Critical Points of Line Restrictions of Signed Point-Charge Potentials
Abstract
Let α>0 and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height (a, b2), delete classes whose coefficient sum is zero, and let m be the number of remaining classes. We prove the following dichotomy. If m=0, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If m ≥ 1, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every α>0. The structural input is a projective paired Haar theorem: for finite β>1, the full $2 m$-dimensional space ∑ Lj / Qjβ, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution p=2 α proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range p ≥ 1 and extends the same conclusion to every p>0. For every n, an explicit positive n-charge configuration attains $2 n-1$ simple critical points.
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BibTeX
@misc{duan2026critical,
title = {{Critical Points of Line Restrictions of Signed Point-Charge Potentials}},
author = {Xiuqing Duan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28376},
eprint = {2608.28376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28376},
}
RIS
TY - UNPB
TI - Critical Points of Line Restrictions of Signed Point-Charge Potentials
AU - Xiuqing Duan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.28376
UR - https://arxiv.org/abs/2608.28376
AB - Let α>0 and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height (a, b^2), delete classes whose coefficient sum is zero, and let m be the number of remaining classes. We prove the following dichotomy. If m=0, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If m ≥ 1, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every α>0. The structural input is a projective paired Haar theorem: for finite β>1, the full $2 m$-dimensional space ∑ L_j / Q_j^β, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution p=2 α proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range p ≥ 1 and extends the same conclusion to every p>0. For every n, an explicit positive n-charge configuration attains $2 n-1$ simple critical points.
ER -