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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Byeong-Kweon Oh
Journal: Proc. Amer. Math. Soc. 128 (2000), 683-689
MSC (1991): Primary 11E12, 11H06
Published electronically: July 6, 1999
MathSciNet review: 1654105
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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
Email: oandhan@math.snu.ac.kr

DOI: http://dx.doi.org/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
Published electronically: July 6, 1999
Additional Notes: The author was partially supported by GARC and BSRI-98-1414
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society