Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T14:35:33.041Z Has data issue: false hasContentIssue false
  • English
  • Français

An analogue of Vosper's theorem for extension fields

Published online by Cambridge University Press:  31 January 2017

CHRISTINE BACHOC ,
ORIOL SERRA  and
GILLES ZÉMOR
CHRISTINE BACHOC
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: [email protected]
ORIOL SERRA
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya and Barcelona Graduate School of Mathematics, Edifici C3, Despatx: 112, C. Jordi Girona, 1–3, 08034 Barcelona, Spain. e-mail: [email protected]
GILLES ZÉMOR
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which

\begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath}
when dimFS, dimFT ⩾ 2 and dimFST ⩽ [L : F] − 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 423 - 452
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bachoc, C., Serra, O. and Zémor, G. Revisiting Kneser's Theorem for Field Extensions. Combinatorica. To appear.Google Scholar
[2] Bannai, E. and Ito, T. Algebraic Combinatorics I: Association Schemes (Benjamin/Cummings Publishing Co., 1984).Google Scholar
[3] Delsarte, P. An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 197, (1973).Google Scholar
[4] Delsarte, P. Bilinear forms over a finite field, with applications to coding theory. J. Combinatorial Theory Ser. A. (3) 25 (1978), 226–241.CrossRefGoogle Scholar
[5] Delsarte, P. and Goethals, J. M. Alternating bilinear forms over GF(q). J. Combinatorial Theory Ser. A. (1) 19 (1975), 26–50.CrossRefGoogle Scholar
[6] Delsarte, P. and Levenshtein, V. Association schemes and coding theory. IEEE Trans. Inform. Theory. (6) 44 (1998), 2477–2504.CrossRefGoogle Scholar
[7] Eliahou, S. and Lecouvey, C. On linear versions of some additive theorems. Linear Multilinear Algebra. 57 (2009), 759–775.CrossRefGoogle Scholar
[8] Feng, R., Wang, Y., Ma, C. and Ma, J.. Eigenvalues of association schemes of quadratic forms. Discrete Maths. 308 (2008), 3023–3047.CrossRefGoogle Scholar
[9] Freiman, G. A. Foundations of a structural theory of set addition. Transl. Math. Monographs 37 (Amer. Math. Soc., Providence, RI, 1973).Google Scholar
[10] In memory of Yahya Ould Hamidoune, special issue of European Journal of Combinatorics, Plagne, Serra and Zémor Eds. vol. 34 (2013).Google Scholar
[11] Hamidoune, Y. O. On the connectivity of Cayley digraphs. European J. Combin. 5 (1984), 309–312.CrossRefGoogle Scholar
[12] Hamidoune, Y. O. An isoperimetric method in additive theory. J. Algebra. 179 (1996), 622–630.CrossRefGoogle Scholar
[13] Hamidoune, Y. O. Some results in additive number theory I: the critical pair theory. Acta Arith. 96 (2000), 97–119.CrossRefGoogle Scholar
[14] Hamidoune, Y. O. Some additive applications of the isoperimetric approach. Ann. Inst. Fourier. 58 (2008), fasc. 6 (2007-2036).CrossRefGoogle Scholar
[15] Hartshorne, R. Algebraic Geometry. GTM 52 (Springer).CrossRefGoogle Scholar
[16] Hou, X., Leung, K.H., and Xiang, Q. A generalization of an addition theorem of Kneser. J. Number Theory 97 (2002), 1–9.CrossRefGoogle Scholar
[17] Kemperman, J.H.B. On complexes in a semigroup. Idag. Math. 18 (1956), 247–254.Google Scholar
[18] Kneser, M. Summenmengen in lokalkompakten abelesche Gruppen. Math. Z. 66 (1956), 88–110.CrossRefGoogle Scholar
[19] Lang, S. Algebra (Springer, 3rd Edition, 2005).Google Scholar
[20] Lang, S. and Weil, A. Number of points of varieties over finite fields. Amer. J. Math. 76 (1954), 819–827.CrossRefGoogle Scholar
[21] Lecouvey, C. Plünnecke and Kneser type theorems for dimension estimates. Combinatorica (3) 34 (2014), 331–358.CrossRefGoogle Scholar
[22] MacWilliams, F.J. and Sloane, N.J.A.. The Theory of Error-Correcting Codes (North-Holland 1977).Google Scholar
[23] Mann, H. B. Addition Theorems: the Addition Theorems of Group Theory and Number Theory (Krieger Publishing Company, 1976).Google Scholar
[24] Nathanson, M.B. Additive Number Theory. Inverse problems and the geometry of sumsets. Grad. Texts in Math. 165 (Springer, 1996).CrossRefGoogle Scholar
[25] Olson, J.E. On the sum of two sets in a group. J. Number Theory 18 (1984), 110–120.CrossRefGoogle Scholar
[26] Plagne, A., Serra, O. and Zémor, G. Yahya Ould Hamidoune's mathematical journey: a critical review of his work. European J. Combin. 34 (2013), 1207–1222.CrossRefGoogle Scholar
[27] Ruzsa, I.Z. An application of graph theory to additive number theory. Sci. Ser. A Math. Sci. (N.S.) 3 (1989), 97–109.Google Scholar
[28] Schmidt, K.-U.. Symmetric bilinear forms over finite fields of even characteristic. J. Combinatorial Theory Ser. A. (8) 117 (2010), 1011–1026.CrossRefGoogle Scholar
[29] Schmidt, K.-U.. Symmetric bilinear forms over finite fields with applications to coding theory. J. Algebraic Combinatorics 42 (2015), 635–670.CrossRefGoogle Scholar
[30] Tao, T. and Vu, V. Additive Combinatorics (Cambridge University Press, 2006).CrossRefGoogle Scholar
[31] Taylor, D. E. The geometry of the classical groups Sigma Series on Pure Mathematics 9 (Heldermann, Berlin 1992).Google Scholar
[32] Vosper, G. The critical pairs of subsets of a group of prime order. J. London Math. Soc. 31 (1956), 200–205.CrossRefGoogle Scholar
[33] Wang, Y., Wang, C., Ma, C. and Ma, J.. Association Schemes of Quadratic Forms and Symmetric Bilinear Forms. J. Algebraic Combin. 17 (2003), 149–161.CrossRefGoogle Scholar