Remarks on the Jacobian conjecture

[摘要] Let F := (F-1,..., F-n) is an element of (C[X(1),..., X(n)])(n) with det(J(F)) is an element of C* and let M(i)(X(i), Y) = m(i0)(Y) + m(i1)(Y)X(i) + ... + m(idi)(Y)X(i)(di) is an element of C[X(i), Y] = C[X(i), Y-1,..., Y-n] be the minimal polynomial of F over C(X(i)). We prove that m(i0)(Y),..., m(idi)(Y) have no common zeros in C-n. As a direct consequence, we obtain flatness of C[F, X(i)] over C[F] for every i. As applications, we obtain simple algebraic proofs of the following two known results: (i) A birational polynomial map from C-n into C-n with det(J(F)) is an element of C* is actually an automorphism; (ii) an injective polynomial map from C-n into C-n is also an automorphism. (C) 1997 Academic Press.

[发布日期] 1997-02-01