We discuss a multiplicative counterpart of Freiman’s theorem in the context of a function field over an algebraically closed field . Such a theorem would give a precise description of subspaces , such that the space spanned by products of elements of satisfies . We make a step in this direction by giving a complete characterisation of spaces such that . We show that, up to multiplication by a constant field element, such a space is included in a function field of genus or . In particular if the genus is then this space is a Riemann–Roch space.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.19
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{\textquoteright}s} {Theorem}}, journal = {Algebraic Combinatorics}, pages = {501--521}, publisher = {MathOA foundation}, volume = {1}, number = {4}, year = {2018}, doi = {10.5802/alco.19}, mrnumber = {3875074}, zbl = {06963902}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.19/} }
TY - JOUR AU - Bachoc, Christine AU - Couvreur, Alain AU - Zémor, Gilles TI - Towards a function field version of Freiman’s Theorem JO - Algebraic Combinatorics PY - 2018 SP - 501 EP - 521 VL - 1 IS - 4 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.19/ DO - 10.5802/alco.19 LA - en ID - ALCO_2018__1_4_501_0 ER -
%0 Journal Article %A Bachoc, Christine %A Couvreur, Alain %A Zémor, Gilles %T Towards a function field version of Freiman’s Theorem %J Algebraic Combinatorics %D 2018 %P 501-521 %V 1 %N 4 %I MathOA foundation %U http://archive.numdam.org/articles/10.5802/alco.19/ %R 10.5802/alco.19 %G en %F 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. http://archive.numdam.org/articles/10.5802/alco.19/
[1] An analogue of Vosper’s theorem for extension fields, Math. Proc. Philos. Soc., Volume 163 (2017), pp. 423-452 | DOI | MR | Zbl
[2] Revisiting Kneser’s theorem for field extensions, Combinatorica (2017) | DOI | Zbl
[3] Additive combinatorics methods in associative algebras (2015) (To appear in Confluentes Math. ArXiv:math/1504.02287) | Zbl
[4] Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No, Hermann, Paris, 1964 | Zbl
[5] On linear versions of some addition theorems, Linear Multilinear Algebra, Volume 57 (2009), pp. 759-775 | DOI | MR | Zbl
[6] Foundations of a structural theory of set addition, Transl. Math. Monogr., 37, Amer. Math. Soc., Providence, R. I., 1973, vii+108 pages | MR | Zbl
[7] 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 | Zbl
[8] An inverse theorem modulo , Acta Arith., Volume 92 (2000), pp. 251-262 | DOI
[9] A generalization of an addition theorem of Kneser, J. Number Theory, Volume 97 (2002), pp. 1-9 | DOI | MR | Zbl
[10] Summenmengen in Lokalkompakten Abelesche Gruppen, Math. Z., Volume 66 (1956), pp. 88-110 | DOI | Zbl
[11] Critical pairs for the product singleton bound, IEEE Trans. Inform. Theory, Volume 61 (2015) no. 9, pp. 4928-4937 | DOI | MR | Zbl
[12] 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
[13] 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 | DOI | MR | Zbl
[14] Construction of Sidon spaces with applications to coding, IEEE Trans. Inform. Theory, Volume 64 (2018) no. 6, pp. 4412-4422 | DOI | MR | Zbl
[15] The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009, xx+513 pages | MR | Zbl
[16] Algebraic function fields and codes, Grad. Texts in Math., 254, Springer-Verlag, Berlin, 2009, xiv+355 pages | MR | Zbl
[17] Additive combinatorics, Cambridge Stud. Adv. Math., 105, Cambridge University Press, Cambridge, 2006, xviii+512 pages | DOI | MR | Zbl
[18] The critical pairs of subsets of a group of prime order, J. London Math. Soc, Volume 31 (1956), pp. 200-205 | DOI | MR | Zbl
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