Universal ℤ-lattices of minimal rank

Oh, Byeong-Kweon

doi:10.1090/S0002-9939-99-05254-5

Universal $\mathbb {Z}$-lattices of minimal rank
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by Byeong-Kweon Oh

Proc. Amer. Math. Soc. 128 (2000), 683-689

DOI: https://doi.org/10.1090/S0002-9939-99-05254-5

Published electronically: July 6, 1999

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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 $6 \le n \le 8$.

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