Isolations of cubic lattices from their proper sublattices

2021/04/09 by Byeong-Kweon Oh, Oh, Byeong-Kweon · 1 citation
Computer Science · Engineering · #11E12 #11E25 #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2104.04308

openalex publication_date 2021/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A (positive definite and integral) quadratic form is called \it an isolation of a quadratic form f if it represents all subforms of f except for f itself. The minimum rank of isolations of a quadratic form f is denoted, if it exists, by Iso(f). In this article, we show that Iso(I₂)=5 and Iso(I₃)=6, where I_n=x₁²+…+x_n² is the sum of n squares for any positive integer n. After proving that there always exists an isolation of I_n for any positive integer n, we provide some explicit lower and upper bounds for Iso(I_n). In particular, we show that Iso(I_n) ∈ Ω(n^\frac32-ε) for any ε>0.

Cited by

• Composition laws of binary quadratic forms and isolations of quadratic forms

Related

• Isolations of the sum of two squares from its proper subforms
• Composition laws of binary quadratic forms and isolations of quadratic forms
• Quadratic forms with a strong regularity property on the representations of squares
• Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one
• The number of representations of squares by integral quaternary quadratic forms