Generic Irreducibility of Bloch Varieties for Periodic Graph Operators

TL;DR

This paper characterizes when dispersion polynomials and Bloch varieties of periodic graph operators are generically irreducible, linking it to the connectivity of the quotient graph.

Contribution

It provides a complete characterization of generic irreducibility for Bloch varieties of periodic graphs based on edge weights and connectivity.

Findings

Irreducibility occurs generically if and only if the quotient graph is connected.

A dichotomy for parameterized Laurent polynomials is established: reducibility either always occurs or almost never.

The problem reduces to minimally connected periodic graphs.

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.