Freiman's $3k-4$ Theorem for Function Fields
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914960494297088 |
|---|---|
| author | Couvreur, Alain Zémor, Gilles |
| author_facet | Couvreur, Alain Zémor, Gilles |
| contents | 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 $S\supset K$ 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 $K(S)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Freiman's $3k-4$ Theorem for Function Fields Couvreur, Alain Zémor, Gilles Number Theory Combinatorics 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 $S\supset K$ 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 $K(S)$. |
| title | Freiman's $3k-4$ Theorem for Function Fields |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2408.00183 |