Rational Realizations of the Minimum Rank of a Sign Pattern Matrix

A sign pattern matrix is a matrix whose entries are from the set {+,−, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n−1 (where A is m×n), the conjecture is shown to hold. Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either −1 or 1 are investigated. AMS classification: 15A36; 15A03; 15A48; 15A24; 11D09; 11P21