Upper bounds for the number of isolated critical points via Thom-Milnor theorem

TL;DR

This paper uses the Thom-Milnor theorem to derive upper bounds on the number of isolated critical points in various physical and mathematical potential problems, including Maxwell's problem and n-body configurations.

Contribution

It introduces a novel application of the Thom-Milnor theorem to establish bounds on critical points across multiple classical problems in physics and mathematics.

Findings

Exponential upper bound for Maxwell's problem

Polynomial bound for even-dimensional potentials

Unified approach for different potential-based problems

Abstract

We apply the Thom-Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges(Maxwell's problem on point charges), (2) critical points of SINR, (3) critical points of a potential generated by several fixed Newtonian point masses augmented with a quadratic term, (4) central configurations in the $n$-body problem. In particular, we get an exponential bound for Maxwell's problem and the polynomial bound for the case of an "even dimensional" potential in Maxwell's problem.