Assume E is an imaginary quadratic field and $\mathcal{O}$ is its ring of integers. For each positive integer m, let $I_m$ be the free Hermitian lattice of rank m over $\mathcal{O}$ having an orthonormal basis. For each positive integer n, let ${\mathfrak{S}}_{\mathcal{O}}(n)$ be the set of all Hermitian lattices of rank n over $\mathcal{O}$ that can be represented by some $I_m$. Denote by $g_{\mathcal{O}}(n)$ the smallest positive integer g such that each Hermitian lattice in ${\mathfrak{S}}_{\mathcal{O}}(n)$ can be represented by $I_g$. In this paper, we shall provide an explicit upper bound for $g_{\mathcal{O}}(n)$ for all imaginary quadratic fields E and positive integers n.

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关于厄米格的沃林问题

假设E是一个虚数二次场，$\mathcal{Ø}$是它的整数环。对于每个正整数m，让${\mathrm{一世}}_{米}$成为m级以上的自由厄米格$\mathcal{Ø}$具有正交基础。对于每个正整数n，让${\mathfrak{小号}}_{\mathcal{Ø}}（ñ）$是集级别的所有埃尔米特格ñ过$\mathcal{Ø}$ 可以用一些代表 ${\mathrm{一世}}_{米}$。表示为$G_{\mathcal{Ø}}（ñ）$最小正整数g，使得每个Hermitian晶格${\mathfrak{小号}}_{\mathcal{Ø}}（ñ）$ 可以用 ${\mathrm{一世}}_G$。在本文中，我们将为$G_{\mathcal{Ø}}（ñ）$对于所有虚数二次场E和正整数n。