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On the Jacobian Conjecture: Reduction of Coefficients

✍ Scribed by J.T. Yu

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 332 KB
Volume: 171
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1995.1024

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✦ Synopsis

We prove that a polynomial map from (\mathbf{R}^{n}) 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 (\mathbf{R}^{n}) 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. 1995 Academic Press, Inc.

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