PreprintCC BY 4.0arXiv
Critical points of point charge potentials along lines
Abstract
We prove Conjecture 1.9 of Gabrielov-Novikov-Shapiro. More precisely, for every p>0, every nonconstant restriction to a line of a potential generated by n point charges has at most 2n−1 critical points. The bound is sharp, even for positive charges. The proof uses a duality with an auxiliary planar potential generated by collinear charges and Morse theory.
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BibTeX
@misc{oliveira2026critical,
title = {{Critical points of point charge potentials along lines}},
author = {Goncalo Oliveira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00336},
eprint = {2609.00336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00336},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical points of point charge potentials along lines
AU - Goncalo Oliveira
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00336
UR - https://arxiv.org/abs/2609.00336
AB - We prove Conjecture 1.9 of Gabrielov-Novikov-Shapiro. More precisely, for every p>0, every nonconstant restriction to a line of a potential generated by n point charges has at most 2n−1 critical points. The bound is sharp, even for positive charges. The proof uses a duality with an auxiliary planar potential generated by collinear charges and Morse theory.
ER -