Towards a function field version of Freiman's Theorem
Résumé
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.