Graphs and the Jacobian conjecture

Abstract

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 〉 = x t y and each α is an isotropic vector i.e. 〈 α , α 〉 = 0 . To a set { α 1 , … , α s } of isotropic vectors, we assign a graph and study its structure in case the corresponding polynomial f = ∑ 〈 α j , x 〉 d has a nilpotent Hessian. The main result of this article asserts that in the case dim ( [ α 1 , … , α s ] ) ≤ 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.