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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
DOI: https://doi.org/10.1090/S0002-9939-99-05254-5
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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  • [1] E. Bannai, Positive definite unimodular lattices with trivial automorphism groups, Ohio State Univ., Thesis, 1988.
  • [2] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369
  • [3] J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. I. Quadratic forms of small determinant, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 17–41. MR 953276
  • [4] J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. V. Integral coordinates for integral lattices, Proc. Roy. Soc. London Ser. A 426 (1989), no. 1871, 211–232. MR 1030461
  • [5] J.H. Conway, W. Schneeberger, A $15$-theorem for universal quadratic forms, to appear.
  • [6] B.M. Kim, M-H. Kim, S. Raghavan, $2$-universal positive definite integral quinary diagonal quadratic forms, Ramanujan J. 1 (1997), 333-337. CMP 98:10
  • [7] B.M. Kim, M-H. Kim, B-K. Oh, $2$-universal positive definite integral quinary quadratic forms, Preprint.
  • [8] Myung-Hwan Kim and Byeong-Kweon Oh, Representations of positive definite senary integral quadratic forms by a sum of squares, J. Number Theory 63 (1997), no. 1, 89–100. MR 1438651, https://doi.org/10.1006/jnth.1997.2069
  • [9] Myung-Hwan Kim and Byeong-Kweon Oh, A lower bound for the number of squares whose sum represents integral quadratic forms, J. Korean Math. Soc. 33 (1996), no. 3, 651–655. MR 1419759
  • [10] C. Ko, On the representation of a sum of squares of linear forms, Quart. J. Math. Oxford 8 (1937), 81-98.
  • [11] -, On the decomposition of quadratic forms in six variables, Acta Arith. 3 (1939), 64-78.
  • [12] J.L. Lagrange, Oeuvres 3 (1869), 189-201.
  • [13] O. T. O’Meara, The integral representations of quadratic forms over local fields, Amer. J. Math. 80 (1958), 843–878. MR 0098064, https://doi.org/10.2307/2372837
  • [14] -, Introduction to quadratic forms, Springer-Verlag, 1973.
  • [15] L.J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math. Oxford 1 (1930), 276-288.
  • [16] S. Ramanujan, On the expression of a number in the form $ax^{2}+by^{2}+cz^{2}+dw^{2}$, Proc Cambridge Phil. Soc. 19 (1917), 11-21.
  • [17] W. Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory 47 (1994), no. 3, 273–283. MR 1278399, https://doi.org/10.1006/jnth.1994.1037
  • [18] G. L. Watson, The class-number of a positive quadratic form, Proc. London Math. Soc. (3) 13 (1963), 549–576. MR 0150104, https://doi.org/10.1112/plms/s3-13.1.549
  • [19] G. L. Watson, One-class genera of positive quadratic forms in at least five variables, Acta Arith. 26 (1974/75), no. 3, 309–327. MR 0379369
  • [20] G. L. Watson, One-class genera of positive quadratic forms in nine and ten variables, Mathematika 25 (1978), no. 1, 57–67. MR 0491499, https://doi.org/10.1112/S0025579300009268
  • [21] M.F. Willerding, Determination of all classes of (positive) quaternary quadratic forms which represent all positive integers, Bull. Amer. Math. Soc. 54 (1948), 334-337. MR 9:571e

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Additional Information

Byeong-Kweon Oh
Email: oandhan@math.snu.ac.kr

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