On indecomposable and non-decomposable quadratic forms
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.
scientific article; zbMATH DE number 4210236
scientific article