Abstract
In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension nâ€3 over a field K of characteristic zero. We do this by extending the following result for nâ€2 by F. Dillen to nâ€3: If f is a polynomial of degree larger than 2 in nâ€3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some TâGLn(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n.The proof of the case detâĄHfâKâ is based on the following result, which in turn is based on the already known case detâĄHf=0: If f is a polynomial in nâ€3 variables such that detâĄHfâ 0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace âpositiveâ by ânontrivialâ above).In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [13]. For polynomials in dimension nâ€3, we generalize this result to arbitrary fields of characteristic zero.
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