阿贝尔Cayley高维扩展图的多对数度数
Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree
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中文总结 AI 辅助
本文构造了显式的无限族二维Cayley复形，其度数多项式且特征值可控，并推广到加权高维情形，利用代数曲线有理点求值实现。
中文摘要 AI 辅助
我们构造了一个显式的无限族简单二维Cayley复形，定义在$\mathbb{F}_2^n$上，其度数为$n$的多项式，且对于每个固定的$\lambda>0$，其非平凡顶点链接特征值位于$[-\lambda,\lambda]$中。对于每个固定的$d\ge2$，我们还获得了定义在$\mathbb{F}_2^n$上的显式无限族加权$d$维Cayley复形，其余维二局部谱范数至多为$1/d$，且Cayley度数为$\Theta_d(n)$。我们的二维构造利用代数曲线上有理点的求值来产生射影方向集以及沿这些方向仿射的许多函数，这可能对进一步的构造和改进有用。
英文摘要
We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-λ,λ]$ for every fixed $λ>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $Θ_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.
发表机构
- Johns Hopkins University（约翰斯·霍普金斯大学）
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