Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Minimum Rank Problem: a counterexample   We provide a counterexample to a recent conjecture that the minimum rank of
every sign pattern matrix can be realized by a rational matrix. We use one of
the equivalences of the conjecture and some results from projective geometry.
As a consequence of the counterexample, we show that there is a graph for which
the minimum rank over the reals is strictly smaller than the minimum rank over
the rationals. We also make some comments on the minimum rank of sign pattern
matrices over different subfields of $\mathbb R$.

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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: astro-ph