The Lie algebra of polynomial vector fields and the Jacobian conjecture期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The Lie algebra of polynomial vector fields and the Jacobian conjecture

Authors:	Herwig Hauser Gerd Müller

Institution:	(1) Mathematisches Institut, Universität Innsbruck, A-6020 Innsbruck, Austria;(2) Fachbereich Mathematik, Universität Mainz, D-55099 Mainz, Germany

Abstract:	The Jacobian conjecture for polynomial maps :K ⁿ K ⁿ is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra $\mathbb{D}$ of formal vector fields inn variables. To be precise, let $\mathbb{D}_0$ be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of $\mathbb{D}_0$ ,H the root spaces corresponding to linear forms onH and $A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda$ . Then every polynomial map :K ⁿ K ⁿ with invertible Jacobian matrix is an automorphism if and only if every automorphism of $\mathbb{D}$ with (A) $\subseteq A$ satisfies (A)=A.

Keywords:	14E09 17B66
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