The Critical Point Degree of a Periodic Graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912740354818048 |
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| author | Faust, Matthew Robinson, Jonah Sottile, Frank |
| author_facet | Faust, Matthew Robinson, Jonah Sottile, Frank |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Critical Point Degree of a Periodic Graph Faust, Matthew Robinson, Jonah Sottile, Frank Spectral Theory Algebraic Geometry 81Q10, 14M25, 47A75, 51B20, 05C50 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. |
| title | The Critical Point Degree of a Periodic Graph |
| topic | Spectral Theory Algebraic Geometry 81Q10, 14M25, 47A75, 51B20, 05C50 |
| url | https://arxiv.org/abs/2512.01226 |