Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture Ii

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix of F . Then the Jacobian Conjecture (which dates back to Keller [9], 1939) asserts that if detJF ∈ C∗, then F is invertible. It was shown in [1] and [12] that it suffices to prove the Jacobian Conjecture for all n ≥ 2 and all polynomial maps of the form F = x + H , where JH is homogeneous and nilpotent (these two conditions imply that detJF = 1); in fact it is even shown that the case where JH is nilpotent and H is homogeneous of degree 3 is sufficient. For n = 3 resp. n = 4 this so-called cubic homogeneous case was proved by Wright resp. Hubbers in [11] resp. [8]. For n = 3, the case F = x + H , where H is not necessarily homogeneous, but of degree 3, was proved by Vistoli in [10]. On the other hand, if H has degree ≥ 4 not much is known; if for example F is of the form x+H where H is homogeneous of degree ≥ 4, then all cases n ≥ 3 remain open. The aim of this paper is to study these type of problems under the additional hypothesis that JH is symmetric. This is no loss of generality since it was recently shown by the authors in [3] that it suffices to prove the Jacobian Conjecture for all polynomial maps F : C → C of the form F = x+H with JH nilpotent, homogeneous of degree ≥ 2 and symmetric. For such maps the conjecture was proved for all n ≤ 4 in [6]. The proof of this result is based on a remarkable theorem of Gordan and Noether, which asserts that if n ≤ 4, then h(f), the Hessian matrix of the homogeneous polynomial f ∈ C[x1, . . . , xn], is singular iff f is degenerate i.e. there exists a linear coordinate change T such that f(Tx) ∈ C[x1, . . . , xn−1]. However if n = 5 such a result does not hold: the polynomial f = x1x3+x1x2x4+x 2 2x5 has a singular Hessian but is not degenerate.