Scattered Behavior Using Modified Cyclotomic Mapping Over Finite Fields Of Odd Characteristic

Introduced by Sheekey in 2016, the study of scattered polynomials over a finite field Fqⁿ 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)=∑ᵢ₌₁ᵏ aᵢx^qrᵢ ∈ Fqⁿ[x], where q is a power of an odd prime, 0<r₁<r₂< ⋯<rₖ<n and a₁, ⋯,aₖ ∈ Fqⁿ^* such that the order of aᵢ's divide (qr₁-1), ∀ i=2,3,⋯,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=r₁, or 0<t<r₁, or r₁<t<n. Starting with the pseudoregulus type of index t ≥ 0, we present conditions to verify scattered behavior of $S(x)$ of index r₁. With some additional conditions, we do the same in case 0<t<r₁ or r₁<t<n. In particular, for S(x)=a₁x^qr₁+a₂x^qr₂ ∈ Fqⁿ[x] with a₁,a₂ ∈ Fqⁿ^* such that |a₂| qr₁-1, we present a necessary and sufficient condition to verify its scattered behavior of index t ∈ ₁,r₂\. 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)=xq+δx^q⁽²ʳ⁺¹⁾ ∈ Fqⁿ[x] of index +1\.