# Minimal $\mathcal{S}$-universality criteria may vary in size

Noam D. Elkies^{[1]}; Daniel M. Kane^{[2]}; Scott Duke Kominers^{[3]}

- [1] Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138
- [2] Department of Mathematics Stanford University Building 380, Sloan Hall Stanford, California 94305
- [3] Society of Fellows Dpt of Economics Program for Evolutionary Dynamics Center for Research on Computation and Society Harvard University One Brattle Square, Suite 6 Cambridge, MA 02138-3758

Journal de Théorie des Nombres de Bordeaux (2013)

- Volume: 25, Issue: 3, page 557-563
- ISSN: 1246-7405

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topElkies, Noam D., Kane, Daniel M., and Kominers, Scott Duke. "Minimal $\mathcal{S}$-universality criteria may vary in size." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 557-563. <http://eudml.org/doc/275777>.

@article{Elkies2013,

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.},

affiliation = {Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138; Department of Mathematics Stanford University Building 380, Sloan Hall Stanford, California 94305; Society of Fellows Dpt of Economics Program for Evolutionary Dynamics Center for Research on Computation and Society Harvard University One Brattle Square, Suite 6 Cambridge, MA 02138-3758},

author = {Elkies, Noam D., Kane, Daniel M., Kominers, Scott Duke},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {universality criteria; quadratic forms},

language = {eng},

month = {11},

number = {3},

pages = {557-563},

publisher = {Société Arithmétique de Bordeaux},

title = {Minimal $\mathcal\{S\}$-universality criteria may vary in size},

url = {http://eudml.org/doc/275777},

volume = {25},

year = {2013},

}

TY - JOUR

AU - Elkies, Noam D.

AU - Kane, Daniel M.

AU - Kominers, Scott Duke

TI - Minimal $\mathcal{S}$-universality criteria may vary in size

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2013/11//

PB - Société Arithmétique de Bordeaux

VL - 25

IS - 3

SP - 557

EP - 563

AB - 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.

LA - eng

KW - universality criteria; quadratic forms

UR - http://eudml.org/doc/275777

ER -

## References

top- 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. Zbl0987.11027MR1803359
- 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. Zbl0987.11026MR1803358
- 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. Zbl1143.11309MR2058677
- 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. Zbl0955.11011MR1732349
- —, A finiteness theorem for representability of quadratic forms by forms. Journal fur die Reine und Angewandte Mathematik 581 (2005), 23–30. Zbl1143.11011MR2132670
- S. D. Kominers, The $8$-universality criterion is unique. Preprint, arXiv:0807.2099, 2008. MR2681001
- —, Uniqueness of the $2$-universality criterion. Note di Matematica 28 (2008), no. 2, 203–206. Zbl1219.11061MR2681001
- B.-K. Oh, Universal $\mathbb{Z}$-lattices of minimal rank. Proceedings of the American Mathematical Society 128 (2000), 683–689. Zbl1044.11015MR1654105

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