An Algebraic Proof of the Jacobian Conjecture

Qianghui Xiao

doi:10.20944/preprints202307.1834.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 25 July 2023 / Approved: 25 July 2023 / Online: 27 July 2023 (09:41:03 CEST)
Version 2 : Received: 30 July 2023 / Approved: 31 July 2023 / Online: 1 August 2023 (09:46:21 CEST)

In this paper, a short survey of the existed results concerning the Jacobian Conjecture is first given. Then the 3-fold linear type polynomial map will be analyzed in detail. The expansion of the Jacobian condition is deduced to obtain its equivalent algebraic equations, and the Jacobian condition will be analyzed to derive two coordinate transformations that can maintain the invariance of the Jacobian condition. Finally, it is proved by mathematical induction method that one general chain expression presented in this paper is just the inverse polynomial map of 3-fold linear type polynomial map, i.e. LJC(n,[3]) holds such that the Jacobian Conjecture holds.

Jacobian Conjecture, 3-fold linear type map, General 3-fold linear type map, Homogeneous type map, Jacobian condition, Coordinate transformation, Equivalent algebraic equations, Invariance of the Jacobian condition, General chain expression, Inverse polynomial map, Injective problem

Computer Science and Mathematics, Algebra and Number Theory

Comment 1

Received: 1 August 2023
Commenter: Qianghui Xiao
Commenter's Conflict of Interests: Author

Comment: 1. Five keywords have been added; 2. The errors in Lemma 3.10, Lemma 3.11 and Theorem 3.12 are corrected.

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An Algebraic Proof of the Jacobian Conjecture

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