Equivalence of polynomials under automorphisms of K[x,y]K[x,y]

Let K[x,y]K[x,y] be the algebra of polynomials in two variables over an arbitrary field KK. We show that if the maximum of the xx- and yy-degrees of a given polynomial p(x,y)p(x,y) cannot be decreased by a single triangular or linear automorphism of K[x,y]K[x,y], then it cannot be decreased by any   automorphism of K[x,y]K[x,y]. If KK is an algebraically closed constructible field, this result yields an algorithm for deciding whether or not two polynomials p,q∈K[x,y]p,q∈K[x,y] are equivalent under an automorphism of K[x,y]K[x,y].We also show that if there is an automorphism of K[x,y]K[x,y] taking pp to qq, then it is “almost” unique. More precisely: if an automorphism αα of K[x,y]K[x,y] is not conjugate to a triangular or linear automorphism, then any polynomial invariant (or even semiinvariant) under αα is a constant.