$r$-fat linearized polynomials over finite fields

TL;DR

This paper introduces and studies $r$-fat linearized polynomials over finite fields, revealing their properties, bounds, and existence conditions, and connecting them to various mathematical objects like clubs and rank metric codes.

Contribution

It generalizes the concept of scattered polynomials to $r$-fat polynomials, providing bounds, non-existence results, and explicit examples, especially for small degrees.

Findings

Non-existence of exceptional $r$-fat polynomials for $r>0$

Complete characterization of $r$-fat polynomials for $n \\leq 4$

New family of 1-fat polynomials for $n=5$

Abstract

In this paper we prove that the property of being scattered for a $\mathbb{F}_q$-linearized polynomial of small $q$-degree over a finite field $\mathbb{F}_{q^n}$ is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate $r$-fat polynomials, a natural generalization of scattered polynomials. An $r$-fat $\mathbb{F}_q$-linearized polynomial defines a linear set of rank $n$ in the projective line of order $q^n$ with $r$ points of weight larger than one. When $r$ equals $1$, the corresponding linear sets are called clubs, and they are related with a number of remarkable mathematical objects like KM-arcs, group divisible designs and rank metric codes. Using techniques on algebraic curves and global function fields, we obtain numerical bounds for $r$ and the non-existence of exceptional $r$-fat polynomials with $r>0$. In the case $n\leq 4$, we completely determine the spectrum of values of $r$ for which an $r$-fat polynomial exists. In the case $n=5$, we provide a new family of $1$-fat polynomials. Furthermore, we determine the values of $r$ for which the so-called LP-polynomials are $r$-fat.