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

dim S 2 = 2 dim S

where S2 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

dim S 2 = 2 dim S .

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.