论文标题
有限的投影线和翻译平面中的分散线性套件
Scattered linear sets in a finite projective line and translation planes
论文作者
论文摘要
Lunardon和Polverino构建了一个翻译平面，从$ \ Mathrm {pg}（1，q^t）$中的伪造型式线性套装开始。在本文中，描述并研究了类似的翻译平面$ \ MATHCAL A_F $从$ \ Mathbb f_ {q^t} [x] [x] $中获得的任何分散的线性多项式$ f（x）$获得的类似结构。定义了产生此类平面的一类Quasifield。用$ u_f $表示$ \ mathbb f_q $ -subspace of $ \ mathbb f_ {q^t}^2 $与$ f（x）$相关。如果$ f（x）$和$ f'（x）$分散了，则$ \ Mathcal a_f $和$ \ Mathcal a_ {f'} $在且仅当$ u_f $和$ u_f $和$ u_f $ and $ u_f'} $属于同一ORBIT的情况下属于$ $ umγ\ Mathrm l（2，Q^t）$时。这引起了与不相等的线性线性多项式相同的不同的翻译平面。 In particular, for any scattered linear set $L$ of maximum rank in $\mathrm{PG}(1,q^t)$ there are $c_Γ(L)$ pairwise non-isomorphic translation planes, where $c_Γ(L)$ denotes the $Γ\mathrm L$-class of $L$, as defined by Csajbók, Marino and Polverino. JHA和Johnson的结果允许描述从Lunardon和Polverino定义的未定义的伪基型类型的线性集合获得的平面的自动形群。
Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$ belong to the same orbit under the action of $Γ\mathrm L(2,q^t)$. This gives rise to as many distinct translation planes as there are inequivalent scattered linearized polynomials. In particular, for any scattered linear set $L$ of maximum rank in $\mathrm{PG}(1,q^t)$ there are $c_Γ(L)$ pairwise non-isomorphic translation planes, where $c_Γ(L)$ denotes the $Γ\mathrm L$-class of $L$, as defined by Csajbók, Marino and Polverino. A result by Jha and Johnson allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined by Lunardon and Polverino.
