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

TL;DR

This work studies the stabilizer $G_f$ of the scattered subspace $U_f$ in $\mathrm{GL}(2,q^n)$ and its implications for the associated translation planes $\mathcal{A}_f$. By applying simultaneous diagonalization results, it derives a canonical standard form for polynomials with large stabilizers, showing the form $h(x)=\sum_{j=0}^{n/t-1} a_j x^{q^{s+jt}}$ is attainable when $|G_f|>q-1$ and is essentially unique. The authors catalog stabilizers for all known maximum scattered subspaces, connect stabilizer size to affine homologies in $\mathcal{A}_f$, and prove that non-pseudoregulus scattered polynomials cannot yield generalized André planes. They further analyze the translation-plane automorphism groups, kernel homologies, and the structure of affine homologies, establishing a cohesive bridge between scattered polynomials, MRD codes, and finite geometry with canonical normal forms and symmetry descriptions.

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 André plane.

Theorems & Definitions (46)

• Theorem 3.1
• Theorem 3.2
• Proposition 3.3
• Remark 3.4
• Theorem 3.5
• Proposition 3.6
• proof
• Proposition 3.7
• proof
• Remark 3.8