On indecomposable and non-decomposable quadratic forms
type: Item
instance of: scholarly article
title: On indecomposable and non-decomposable quadratic forms
zbMATH Open document ID: 0732.11019
author: Fuzu Zhu
published in: Southeast Asian Bulletin of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: In the present paper the following two theorems are proved. Theorem 2.1: If the positive definite quadratic form f(X) over \({\mathbb{Z}}\) is unimodular, then f(X) is non-decomposable if and only it is indecomposable. Theorem 2.2: For any natural number n there exists an n- ary non-decomposable positive definite unimodular quadratic form over \({\mathbb{Z}}\) but for the following ten exceptions: \(n=2,3,4,5,6,7,9,10,11,13\). In these exceptional cases, there are no forms with the desired properties. Here some words for the terminologies: \({\mathbb{Z}}\) is the ring of rational integers. The quadratic form \(f(X)=f(X_ 1,...,X_ n)=\sum^{n}_{i,j=1}a_{ij}X_ iX_ j,\quad a_{ij}=a_{ji}\in {\mathbb{Z}}\) which is positive definite, is called non-decomposable if there is no expression of the type \(f(X)=g(X)+h(X)\) with g(X) and h(X) non- negative over \({\mathbb{Z}}\). A quadratic form f(X) is called indecomposable if the lattice L associated with f(X) does not admit an orthogonal splitting \(L=P\perp R\) into two non-zero sublattices. The latter theorem is already obtained by previous papers (including one by the present author), but the proof in the present paper is largely shortened.
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 4210236
zbMATH Keywords: unimodular quadratic form
zbMATH Keywords: non-decomposable
zbMATH Keywords: indecomposable
reviewed by: Michio Ozeki
MaRDI profile type: Publication
Recommended article: Q3825147
Recommended article: Decomposable quadratic forms and involutions
Recommended article: Indecomposable Positive Quadratic Forms
Recommended article: On the composition of nondegenerate quadratic forms with an arbitrary index
Recommended article: Indecomposable binary quadratic forms
Recommended article: On quadratic forms over inseparable quadratic extensions
Recommended article: The construction of indecomposable positive definite quadratic forms.
Recommended article: On the theory of indefinite quadratic forms
description: scientific article; zbMATH DE number 4210236