Nilpotent Jacobian maps in dimension three and stable tameness in block extensions
Abstract
We study polynomial maps in three variables with nilpotent Jacobian over a field of characteristic zero. The proposed classification reduces maps with linearly independent components to a family determined by a univariate polynomial evaluated at a quadratic coordinate. The geometric part of the argument produces two algebraically dependent constant linear combinations of the components. A derivation argument then yields the normal form over the original field. We obtain explicit polynomial inverses and tame factorizations. We then study higher-dimensional maps in which all but the last three components depend on three variables. For this class, we give separate normal forms for the two three-variable blocks and deduce stable tameness from the residue-field criterion of Berson, van den Essen, and Wright.
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BibTeX
@misc{wang2026nilpotent,
title = {{Nilpotent Jacobian maps in dimension three and stable tameness in block extensions}},
author = {Jie Wang and Dan Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16580},
eprint = {2609.16580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16580},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nilpotent Jacobian maps in dimension three and stable tameness in block extensions
AU - Jie Wang
AU - Dan Yan
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16580
UR - https://arxiv.org/abs/2609.16580
AB - We study polynomial maps in three variables with nilpotent Jacobian over a field of characteristic zero. The proposed classification reduces maps with linearly independent components to a family determined by a univariate polynomial evaluated at a quadratic coordinate. The geometric part of the argument produces two algebraically dependent constant linear combinations of the components. A derivation argument then yields the normal form over the original field. We obtain explicit polynomial inverses and tame factorizations. We then study higher-dimensional maps in which all but the last three components depend on three variables. For this class, we give separate normal forms for the two three-variable blocks and deduce stable tameness from the residue-field criterion of Berson, van den Essen, and Wright.
ER -