ALGEBRAIC COMBINATORICS

no. 4

Towards a function field version of Freiman’s Theorem
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 501-521.

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 dimS 2 ≤3dimS-4. We make a step in this direction by giving a complete characterisation of spaces S such that dimS 2 =2dimS. 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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.19
Classification: 11R58,  11P99,  05E40,  14H05
Keywords: Additive combinatorics, function fields 
@article{ALCO_2018__1_4_501_0,
     author = {Bachoc, Christine and Couvreur, Alain and Z\'emor, Gilles},
     title = {Towards a function field version of Freiman{\textquoteright}s Theorem},
     journal = {Algebraic Combinatorics},
     pages = {501--521},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     doi = {10.5802/alco.19},
     mrnumber = {3875074},
     zbl = {06963902},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.19/}
}
Bachoc, Christine; Couvreur, Alain; Zémor, Gilles. Towards a function field version of Freiman’s Theorem. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 501-521. doi : 10.5802/alco.19. https://alco.centre-mersenne.org/articles/10.5802/alco.19/

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