On the unique representability of spikes over prime fields

For an integer n⩾3n⩾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}{1,2,…,n-1}, the union of every set of k   of these lines has rank k+1k+1. Spikes are very special and important in matroid theory. Wu [On the number of spikes over finite fields, Discrete Math. 265 (2003) 261–296] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number p  , a GF(pGF(p) representable n-spike is only representable on fields with characteristic p   provided that n⩾2p-1n⩾2p-1. Moreover, M   is uniquely representable over GF(p)GF(p).