Polynomial automorphisms and gröbner reductions

Abstract

LetPn=K[x1,...,xn] be the polynomial algebra over a fieldKof characteristic 0. We show that applying an automorphism to a given polynomialp∈Pnis mimicked by Gröbner transformations of a basis of the ideal ofPngenerated by partial derivatives of this polynomial. In the case ofP2, this yields a miraculously simple algorithm for deciding whether or not a given polynomial fromP2is part of a basis. Another application is an algorithm which, given a polynomialp∈P2that is part of a basis, finds a sequence of elementary automorphisms that reducesptox1. We also speculate on how our method may be used for constructing a possible counterexample to the Jacobian conjecture in higher dimensions. © 1997 Academic Press.