Scattered polynomials: an overview on their properties, connections and applications

TL;DR

This survey reviews the properties, classifications, and applications of scattered polynomials over finite fields, highlighting their connections to maximum rank-metric codes, linear sets, and translation planes, and discussing recent advances and open problems.

Contribution

It provides a comprehensive overview of scattered polynomials, their known examples, classifications, and their links to various geometric and coding theoretic structures, summarizing recent progress.

Findings

Known examples of scattered polynomials are summarized.

Classification results for small n are discussed.

Connections to translation planes and rank-metric codes are reviewed.

Abstract

The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as in [A. Blokhuis, M. Lavrauw. Scattered spaces with respect to a spread in $\mathrm{PG}(n, q)$. Geometriae Dedicata 81(1) (2000), 231-243] and [G. Lunardon, O. Polverino. Blocking sets and derivable partial spreads. J. Algebraic Combin. 14 (2001), 49-56]. Recently, their connection to maximum rank-metric codes was brought to light in [J. Sheekey. MRD codes: Constructions and connections. In K.-U. Schmidt and A. Winterhof, editors, Combinatorics and Finite Fields, De Gruyter (2019), 255-286]. This link has significantly advanced their study and investigation, sparking considerable interest in recent years. Here, we will explore their relationship with certain subsets of the finite projective line $\mathrm{PG}(1, q^n)$ known as maximum scattered linear sets, as well as with codes made up of square matrices of order $n$ equipped with the rank metric. We will review the known examples of scattered polynomials up to date and discuss some of their key properties. We will also address the classification of maximum scattered linear sets of the finite projective line $\mathrm{PG}(1, q^n)$ for small values of $n$ and discuss characterization results for the examples known so far. Finally, we will retrace how each scattered polynomial gives rise to a translation plane, as discussed in [V. Casarino, G. Longobardi, C. Zanella. Scattered linear sets in a finite projective line and translation planes, Linear Algebra Appl. 650 (2022), 286-298] and in [G. Longobardi, C. Zanella, A standard form for scattered linearized polynomials and properties of the related translation planes, J. Algebr. Comb. 59(4) (2024), 917-937].