Critical points of point charge potentials along lines
Goncalo Oliveira
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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