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An Analytic and Probabilistic Approach to the Problem of Matroid Representibility
We introduce various quantities that can be defined for an arbitrary matroid,
and show that certain conditions on these quantities imply that a matroid is
not representable over . Mostly, for a matroid of rank , we
examine the proportion of size- subsets that are dependent, and give
bounds, in terms of the cardinality of the matroid and a prime power, for
this proportion, below which the matroid is not representable over
. We also explore connections between the defined quantities and
demonstrate that they can be used to prove that random matrices have high
proportions of subsets of columns independent
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