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The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors
Abstract
We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if f∈ (x1,…,xn)2subsetneq k[[x1,…,xn]] has an isolated critical point and chark=0, then its Euler conductor Jf:f is not contained in the Tjurina ideal (Jf,f). More generally, if A=R/I is a nonzero Noetherian k-algebra and a=[f]∈ A is nilpotent with dA/ka=0, then I:fnsubseteq(I,f). For a local ring (R,n) we obtain the stronger noncontainment I:fnsubseteq(I,f)+n(I:f). The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies Jf:f⊆Jf̅, disproving the integral-closure strengthening proposed by Ma and Zuo.
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BibTeX
@misc{zhang2026hassanzadehnasrollah,
title = {{The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors}},
author = {Yizhi Zhang and Huaiqing Zuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09833},
eprint = {2610.09833},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09833},
}
RIS
TY - UNPB
TI - The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors
AU - Yizhi Zhang
AU - Huaiqing Zuo
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09833
UR - https://arxiv.org/abs/2610.09833
AB - We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if f∈ (x_1,…,x_n)^2subsetneq k[[x_1,…,x_n]] has an isolated critical point and chark=0, then its Euler conductor J_f:f is not contained in the Tjurina ideal (J_f,f). More generally, if A=R/I is a nonzero Noetherian k-algebra and a=[f]∈ A is nilpotent with d_A/ka=0, then I:fnsubseteq(I,f). For a local ring (R,n) we obtain the stronger noncontainment I:fnsubseteq(I,f)+n(I:f). The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies J_f:f⊆J_f̅, disproving the integral-closure strengthening proposed by Ma and Zuo.
ER -