Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918429785587712 |
|---|---|
| author | Xiong, Maosheng Yip, Chi Hoi |
| author_facet | Xiong, Maosheng Yip, Chi Hoi |
| contents | Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes Xiong, Maosheng Yip, Chi Hoi Number Theory Combinatorics 11B30, 11T30, 05C25 Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip. |
| title | Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes |
| topic | Number Theory Combinatorics 11B30, 11T30, 05C25 |
| url | https://arxiv.org/abs/2604.04126 |