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The Jacobian conjecture and injectivity conditions. (English) Zbl 1406.14044

In the paper, the authors first give some sufficient conditions for global injectivity. They prove that \(f\) is injective in a convex domain \(D\) if the determinant of \(Jf(X_{i,j})\) is not equal to zero, where \(X_{i,j}\) belong to \(L \cap D\) and \(L\) in \(\mathbb R^n\). Thus, they give a sufficient condition for the analytic function f to be injective in \(D\), where \(D\) is a convex domain in \(C\). Then they give an affirmative answer to the Jacobian conjecture for a class of polynomial maps \(F=x+H\) with rank \(JH\leq 1\).
Reviewer: Yan Dan (Changsha)

MSC:

14R15 Jacobian problem
32A10 Holomorphic functions of several complex variables
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31C10 Pluriharmonic and plurisubharmonic functions
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