The minimum size of a linear set

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:164

Article

De Beule, Jan¹  Van de Voorde, Geertrui² 

[1] Vrije Univ Brussel, Dept Math, Pl Laan 2, B-1050 Brussels, Belgium

[2] Univ Canterbury, Sch Math & Stat, Private Bag 4800, Christchurch 8140, New Zealand

关键词: Linear set;    Linearised polynomial;    Directions determined by a point set;   

DOI  :  10.1016/j.jcta.2018.12.008

来源: Elsevier

【 摘 要 】

In this paper, we first determine the minimum possible size of an F-q-linear set of rank k in PG(1, q(n)). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F-q-linear set of rank k in PG(2, q(n)). In the case k = n, this confirms a conjecture by Sziklai in [9]. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free