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Universal $\mathbb{Z}$-lattices of minimal rank

Author(s): Byeong-Kweon Oh
Journal: Proc. Amer. Math. Soc. 128 (2000), 683-689.
MSC (1991): Primary 11E12, 11H06
Posted: 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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Additional Information:

Byeong-Kweon Oh
Affiliation: Department of Mathematics, Seoul National University, Seoul, 151-742, Korea
Email: oandhan@math.snu.ac.kr

DOI: 10.1090/S0002-9939-99-05254-5
PII: S 0002-9939(99)05254-5
Keywords: $n$-universal lattice, $U_{\mathbb{Z}}(n)$, root lattice, additively indecomposable
Received by editor(s): April 27, 1998
Posted: July 6, 1999
Additional Notes: The author was partially supported by GARC and BSRI-98-1414
Communicated by: David E. Rohrlich
Copyright of article: Copyright 1999, American Mathematical Society

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