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ON THE LND CONJECTURE

Part of: Affine geometry Differential algebra

Published online by Cambridge University Press:  03 July 2023

XIAOSONG SUN  and
BEINI WANG
XIAOSONG SUN*
Affiliation:
School of Mathematical Sciences, Jilin University, Changchun, Jilin 130012, PR China
BEINI WANG
Affiliation:
School of Mathematical Sciences, Jilin University, Changchun, Jilin 130012, PR China e-mail: wangbn21@jlu.edu.cn
*
e-mail: sunxs@jlu.edu.cn
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Abstract

Let k be a field of characteristic zero and $k^{[n]}$ the polynomial algebra in n variables over k. The LND conjecture concerning the images of locally nilpotent derivations arose from the Jacobian conjecture. We give a positive answer to the LND conjecture in several cases. More precisely, we prove that the images of rank-one locally nilpotent derivations of $k^{[n]}$ acting on principal ideals are MZ-subspaces for any $n\geq 2$, and that the images of a large class of locally nilpotent derivations of $k^{[3]}$ (including all rank-two and homogeneous rank-three locally nilpotent derivations) acting on principal ideals are MZ-subspaces.

Keywords

locally nilpotent derivationsJacobian conjectureLND conjectureMZ-subspaces

MSC classification

Primary: 14R15: Jacobian problem
Secondary: 13N15: Derivations 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 3 , December 2023 , pp. 412 - 421
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by the EDJPC (JJKH20220962KJ), NSFJP (20210101469JC) and NSFC (12171194).

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