Minimal 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

Abstract

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

How to cite

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Elkies, 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

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  1. 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
  2. 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
  3. 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
  4. 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
  5. —, A finiteness theorem for representability of quadratic forms by forms. Journal fur die Reine und Angewandte Mathematik 581 (2005), 23–30. Zbl1143.11011MR2132670
  6. S. D. Kominers, The 8 -universality criterion is unique. Preprint, arXiv:0807.2099, 2008. MR2681001
  7. —, Uniqueness of the 2 -universality criterion. Note di Matematica 28 (2008), no. 2, 203–206. Zbl1219.11061MR2681001
  8. B.-K. Oh, Universal ℤ -lattices of minimal rank. Proceedings of the American Mathematical Society 128 (2000), 683–689. Zbl1044.11015MR1654105

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