# 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 $dim{S}^{2}\le 3dimS-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $dim{S}^{2}=2dimS$. 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.
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
@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. https://alco.centre-mersenne.org/item/ALCO_2018__1_4_501_0/

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