ALGEBRAIC COMBINATORICS

no. 4
Towards a function field version of Freiman’s Theorem
Bachoc, Christine1; Couvreur, Alain2; Zémor, Gilles1
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
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 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.19
Classification: 11R58, 11P99, 05E40, 14H05
Keywords: Additive combinatorics, function fields
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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