The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture

By A. Belov-Kanel, M. Kontsevich. 2005

127 citations recorded by the source corpus.

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra (the algebra of polynomial differential operators in $n$ variables) is invertible. We prove that the Jacobian conjecture in dimension $2n$ implies the Dixmier conjecture in rank $n$. Together with a well-known implication in the opposite direction, it shows that the stable Jacobian and Dixmier conjectures are equivalent. The main tool of the proof is the reduction to finite characteristic. After the paper was finished we have learned that the main result was already published by Y.Tsuchimoto in Osaka Journal of Mathematics Volume 42, Number 2 (June 2005). His proof is different.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d287f72d88e605510f04cb269f9b6c964a17aa86

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In Geometry and Topology

Citations in the corpus

Universal enveloping algebras and universal derivations of Poisson algebras (2011, background): “This problem was posed in [4] for fields of characteristic zero but it makes sense in positive characteristic also [1, 5, 30, 31].”
Images of commuting differential operators of order one with constant leading coefficients (2009, background): “therein. Recently, the equivalence of the JC with the Dixmier conjecture proposed by J. Dixmier [D] in 1968 has been established by V. Kac (unpublished), Y. Tsuchimoto [T], A. Belov and M. Kontsevich [BK] and P. K. Adjamagbo and A. van den Essen [AE]. The Dixmier conjecture claims that any homomorph”
Automorphisms of the Weyl Algebra (2005, unknown): “R”
Mathieu Subspaces of Associative Algebras (2010, result): “tures that are directly or indirectly related with Mathieu subspaces. For example, the Dixmier conjecture [D] which as shown ﬁrst by Y. Tsuchimoto [T] in 2005, and later by A. Belov and M. Kontsevich [BK] and P. K. Adjamagbo and A. van den Essen [AE] in 2007 is actually equivalent to the Jacobian co”