Graphs and the Jacobian conjecture

It is proved in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proceedings of the AMS 133 (8) (2005) 2201-2205] that it suffices to study the Jacobian Conjecture for maps of the form x+∇f, where f is a homogeneous polynomial of degree d(=4). The Jacobian Condition implies that f is a finite sum of d-th powers of linear forms, 〈α,x〉d, where 〈x,y〉=xty and each α is an isotropic vector i.e. 〈α,α〉=0. To a set {α1,…,αs} of isotropic vectors, we assign a graph and study its structure in case the corresponding polynomial f=∑〈αj,x〉d has a nilpotent Hessian. The main result of this article asserts that in the case dim([α1,…,αs])≤2 or ≥s−2, the Jacobian Conjecture holds for the maps x+∇f. In fact, we give a complete description of the graphs of such f's, whose Hessian is nilpotent. As an application of the result, we show that lines and cycles cannot appear as graphs of HN polynomials.