发表机构
Bielefeld University of Applied Sciences（比勒费尔德应用科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明度量图tau不变量的正下界与总长度成正比，并构造反例否证通用常数1/108，该下界与亏格、边数及边长分布无关，并应用于雅可比簇高度公式。
AI 中文摘要
我们证明了Baker--Rumely猜想，即度量图的tau不变量存在一个正下界，该下界与其总长度成正比，且比例常数为绝对常数。我们还构造了简单的三次度量图，其归一化tau不变量趋于$59/7260<1/108$，从而否证了提出的通用常数$1/108$。该下界与亏格、边数以及边长的分布无关。其证明结合了欧几里得格上的二阶矩不等式与将循环格的边坐标划分为三个独立集合的方法。反例仅具有两种边长，并允许进行初等的电阻计算。通过热带矩恒等式，该下界还为雅可比簇高度公式中的非阿基米德项提供了均匀估计。
英文摘要
We prove the Baker--Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.
