Work

Critical points of point charge potentials along lines

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.00336

Author: Oliveira, Goncalo

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Fundação para a Ciência e a Tecnologia

Awards: Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

OpenAlex ID: W7206207411

Cites: 3

Cited by: 1

Related to: 10