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 $\dim S^2\le 3\dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2=2\dim S$. 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.

References in corpus (2)

• An analogue of Vosper's Theorem for Extension Fields
• Revisiting Kneser's Theorem for Field Extensions