over F _q . A projective curve is defined as a set of points lying in the projective plane and nullifying the form F(X, Y, Z).