Recognizing and Parametrizing Curves without Affine Singularities

Some time ago, Shpilrain and Yu reported an algorithm for deciding whether or not a polynomial p ∈ K[x, y] is a coordinate, or, equivalently, whether or not a plane curve p(x, y) = 0 is isomorphic to a line. Here K is any constructible field of characteristic 0. In this paper, we show that their algorithm requires O(n log2 n) field operations, where n is the degree of a given polynomial. We also show how their algorithm can be used to find a polynomial parametrization of a plane curve p(x, y) = 0 which is isomorphic to a line (or, equivalently, has no affine singularities). This requires O(n) field operations.