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{\bf Arun P.\ Mani }
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{\bf An Extension of Matroid Rank Submodularity and the $Z$-Rayleigh Property}
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We define an extension of matroid rank submodularity called
$R$-submodularity, and introduce a minor-closed class of matroids
called extended submodular matroids that are well-behaved with respect
to $R$-submodularity. We apply $R$-submodularity to study a class of
matroids with negatively correlated multivariate Tutte polynomials
called the $Z$-Rayleigh matroids. First, we show that the class of
extended submodular matroids are $Z$-Rayleigh. Second, we
characterize a minor-minimal non-$Z$-Rayleigh matroid using its
$R$-submodular properties. Lastly, we use $R$-submodularity to show
that the Fano and non-Fano matroids (neither of which is extended
submodular) are $Z$-Rayleigh, thus giving the first known examples of
$Z$-Rayleigh matroids without the half-plane property.
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