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

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 $6 \le n \le 8$.

References

E. Bannai, Positive definite unimodular lattices with trivial automorphism groups, Ohio State Univ., Thesis, 1988.
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, DOI 10.1007/978-1-4757-2016-7
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
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
J.H. Conway, W. Schneeberger, A $15$-theorem for universal quadratic forms, to appear.
B.M. Kim, M-H. Kim, S. Raghavan, $2$-universal positive definite integral quinary diagonal quadratic forms, Ramanujan J. 1 (1997), 333-337.
B.M. Kim, M-H. Kim, B-K. Oh, $2$-universal positive definite integral quinary quadratic forms, Preprint.
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, DOI 10.1006/jnth.1997.2069
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
C. Ko, On the representation of a sum of squares of linear forms, Quart. J. Math. Oxford 8 (1937), 81-98.
—, On the decomposition of quadratic forms in six variables, Acta Arith. 3 (1939), 64-78.
J.L. Lagrange, Oeuvres 3 (1869), 189-201.
O. T. O’Meara, The integral representations of quadratic forms over local fields, Amer. J. Math. 80 (1958), 843–878. MR 98064, DOI 10.2307/2372837
—, Introduction to quadratic forms, Springer-Verlag, 1973.
L.J. Mordell, A new Waring’s problem with squares of linear forms, Quart. J. Math. Oxford 1 (1930), 276-288.
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.
W. Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory 47 (1994), no. 3, 273–283. MR 1278399, DOI 10.1006/jnth.1994.1037
G. L. Watson, The class-number of a positive quadratic form, Proc. London Math. Soc. (3) 13 (1963), 549–576. MR 150104, DOI 10.1112/plms/s3-13.1.549
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 379369, DOI 10.4064/aa-26-3-309-327
G. L. Watson, One-class genera of positive quadratic forms in nine and ten variables, Mathematika 25 (1978), no. 1, 57–67. MR 491499, DOI 10.1112/S0025579300009268
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945