论文标题
相关翻译平面的散射线性多项式和性能的标准形式
A standard form for scattered linearized polynomials and properties of the related translation planes
论文作者
论文摘要
在本文中，我们介绍了有关稳定剂$ g_f $的结果f_ {q^n} [x] $。每个$ g_f $都包含$ q-1 $映射$（x，y）\ mapsto（ax，ay）$，$ a \ in \ mathbb f_ {q}^*$。借助Beard（1972）和Willett（1973）的结果，$ g_f $中的矩阵同时可对角线。这有几个后果：$（i）$多项式使得$ | g_f |> q-1 $具有标准形式的$ \ sum_ {j = 0}^{n/t-1}^{n/t-1} a_jx^a_jx^{q^{q^{s+jt {s+jt}} $，对于某些$ s $和$ t $ t $（$ t $） $（ii）$此标准表格本质上是独一无二的； $(iii)$ 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; $（iv）$无类型$ \ cal a_f $，$ f（x）$ a散射的多项式而不是伪古鲁斯类型的平面是一台广义的安德烈平面。
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.
