PreprintarXiv licencearXiv
When is a polynomial in three variables cylindrical?
Abstract
We prove that a nonconstant polynomial f∈ℂ[x1,x2,x3] is cylindrical (that is, f∈ℂ[x1,x2] after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over ℝ. We also give explicit counterexamples to this criterion in every dimension n≥4.
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BibTeX
@misc{lin2026when,
title = {{When is a polynomial in three variables cylindrical?}},
author = {Xiaojin Lin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06465},
eprint = {2609.06465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06465},
}
RIS
TY - UNPB
TI - When is a polynomial in three variables cylindrical?
AU - Xiaojin Lin
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06465
UR - https://arxiv.org/abs/2609.06465
AB - We prove that a nonconstant polynomial f∈ℂ[x_1,x_2,x_3] is cylindrical (that is, f∈ℂ[x_1,x_2] after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over ℝ. We also give explicit counterexamples to this criterion in every dimension n≥4.
ER -