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The Critical Point Degree of a Periodic Graph

Matthew Faust, Jonah Robinson, Frank Sottile

TL;DR

The paper develops a toric-geometry framework to bound the critical point degree cpdeg of the Bloch variety for Z^d-periodic graphs, refining the classical BKK-type bound cpdeg ≤ nvol(A). By introducing and quantifying two structural correction terms, N_vert from vertical faces and N_disc from asymptotic/disconnected initial graphs, it shows cpdeg ≤ nvol(A) − N_vert − N_disc for dimensions d = 1, 2, 3. The approach connects the asymptotic behavior of BV to faces of the Newton polytope via initial forms and initial graphs, and leverages projective toric varieties to interpret critical points as linear-section intersections with multiplicities. Concrete examples illustrate how graph structure induces asymptotic critical points and how singularities contribute to multiplicities, with implications for nonlinear optimization and spectral-edge analysis in periodic media.

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.

The Critical Point Degree of a Periodic Graph

TL;DR

The paper develops a toric-geometry framework to bound the critical point degree cpdeg of the Bloch variety for Z^d-periodic graphs, refining the classical BKK-type bound cpdeg ≤ nvol(A). By introducing and quantifying two structural correction terms, N_vert from vertical faces and N_disc from asymptotic/disconnected initial graphs, it shows cpdeg ≤ nvol(A) − N_vert − N_disc for dimensions d = 1, 2, 3. The approach connects the asymptotic behavior of BV to faces of the Newton polytope via initial forms and initial graphs, and leverages projective toric varieties to interpret critical points as linear-section intersections with multiplicities. Concrete examples illustrate how graph structure induces asymptotic critical points and how singularities contribute to multiplicities, with implications for nonlinear optimization and spectral-edge analysis in periodic media.

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.

Paper Structure

This paper contains 13 sections, 32 theorems, 112 equations, 11 figures.

Table of Contents

  1. Algebraic Aspects of Discrete Periodic Operators
  2. Discrete Periodic Operators
  3. Bloch and Fermi Varieties
  4. Critical Points of Discrete Periodic Operators
  5. The Initial Graph of a periodic operator
  6. Cancellation-free determinants
  7. Initial matrix
  8. The initial graph
  9. Toric varieties
  10. Toric varieties and critical points
  11. Structure of $X_{\mathcal{A}}$
  12. Vertical faces
  13. Initial graph disconnected

Key Result

Proposition 1.1

A point $(z,\lambda)\in({\mathbb C}^\times)^d\times {\mathbb C}$ is a critical point of the function $\lambda$ on the Bloch variety if and only if it is a solution to the system of equations

Figures (11)

  • Figure 1: Three periodic graphs.
  • Figure 2: The hexagonal lattice and a labeling in a neighborhood of $W$.
  • Figure 3: Three Bloch varieties for the hexagonal lattice
  • Figure 4: Three more Bloch varieties
  • Figure 5: Some Newton polytopes
  • ...and 6 more figures

Theorems & Definitions (52)

  • Proposition 1.1: Prop. 2.1 of FS
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • proof
  • Proposition 1.5: Cor. 2.5 FS
  • Lemma 1.6
  • proof
  • Corollary 1.7
  • proof
  • ...and 42 more