Recognizing and parametrizing curves isomorphic to a line

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(n2) 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. This requires O(n2log2n) field operations.