Constant polynomial Hessian determinants in dimension three

Details Constant polynomial Hessian determinants in dimension three Constant polynomial Hessian determinants in dimension three

2012-03-29

arxiv.org/abs/1203.6605v1

Mathematics

Algebraic Geometry

15 pages

Scientific paper

In this paper, we extend the following result for n <= 2 by F. Dillen to n <= 3: if f is a polynomial of degree larger than two in n <= 3 variables such that the Hessian determinant is constant, then after a suitable linear transformation the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n. The proof of the case det H f in K* is based on the following result, which in turn is based on the already known case det H f = 0: if f is a polynomial in n <= 3 variables such that det H f is nonzero, 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 allow any nontrivial w. As a consequence, we prove that the Jacobian conjecture holds for gradient maps of Keller type in dimension n <= 3, and that such gradient maps have linearly triangularizable and hence unipotent Jacobians in case their linear part is the identity map. 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 corresponding Keller maps are translations. We do this by showing that this problem is equivalent to the main result of [Pog], which we generalize to arbitrary fields of characteristic zero in dimension n <= 3.

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