Equivalence of polynomials under automorphisms of image

We provide an effective and efficient algorithm to decide, given two polynomials in image, whether there is an automorphism of image taking p to q, and to find this automorphism if it exists. For p(x,y)=x an algorithm is implicit in the proofs of the Abhyankar–Moh/Suzuki embedding theorem, and a different and interesting explicit algorithm was given in 1997 by Shpilrain and Yu (J. Alg. 197 (2) (1997) 546–558). Other special cases are done in a preprint of Shpilrain and Yu (to appear in J. Alg.).