Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over $\mathcal{O}$ with an orthonormal basis, let $\mathfrak{S}_d(1)$ be the set consisting of all positive definite integral unary Hermitian lattices over $\mathcal{O}$ that can be represented by some $I_m$, and let $g_d(1)$ be the least positive integer such that all Hermitian lattices in $\mathfrak{S}_d(1)$ can be uniformly represented by $I_{g_d(1)}$. The main results of this work provide an algorithm to calculate the explicit form of $\mathfrak{S}_d(1)$ and the exact value of $g_d(1)$ for every imaginary quadratic field $E$, which can be viewed as a natural extension of the Pythagoras number in the lattice setting.

翻译：设$E=\mathbb{Q}\big(\sqrt{-d}\big)$为虚二次域，其中$d$为无平方因子正整数，$\mathcal{O}$为其整数环。对每个正整数$m$，令$I_m$为$\mathcal{O}$上具有标准正交基的自由Hermite格子，令$\mathfrak{S}_d(1)$为所有可由某个$I_m$表示的$\mathcal{O}$上正定整一元Hermite格子构成的集合，并令$g_d(1)$为最小正整数，使得$\mathfrak{S}_d(1)$中所有Hermite格子均可被$I_{g_d(1)}$一致表示。本研究的主要结果为每个虚二次域$E$提供了一种计算$\mathfrak{S}_d(1)$显式形式及$g_d(1)$精确值的算法，这可视为格子设定下Pythagoras数的自然推广。