On the Jacobian Conjecture: Reduction of Coefficients

Abstract

We prove that a polynomial map from Rn to itself with non-zero constant Jacobian determinant is a stably tame automorphism if its linear part is the identity and all the coefficients of its higher order terms are non-positive. We also prove that the Jacobian conjecture holds for any number of variables and any field of characteristic zero, if one can show that every polynomial map of Rn to itself is injective when it has a non-zero constant Jacobian determinant and has linear part the identity, and all the coefficients of higher order terms are non-negative. The proofs use special properties of matrices with non-positive off-diagonal elements and non-negative principal minors, and of matrices with vanishing principal minors. Furthermore we reduce the Jacobian conjecture to a polynomial matrix problem. Moreover, if the matrix has a positive answer, then every real polynomial automorphism is stably tame.