# ALGEBRAIC COMBINATORICS

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 $dim{S}^{2}\le 3dimS-4$. 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:
Accepted:
Published online:
DOI: 10.5802/alco.19
Classification: 11R58,  11P99,  05E40,  14H05
Bachoc, Christine 1; Couvreur, Alain 2; Zémor, Gilles 1

1 Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France
2 INRIA & Laboratoire LIX, CNRS UMR 7161, École Polytechnique, Université Paris Saclay, 91128 Palaiseau Cedex, France
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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/articles/10.5802/alco.19/

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