Towards the classification of scattered binomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917925442551808 |
|---|---|
| author | Bartoli, Daniele Ghiandoni, Francesco Giannoni, Alessandro Marino, Giuseppe |
| author_facet | Bartoli, Daniele Ghiandoni, Francesco Giannoni, Alessandro Marino, Giuseppe |
| contents | Let \( q \) be a prime power and \( n \) an integer. An \( \mathbb{F}_q \)-linearized polynomial \( f \) is said to be scattered if it satisfies the condition that for all \( x, y \in \mathbb{F}_q^n \setminus \{ 0 \} \), whenever \( \frac{f(x)}{x} = \frac{f(y)}{y} \), it follows that \( \frac{x}{y} \in \mathbb{F}_q \).
In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by $f(x) = δx^{q^s} + x^{q^{n-s}},$ and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by $f(x) = δx^{q^s} + x^{q^{s + n/2}},$ where \( n = 6 \) or \( n = 8 \).
Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when \( q \) is sufficiently large and \( n \) is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in $\Fn$ when $n\leq8$ and $q$ is large enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards the classification of scattered binomials Bartoli, Daniele Ghiandoni, Francesco Giannoni, Alessandro Marino, Giuseppe Combinatorics Let \( q \) be a prime power and \( n \) an integer. An \( \mathbb{F}_q \)-linearized polynomial \( f \) is said to be scattered if it satisfies the condition that for all \( x, y \in \mathbb{F}_q^n \setminus \{ 0 \} \), whenever \( \frac{f(x)}{x} = \frac{f(y)}{y} \), it follows that \( \frac{x}{y} \in \mathbb{F}_q \). In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by $f(x) = δx^{q^s} + x^{q^{n-s}},$ and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by $f(x) = δx^{q^s} + x^{q^{s + n/2}},$ where \( n = 6 \) or \( n = 8 \). Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when \( q \) is sufficiently large and \( n \) is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in $\Fn$ when $n\leq8$ and $q$ is large enough. |
| title | Towards the classification of scattered binomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.11666 |