Algorithms for recognizing coordinates in two variables over UFD's

We give an easy and efficient algorithm to check whether a given polynomial <i>f</i> in <i>K</i>[<i>x,y</i>] is a coordinate, where <i>K</i> be a commutative field of characteristic zero, and if so to compute a coordinate's mate of <i>f</i>. Then we treat the same problem replacing the ground field <i>K</i> by a unique factorization domain <i>A</i> of characteristic zero. A notable feature of our method is that it always produces a mate of minimum degree.