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{\bf Avi Berman, Shmuel Friedland, Leslie Hogben, Uriel G. Rothblum and Bryan Shader}
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{\bf Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers}
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We use a technique based on matroids to construct two nonzero patterns
$Z_1$ and $Z_2$ such that the minimum rank of matrices described by
$Z_1$ is less over the complex numbers than over the real numbers, and
the minimum rank of matrices described by $Z_2$ is less over the real
numbers than over the rational numbers.  The latter example provides a
counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about
rational realization of minimum rank of sign patterns. Using $Z_1$ and
$Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and
$G_2$, with the analogous minimum rank properties. We also discuss
issues of computational complexity related to minimum rank.


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