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Universal -lattices of minimal rank
Authors:Byeong-Kweon Oh
Affiliation:Department of Mathematics, Seoul National University, Seoul, 151-742, Korea
Abstract:Let $U_{mathbb{Z}}(n)$ be the minimal rank of $n$-universal $mathbb{Z}$-lattices, by which we mean positive definite $mathbb{Z}$-lattices which represent all positive $mathbb{Z}$-lattices of rank $n$. 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 $n$-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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