An algorithm for $g$-invariant on unary Hermitian lattices over imaginary quadratic fields

TL;DR

This paper develops an algorithm to explicitly determine certain unary Hermitian lattices over imaginary quadratic fields and computes the minimal universal rank for these lattices, extending classical number theory concepts.

Contribution

It introduces a method to compute the set of representable unary Hermitian lattices and determines the minimal rank needed for universal representation over all imaginary quadratic fields.

Findings

Provides an explicit algorithm for classifying unary Hermitian lattices

Calculates the exact minimal rank for universal representation in each field

Extends the concept of Pythagoras number to Hermitian lattice setting

Abstract

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.