Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

TL;DR

This paper introduces a generalized family of scattered quadrinomials over finite fields, providing conditions for their scattering property, and explores their relationships with previously known families, advancing the understanding of scattered polynomials.

Contribution

It generalizes existing scattered quadrinomials by establishing new sufficient conditions and analyzes their equivalences with known families.

Findings

Provided sufficient conditions for scatteredness of generalized quadrinomials

Unified and extended previous scattered polynomial families

Explored equivalences between new and known polynomial families

Abstract

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $\psi_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.