Scattered polynomials: an overview on their properties, connections and applications
Giovanni Longobardi
TL;DR
This survey systematizes scattered polynomials over $\mathbb{F}_{q^n}$, linking linearized polynomials to maximum rank-metric (MRD) codes and to maximum scattered linear sets in $\mathrm{PG}(1,q^n)$. It catalogs known families of scattered polynomials, establishes their standard forms, and surveys equivalence and adjoint relations, idealisers, and weight distributions. It further connects these algebraic objects to geometric constructions via linear sets, Desarguesian spreads, and projection configurations, and shows how they give rise to translation planes through planar spreads and quasifields, including André planes. The work synthesizes coding-theoretic and geometric perspectives, summarizes complete classifications for small $n$, and highlights the role of projecting configurations and automorphism groups in the broader landscape and in guiding open classification questions.
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].