Generic Irreducibility of Bloch Varieties for Periodic Graph Operators

Matthew Faust; Wencai Liu

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arXiv · 2605.01212

Generic Irreducibility of Bloch Varieties for Periodic Graph Operators

Matthew Faust·Wencai Liu

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Abstract

We give a complete characterization of generic irreducibility for dispersion polynomials and Bloch varieties of periodic graph operators. More precisely, we prove that for a generic choice of edge weights and potentials, the dispersion polynomial/Bloch variety of a nontrivial periodic graph is irreducible if and only if the quotient graph is connected. Our proof uses a strong dichotomy for parameterized Laurent polynomials: reducibility either occurs for every parameter or fails on a nonempty Zariski-open set. After establishing this dichotomy, we reduce the problem to minimally connected periodic graphs.

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BibTeX RIS

Matthew Faust, Wencai Liu. 2026-05-02. Generic Irreducibility of Bloch Varieties for Periodic Graph Operators. https://arxiv.org/abs/2605.01212

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