ALGEBRAIC COMBINATORICS

Towards a function field version of Freiman’s Theorem
Algebraic Combinatorics, Volume 1 (2018) no. 4, p. 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},
     zbl = {06963902},
     language = {en},
     url = {https://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. 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

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

[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, Hermann, Paris, Actualités Scientifiques et Industrielles, No (1964)

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

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

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

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

[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

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

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

[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

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

[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

[15] Silverman, Joseph H. The arithmetic of elliptic curves, Springer, Dordrecht, Graduate Texts in Mathematics, Volume 106 (2009), xx+513 pages

[16] Stichtenoth, Henning Algebraic function fields and codes, Springer-Verlag, Berlin, Grad. Texts in Math., Volume 254 (2009), xiv+355 pages

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

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