Remarks on the Jacobian Conjecture

Abstract

LetF(F1,...,Fn)∈(C[X1,...,X n])nwith det(J(F))∈C* and letMi(Xi,Y)=mi0(Y)+mi1(Y)X i+···+midi(Y)X i d i∈C[Xi,Y]C[X i,Y1,...,Yn] be the minimal polynomial ofFoverC(Xi). We prove thatmi0(Y),...,midi(Y) have no common zeros inCn. As a direct consequence, we obtain flatness ofC[F,Xi] overC[F] for everyi. As applications, we obtain simple algebraic proofs of the following two known results: (i) A birational polynomial map fromCnintoCnwith det(J(F))∈C* is actually an automorphism; (ii) an injective polynomial map fromCnintoCnis also an automorphism. © 1997 Academic Press.