ALGEBRAIC COMBINATORICS

no. 4
Towards a function field version of Freiman’s Theorem
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 501-521.

We discuss a multiplicative counterpart of Freiman’s 3k-4 theorem in the context of a function field F over an algebraically closed field K. Such a theorem would give a precise description of subspaces S, such that the space S 2 spanned by products of elements of S satisfies dimS 2 ≤3dimS-4. We make a step in this direction by giving a complete characterisation of spaces S such that dimS 2 =2dimS. We show that, up to multiplication by a constant field element, such a space S is included in a function field of genus 0 or 1. In particular if the genus is 1 then this space is a Riemann–Roch space.

Received: 2017-09-15
Revised: 2018-04-18
Accepted: 2018-05-07
Published online: 2018-09-10
DOI: https://doi.org/10.5802/alco.19
Classification: 11R58,  11P99,  05E40,  14H05
Keywords: Additive combinatorics, function fields 
@article{ALCO_2018__1_4_501_0,
     author = {Bachoc, Christine and Couvreur, Alain and Z\'emor, Gilles},
     title = {Towards a function field version of Freiman's Theorem},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     pages = {501-521},
     doi = {10.5802/alco.19},
     mrnumber = {3875074},
     zbl = {06963902},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2018__1_4_501_0/}
}
Bachoc, Christine; Couvreur, Alain; Zémor, Gilles. Towards a function field version of Freiman’s Theorem. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 501-521. doi : 10.5802/alco.19. https://alco.centre-mersenne.org/item/ALCO_2018__1_4_501_0/

[1] Bachoc, C.; Serra, C.; Zémor, G. An analogue of Vosper’s theorem for extension fields, Math. Proc. Philos. Soc., Volume 163 (2017), pp. 423-452 | Article | MR 3708518 | Zbl 06797071

[2] Bachoc, Christine; Serra, Oriol; Zémor, Gilles Revisiting Kneser’s theorem for field extensions, Combinatorica (2017) | Article | Zbl 06941935

[3] Beck, V.; Lecouvey, C. Additive combinatorics methods in associative algebras (2015) (To appear in Confluentes Math. ArXiv:math/1504.02287) | Zbl 06914192

[4] Bourbaki, N. Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No, Hermann, Paris, 1964 | Zbl 0205.34302

[5] Eliahou, S.; Lecouvey, C. On linear versions of some addition theorems, Linear Multilinear Algebra, Volume 57 (2009), pp. 759-775 | Article | MR 2571853 | Zbl 1263.11015

[6] Freiman, G. A. Foundations of a structural theory of set addition, Transl. Math. Monogr., Volume 37, Amer. Math. Soc., Providence, R. I., 1973, vii+108 pages | MR 360496 | Zbl 0271.10044

[7] Fulton, William Algebraic curves, Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, xxii+226 pages (An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Reprint of 1969 original) | MR 1042981 | Zbl 0681.14011

[8] Hamidoune, Y. O.; Rødseth., Ø. An inverse theorem modulo p, Acta Arith., Volume 92 (2000), pp. 251-262 | Article

[9] Hou, X.; Leung, K. H.; Xiang, Q. A generalization of an addition theorem of Kneser, J. Number Theory, Volume 97 (2002), pp. 1-9 | Article | MR 1939132 | Zbl 1034.11020

[10] Kneser, M. Summenmengen in Lokalkompakten Abelesche Gruppen, Math. Z., Volume 66 (1956), pp. 88-110 | Article | Zbl 0073.01702

[11] Mirandola, Diego; Zémor, Gilles Critical pairs for the product singleton bound, IEEE Trans. Inform. Theory, Volume 61 (2015) no. 9, pp. 4928-4937 | Article | MR 3386492 | Zbl 1359.94724

[12] Mumford, David Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100 | Zbl 0198.25801

[13] Randriambololona, Hugues On products and powers of linear codes under componentwise multiplication, Algorithmic arithmetic, geometry, and coding theory (Contemp. Math.) Volume 637, Amer. Math. Soc., Providence, RI, 2015, pp. 3-78 | Article | MR 3364442 | Zbl 06592375

[14] Roth, R. M.; Raviv, N.; Tamo, I. Construction of Sidon spaces with applications to coding, IEEE Trans. Inform. Theory, Volume 64 (2018) no. 6, pp. 4412-4422 | Article | MR 3809749 | Zbl 1395.94237

[15] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 106, Springer, Dordrecht, 2009, xx+513 pages | MR 2514094 | Zbl 1194.11005

[16] Stichtenoth, Henning Algebraic function fields and codes, Grad. Texts in Math., Volume 254, Springer-Verlag, Berlin, 2009, xiv+355 pages | MR 2464941 | Zbl 1155.14022

[17] Tao, Terence; Vu, Van Additive combinatorics, Cambridge Stud. Adv. Math., Volume 105, Cambridge University Press, Cambridge, 2006, xviii+512 pages | Article | MR 2289012 | Zbl 1127.11002

[18] Vosper, G. The critical pairs of subsets of a group of prime order, J. London Math. Soc, Volume 31 (1956), pp. 200-205 | Article | MR 77555 | Zbl 0072.03402