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Towards a function field version of Freiman’s theorem. (English) Zbl 1439.11283
Summary: 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 \(\mathrm{dim} S^2\leq 3 \mathrm{dim} S-4\). We make a step in this direction by giving a complete characterisation of spaces \(S\) such that \(\mathrm{dim} S^2=2 \mathrm{dim} S\). 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.
MSC:
11R58 Arithmetic theory of algebraic function fields
11P99 Additive number theory; partitions
05E40 Combinatorial aspects of commutative algebra
14H05 Algebraic functions and function fields in algebraic geometry
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