2024/07/31 by Alain Couvreur, Couvreur, Alain, Gilles Zémor +1
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2408.00183
openalex publication_date 2024/07/31 · openalex created_date 2024/08/04 · openalex updated_date 2026/07/28
Freiman's 3k-4 Theorem states that if a subset A of k integers has a Minkowski sum A+A of size at most 3k-4, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if K is a perfect field and if S⊃ K is a vector space of dimension k inside an extension F/K in which~K is algebraically closed, and if the K-vector space generated by all products of pairs of elements of S has dimension at most 3k-4, then K(S) is a function field of small genus, and S is of small codimension inside a Riemann-Roch space of K(S).