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: 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}, pages = {501--521}, publisher = {MathOA foundation}, volume = {1}, number = {4}, year = {2018}, doi = {10.5802/alco.19}, mrnumber = {3875074}, 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. https://alco.centre-mersenne.org/item/ALCO_2018__1_4_501_0/
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