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Universal -lattices of minimal rank

Authors:	Byeong-Kweon Oh

Institution:	Department of Mathematics, Seoul National University, Seoul, 151-742, Korea

Abstract:	Let $U_{\mathbb{Z}}(n)$ be the minimal rank of -universal $\mathbb{Z}$ -lattices, by which we mean positive definite $\mathbb{Z}$ -lattices which represent all positive $\mathbb{Z}$ -lattices of rank . It is a well known fact that $U_{\mathbb{Z}}(n)=n+ 3$ for $1 \le n \le 5$ . In this paper, we determine $U_{\mathbb{Z}}(n)$ and find all -universal lattices of rank $U_{\mathbb{Z}}(n)$ for $6 \le n \le 8$ .

Keywords:	$n$-universal lattice $U_{\mathbb{Z}}(n)$ root lattice additively indecomposable

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