Minimal 𝒮-universality criteria may vary in size
Noam D. Elkies ; Daniel M. Kane ; Scott Duke Kominers
Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 557-563.

Nous donnons des exemples simples d’ensembles 𝒮 de formes quadratiques qui ont des critères d’universalité minimaux de plusieurs cardinalités. Nous donnons ainsi une réponse négative à une question de Kim, Kim et Oh [KKO05].

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.

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Publié le :
DOI : https://doi.org/10.5802/jtnb.848
Classification : 11E20,  11E25
Mots clés : universality criteria, quadratic forms 
@article{JTNB_2013__25_3_557_0,
     author = {Noam D. Elkies and Daniel M. Kane and Scott Duke Kominers},
     title = {Minimal $\mathcal{S}$-universality criteria may vary in size},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {557--563},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {3},
     year = {2013},
     doi = {10.5802/jtnb.848},
     zbl = {1286.11046},
     mrnumber = {3179676},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/}
}
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, Tome 25 (2013) no. 3, pp. 557-563. doi : 10.5802/jtnb.848. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/

[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 ℤ-lattices of minimal rank. Proceedings of the American Mathematical Society 128 (2000), 683–689. | MR 1654105 | Zbl 1044.11015