On the unique representability of spikes over prime fields

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 ∈ {1, 2, . . . , n− 1}, the union of every set of k of these lines has rank k + 1. Spikes are very special and important in matroid theory. Wu [13] 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 (p) representable n-spike M is only representable on fields with characteristic p provided that n ≥ 2p− 1. Moreover, M is uniquely representable over GF (p).