Remarks on the Jacobian Conjecture Original Research Article

LetF colon, equals (F1,…,Fn) set membership, variant (C[X1,…,Xn])nwith det (J(F)) set membership, variant C* and letMi(Xi, Y) = mi0(Y) + mi1(Y)Xi + ··· + midi(Y)Xidi set membership, variant C[Xi, Y] colon, equals C[Xi, 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)) set membership, variant C* is actually an automorphism; (ii) an injective polynomial map fromCnintoCnis also an automorphism.