Towards a function field version of Freiman's Theorem

Abstract : 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 ≤ 3 dimS − 4. We make a step in this direction by giving a complete characterisation of spaces S such that dimS^2 = 2 dimS. 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.
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https://hal.archives-ouvertes.fr/hal-01584034
Contributor : Alain Couvreur <>
Submitted on : Friday, September 8, 2017 - 11:26:26 AM
Last modification on : Wednesday, March 27, 2019 - 4:41:28 PM

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Christine Bachoc, Alain Couvreur, Gilles Zémor. Towards a function field version of Freiman's Theorem. Algebraic Combinatorics, Akihiro Munemasa, Satoshi Murai, Hugh Thomas, Hendrik Van Maldeghem, 2018, 1 (4), pp.501-521. ⟨10.5802/alco.19⟩. ⟨hal-01584034⟩

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