Section: New Results

Towards a function field version of Freiman's theorem

In a collaboration with Christine Bachoc and Gilles Zémor (University of Bordeaux), A. Couvreur obtained a characterisation of subspaces $S$ of a function field $F$ over an algebraically closed field satisfiying

where $S^2$ denotes the space spanned by all the products of two elements of $S$. They obtained the following result [18]:

Theorem. Let $F$ be a function field over an algebraically closed field, and $S$ be a finite dimensional subspace of $F$ which spans $F$ as an algebra and such that

Then $F$ is a function field of transcendence degree 1 and

• either $F$ has genus 1 and $S$ is a Riemann Roch space

• or $F$ has genus 0 and $S$ is a subspace of codimension 1 in a Riemann Roch space.