Polynomials with constant Hessian determinants in dimension three

Abstract

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= 3 over a field K of characteristic zero. We do this by extending 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 of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some T in GL_n(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 det Hf in K* is based on the following result, which in turn is based on the already known case det Hf = 0: if f is a polynomial in n <= 3 variables such that det Hf <> 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 `positive' by `nontrivial' 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 [Pog]. For polynomials in dimension n <= 3, we generalize this result to arbitrary fields of characteristic zero.