发表机构
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了所有势函数下色散多项式可约的周期图算子，并通过两步归约证明其他算子的色散多项式在一般势下不可约。
AI 中文摘要
我们给出了周期图算子的完整刻画，这些算子的色散多项式对每个势函数都是可约的。因此，参数相关洛朗多项式的可约性二分法意味着，对于任何其他周期图算子，其色散多项式（从而布洛赫簇）在一般势函数下是不可约的。我们的证明通过两步归约进行。首先，我们利用色散多项式根关于势参数的单值性，将问题归约为分裂情形。然后，我们证明分裂性质传递到Floquet矩阵的主子矩阵，从而进一步将问题归约为基本域包含两个顶点的情况。
英文摘要
We give a complete characterization of periodic graph operators whose dispersion polynomials are reducible for every potential. Consequently, the reducibility dichotomy for parameter-dependent Laurent polynomials implies that, for every other periodic graph operator, the dispersion polynomial, and hence the Bloch variety, is irreducible for generic potentials. Our proof proceeds through two reductions. We first use the monodromy of the roots of the dispersion polynomial with respect to the potential parameters to reduce the problem to the splitting case. We then show that the splitting property passes to the principal submatrices of the Floquet matrix, reducing the problem further to the case that the fundamental domain contains two vertices.
Comments25 pages
