Finite prime-field characteristic sets for planar configurations

Abstract

Examples are constructed of planar matroids with finite prime-field characteristic sets (i.e. matroids representable over a finite set of prime fields but over fields of no other characteristic). In particular, for any n>3, a projectively unique integer matrix is constructed with 2lsqblog 2 nrsqb+6 columns which often gives nonsingleton characteristic sets and, when n is prime, has characteristic set { n}. Many finite subsets of primes are shown to be characteristic sets, including {23,59} (the smallest pair found using these methods), all pairs of primes { p, p′:67⩽ p< p′⩽293}, and the seventeen largest five-digit primes. Probabilistic arguments are presented to support the conjecture that prime-field characteristic sets exist of every finite cardinality. For p>3, AG(2, p) is shown to be a subset of PG(2, q) only for q= p s . Another general construction technique suggests that when P={p 1,…,p k} are the primitive prime divisors of 2 n ±1 ( n sufficiently large), then there is a matroid with O (log n) points whose characteristic set is P . We remark that although only one finite nonsingleton characteristic set (due to R. Reid) was known prior to this paper, a new technique by J. Kahn has shown that every finite set of primes forms a (non-prime-field) characteristic set.