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 is an element of K vertical bar x,y vertical bar 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 0 (112) 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 0(172 1092 11) field operations. (c) 2007 Elsevier Ltd. All rights reserved.
polynomial algebras automorphisms parametrization plane curves complexity