The Critical Point Degree of a Periodic Graph

TL;DR

This paper investigates the critical point degree of a periodic graph operator's complex Bloch variety, providing bounds and insights relevant to spectral theory and nonlinear optimization.

Contribution

It identifies contributions from asymptotic critical points and refines bounds on the critical point degree based on the graph's structure.

Findings

Bound on critical point degree using Newton polytope volume

Identification of asymptotic critical point contributions

Implications for spectral edges conjecture and optimization

Abstract

The critical point degree of a periodic graph operator is the number of critical points of its complex Bloch variety. Determining it is a step towards the spectral edges conjecture and more generally understanding Bloch varieties. Previous work showed that it is bounded above by the volume of the Newton polytope of the graph, and that the inequality is strict when there are asymptotic critical points. We identify contributions from asymptotic critical points that arise from the structure of the graph, and show that the critical point degree is bounded above by the difference of the volume of the Newton polytope and these contributions. These results have implications for nonlinear optimization.