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

2023/09/28 by Jingbo Liu, Liu, Jingbo
Mathematics · #11Y40 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Primary 11E39. Secondary 11Y16

paper · pdf · doi:10.48550/arxiv.2309.16138

openalex publication_date 2023/09/28 · openalex created_date 2023/10/01 · openalex updated_date 2026/07/28

Abstract

Let E=ℚ(√(-d)) be an imaginary quadratic field for a square-free positive integer d, and let O be its ring of integers. For each positive integer m, let I_m be the free Hermitian lattice over O with an orthonormal basis, let \mathfrakS_d(1) be the set consisting of all positive definite integral unary Hermitian lattices over 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 \mathfrakS_d(1) can be uniformly represented by I_gd(1). The main results of this work provide an algorithm to calculate the explicit form of \mathfrakS_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.

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