On the number of spikes over finite fields

Abstract

For an integer n⩾3, a rank- n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k in {1,2,…, n−1}, the union of every set of k of these lines has rank k+1. It is well known that there is a unique binary n-spike for each integer n⩾3. In this paper, we first prove that, for each integer n⩾3, there are exactly two distinct ternary n-spikes, and there are exactly ⌊( n 2+6 n+24)/12⌋ quaternary n-spikes. Then we prove that, for each integer n⩾4, there are exactly n+2+⌊ n/2⌋ quinternary n-spikes and, for each integer n⩾18, the number of n-spikes representable over GF(7) is ⌊(2 n 2+6 n+6)/3⌋. Finally, for each q>7, we find the asymptotic value of the number of distinct rank- n spikes that are representable over GF( q).