Division and noetherianity in quasianalytic Denjoy-Carleman classes
Abstract
We show that the local rings of quasianalytic Denjoy-Carleman classes need not be Noetherian. More precisely, we construct quasianalytic Denjoy-Carleman classes CM whose local rings CM(ℝd,0) are not Noetherian for any d≥ 2. Conversely, we exhibit quasianalytic classes whose local rings in dimension two are regular and Noetherian. Our approach is based on the problem of exact division: if f=φg, where f∈ CM, φ is real-analytic, and g∈ C∞, under which conditions does it follow that g∈ CM? We introduce a quantitative scale-growth condition on the defining weight sequence M and prove that exact division holds in two variables whenever this condition is sufficiently strong relative to the rate at which the nonreal branches of the complex zero set of φ approach the real plane. We also construct quasianalytic weight sequences with prescribed growth thresholds and establish corresponding counterexamples to exact division. These yield nonclosed principal ideals and, consequently, non-Noetherian local rings.
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BibTeX
@misc{silva2026division,
title = {{Division and noetherianity in quasianalytic Denjoy-Carleman classes}},
author = {André Belotto da Silva and Avner Kiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40265},
eprint = {2610.40265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40265},
}
RIS
TY - UNPB
TI - Division and noetherianity in quasianalytic Denjoy-Carleman classes
AU - André Belotto da Silva
AU - Avner Kiro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40265
UR - https://arxiv.org/abs/2610.40265
AB - We show that the local rings of quasianalytic Denjoy-Carleman classes need not be Noetherian. More precisely, we construct quasianalytic Denjoy-Carleman classes C^M whose local rings C^M(ℝ^d,0) are not Noetherian for any d≥ 2. Conversely, we exhibit quasianalytic classes whose local rings in dimension two are regular and Noetherian. Our approach is based on the problem of exact division: if f=φg, where f∈ C^M, φ is real-analytic, and g∈ C^∞, under which conditions does it follow that g∈ C^M? We introduce a quantitative scale-growth condition on the defining weight sequence M and prove that exact division holds in two variables whenever this condition is sufficiently strong relative to the rate at which the nonreal branches of the complex zero set of φ approach the real plane. We also construct quasianalytic weight sequences with prescribed growth thresholds and establish corresponding counterexamples to exact division. These yield nonclosed principal ideals and, consequently, non-Noetherian local rings.
ER -