分散线性化多项式的标准形式和相关平移平面的性质

IF 0.6 3区 数学 Q3 MATHEMATICS
Giovanni Longobardi, Corrado Zanella
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引用次数: 0

摘要

在本文中,我们提出了关于子空间 \(U_f=\{(x,f(x)):x in \mathbb {F}_{q^n}\}), f(x) a scattered linearized polynomial in \(\mathbb {F}_{q^n}[x]\).每个 G_f\ 都包含(q-1\)映射((x,y)映射到(ax,ay)),(a 在 \mathbb {F}_{q}^*\) 中)。根据 Beard (Duke Math J, 39:313-321, 1972) 和 Willett (Duke Math J 40(3):701-704, 1973) 的结果,\(G_f\) 中的矩阵是同时可对角的。这有几个后果:(i) \(|G_f|>q-1/)中的多项式对于某些 s 和 t 具有标准的 \(\sum_{j=0}^{n/t-1}a_jx^{q^{s+jt}}/)类型,即 \((s,t)=1/),\(t>1/)是 n 的除数;(iii) 对于 \(n>2\) 和 \(q>3\), 与 f(x) 相关联的平移平面 \(\mathcal {A}_f\) 允许非对称仿射同调,当且仅当 \(|G_f|>;q-1\),在这种情况下,那些轴通过原点的平面会形成两个交换轴和同轴的心数为\((q^t-1)/(q-1)\)的群;(iv) 没有一个 f(x) 散点多项式不属于伪多径类型的 \(\mathcal {A}_f\) 型平面是广义的安德烈平面。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A standard form for scattered linearized polynomials and properties of the related translation planes

In this paper, we present results concerning the stabilizer \(G_f\) in \({{\,\mathrm{{GL}}\,}}(2,q^n)\) of the subspace \(U_f=\{(x,f(x)):x\in \mathbb {F}_{q^n}\}\), 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 (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 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 \(\mathcal {A}_f\) 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 \(\mathcal {A}_f\), f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.

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来源期刊
CiteScore
1.50
自引率
12.50%
发文量
94
审稿时长
6-12 weeks
期刊介绍: The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics. The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.
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