PreprintCC BY 4.0arXiv
Noncommutative algebraic sets are not generically nc manifolds
Abstract
A consequence of implicit function theorem is that any algebraic set is a complex manifold except at some singular points which form a strictly lower dimension algebraic subset. Accordingly, an non commutative implicit function theorem of Agler-McCarthy might suggest that a nc algebraic set is generically an nc manifold. However, in this paper, a type of nc polynomials is shown, whose zero locus has larger dimension than the nc manifold given by implicit function theorem.
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BibTeX
@misc{lu2026noncommutative,
title = {{Noncommutative algebraic sets are not generically nc manifolds}},
author = {Jiachen Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33425},
eprint = {2610.33425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncommutative algebraic sets are not generically nc manifolds
AU - Jiachen Lu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33425
UR - https://arxiv.org/abs/2610.33425
AB - A consequence of implicit function theorem is that any algebraic set is a complex manifold except at some singular points which form a strictly lower dimension algebraic subset. Accordingly, an non commutative implicit function theorem of Agler-McCarthy might suggest that a nc algebraic set is generically an nc manifold. However, in this paper, a type of nc polynomials is shown, whose zero locus has larger dimension than the nc manifold given by implicit function theorem.
ER -