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Noam D. Elkies; Daniel M. Kane; Scott Duke Kominers
Minimal $\mathcal{S}$-universality criteria may vary in size
Journal de théorie des nombres de Bordeaux, 25 no. 3 (2013), p. 557-563, doi: 10.5802/jtnb.848
Article PDF | Reviews MR 3179676 | Zbl 1286.11046
Class. Math.: 11E20, 11E25

Résumé - Abstract

In this note, we give simple examples of sets $\mathcal{S}$ of quadratic forms that have minimal $\mathcal{S}$-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

Bibliography

[Bha00] M. Bhargava, On the Conway-Schneeberger fifteen theorem. Quadratic forms and their applications: Proceedings of the Conference on Quadratic Forms and Their Applications, July 5–9, 1999, University College Dublin, Contemporary Mathematics, vol. 272, American Mathematical Society, 2000, pp. 27–37.  MR 1803359 |  Zbl 0987.11027
[Con00] J. H. Conway, Universal quadratic forms and the fifteen theorem. Quadratic forms and their applications: Proceedings of the Conference on Quadratic Forms and Their Applications, July 5–9, 1999, University College Dublin, Contemporary Mathematics, vol. 272, American Mathematical Society, 2000, pp. 23–26.  MR 1803358 |  Zbl 0987.11026
[Kim04] M.-H. Kim, Recent developments on universal forms. Algebraic and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, vol. 344, American Mathematical Society, 2004, pp. 215–228.  MR 2058677 |  Zbl 1143.11309
[KKO99] B. M. Kim, M.-H. Kim, and B.-K. Oh, $2$-universal positive definite integral quinary quadratic forms. Integral quadratic forms and lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, Contemporary Mathematics, vol. 249, American Mathematical Society, 1999, pp. 51–62.  MR 1732349 |  Zbl 0955.11011
[KKO05] —, A finiteness theorem for representability of quadratic forms by forms. Journal fur die Reine und Angewandte Mathematik 581 (2005), 23–30.  MR 2132670 |  Zbl 1143.11011
[Kom08a] S. D. Kominers, The $8$-universality criterion is unique. Preprint, arXiv:0807.2099, 2008.  MR 2681001
[Kom08b] —, Uniqueness of the $2$-universality criterion. Note di Matematica 28 (2008), no. 2, 203–206.  MR 2681001
[Oh00] B.-K. Oh, Universal $\mathbb{Z}$-lattices of minimal rank. Proceedings of the American Mathematical Society 128 (2000), 683–689.  MR 1654105 |  Zbl 1044.11015