Exceptional Scattered Polynomials

TL;DR

This paper classifies exceptional scattered polynomials of certain indices over finite fields, revealing their rarity and characterizing their structure through algebraic geometry techniques.

Contribution

It provides the first complete classification of exceptional scattered monic polynomials of indices 0 (for q>5) and 1, using algebraic curves and intersection theory.

Findings

Complete classification of exceptional scattered monic polynomials of index 0 for q>5.

Complete classification of exceptional scattered monic polynomials of index 1.

Demonstrates the rarity of such polynomials through algebraic geometric methods.

Abstract

Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.