Graphs and the Jacobian conjecture

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• 作者：Arno van den Essen ; Roel Willems
• 关键词：Context independence ; Weakened WARP ; Top cycle
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2008
• 出版时间：March 2008
• 年：2008
• 卷：212
• 期：3
• 页码：578-598
• 全文大小：412 K

文摘

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 90b"">. The Jacobian Condition implies that f is a finite sum of d-th powers of linear forms, α,x^d, where and each α is an isotropic vector i.e. α,α=0. To a set {α₁,…,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([α₁,…,α_s])≤2 or ≥s−2, the Jacobian Conjecture holds for the maps 0b36fdabcf0562"" title=""Click to view the MathML source"">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.