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Counterexamples to GNS Conjecture 1.8
Abstract
We disprove both parts of Conjecture 1.8 of Gabrielov, Novikov, and Shapiro (GNS). For the Coulomb potential, a generic rational configuration of 24 unit charges in ℝ3 has at least 18 nondegenerate critical points of Morse index one but exactly 14 effective Voronoi one-cells. A separate proper-line construction has at least two nondegenerate minima but only one relatively effective zero-dimensional cell intersection. For each positive subspace dimension, product suspension gives a proper-subspace counterexample.
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BibTeX
@misc{duan2026counterexamples,
title = {{Counterexamples to GNS Conjecture 1.8}},
author = {Xiuqing Duan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09637},
eprint = {2609.09637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09637},
}
RIS
TY - UNPB
TI - Counterexamples to GNS Conjecture 1.8
AU - Xiuqing Duan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09637
UR - https://arxiv.org/abs/2609.09637
AB - We disprove both parts of Conjecture 1.8 of Gabrielov, Novikov, and Shapiro (GNS). For the Coulomb potential, a generic rational configuration of 24 unit charges in ℝ^3 has at least 18 nondegenerate critical points of Morse index one but exactly 14 effective Voronoi one-cells. A separate proper-line construction has at least two nondegenerate minima but only one relatively effective zero-dimensional cell intersection. For each positive subspace dimension, product suspension gives a proper-subspace counterexample.
ER -