A standard form for scattered linearized polynomials and properties of the related translation planes

TL;DR

This paper investigates the stabilizer groups of scattered linearized polynomials, revealing a standard form, uniqueness, and their connection to translation planes and affine homologies, advancing understanding of their algebraic and geometric properties.

Contribution

It introduces a standard form for certain scattered linearized polynomials, proves its essential uniqueness, and explores their associated translation planes and symmetry groups.

Findings

Polynomials with large stabilizer groups have a specific standard form.

The standard form of these polynomials is essentially unique.

Translation planes linked to these polynomials have particular symmetry properties.

Abstract

In this paper we present results concerning the stabilizer $G_f$ in $\mathrm{GL}(2,q^n)$ of the subspace $U_f=\{(x,f(x))\colon x\in\mathbb F_{q^n}[x]\}$, $f(x)$ a scattered linearized polynomial in $\mathbb F_{q^n}[x]$. Each $G_f$ contains the $q-1$ maps $(x,y)\mapsto(ax,ay)$, $a\in\mathbb F_{q}^*$. By virtue of the results of Beard (1972) and Willett (1973), the matrices in $G_f$ are simultaneously diagonalizable. This has several consequences: $(i)$ the polynomials such that $|G_f|>q-1$ have a standard form of type $\sum_{j=0}^{n/t-1}a_jx^{q^{s+jt}}$ for some $s$ and $t$ such that $(s,t)=1$, $t>1$ a divisor of $n$; $(ii)$ this standard form is essentially unique; $(iii)$ for $n>2$ and $q>3$, the translation plane $\cal A_f$ associated with $f(x)$ admits nontrivial affine homologies if and only if $|G_f|>q-1$, and in that case those with axis through the origin form two groups of cardinality $(q^t-1)/(q-1)$ that exchange axes and coaxes; $(iv)$ no plane of type $\cal A_f$, $f(x)$ a scattered polynomial not of pseudoregulus type, is a generalized Andr\'e plane.