Scattered Behavior Using Modified Cyclotomic Mapping Over Finite Fields Of Odd Characteristic
Suman Mondal
TL;DR
This work addresses the scattered polynomial problem over $\mathbb{F}_{q^n}$ by focusing on $ S(x)=\sum_{i=1}^k a_i x^{q^{r_i}}$ with $0<r_1<\cdots<r_k<n$ under order constraints on the coefficients, and establishing a link to a modified $r$-th order cyclotomic mapping. By representing $S(x)$ as $S(x)=x^{q^{r_1}} f(x^{s q^{r_1}})$ and identifying it with the modified cyclotomic mapping $f^{r}_{A_0,\dots,A_{l-1}}$, the authors translate scatteredness questions into cyclotomic-coset analyses and permutation conditions on transformed polynomials. They derive necessary and sufficient conditions for scatteredness in the index regimes $t=r_1$, $0<t<r_1$, and $r_1<t<n$, and connect binomial cases to Lunardon-Polverino type polynomials, providing transformations that relate these families. Finally, they construct a new exceptional scattered binomial $S(x)=x^q+\delta x^{q^{(2r+1)}}$ under specific arithmetical constraints, demonstrating exceptional behavior across infinitely many extensions and broadening the catalog of exceptional scattered polynomials with LP-like features.
Abstract
Introduced by Sheekey in 2016, the study of scattered polynomials over a finite field $\mathbb{F}_{q^n}$ has been increasing regarding the classification of those that are exceptional, i.e., polynomials which are scattered over infinite field extensions, are limited to the cases where their index $t$ is small, or a prime number larger than the q-degree k of the polynomial, or an integer smaller than k in the case where k is a prime. In this paper, we focus on the scattered behavior of $S(x)=\sum_{i=1}^k a_ix^{q^{r_i}} \in \mathbb{F}_{q^n}[x]$, where $q$ is a power of an odd prime, $0<r_1<r_2< \cdots<r_k<n$ and $a_1, \cdots,a_k \in \mathbb{F}_{q^n}^*$ such that the order of $a_i$'s divide $(q^{r_1}-1)$, $\forall i=2,3,\cdots,k $. We explore a connection between $S(x)$ and the cyclotomic mapping polynomial. As an application, in three parts, we discuss the scattered behavior of $S(x)$ of index $t$ where $t=r_1$, or $0<t<r_1$, or $r_1<t<n$. Starting with the pseudoregulus type of index $t \geq 0$, we present conditions to verify scattered behavior of $S(x)$ of index $r_1$. With some additional conditions, we do the same in case $0<t<r_1$ or $r_1<t<n$. In particular, for $S(x)=a_1x^{q^{r_1}}+a_2x^{q^{r_2}} \in \mathbb{F}_{q^n}[x]$ with $a_1,a_2 \in \mathbb{F}_{q^n}^*$ such that $|a_2| \mid q^{r_1}-1$, we present a necessary and sufficient condition to verify its scattered behavior of index $t \in \{r_1,r_2\}$. We also connect such scattered binomials with the well known Lunardon-Polverino polynomial. With conditions on $δ, q,n$, and $r$; we present a new family of exceptional scattered polynomial $S(x)=x^q+δx^{q^{(2r+1)}} \in \mathbb{F}_{q^n}[x]$ of index $\{r+1\}$.