  This paper studies systems of polynomial equations that provide information
about orientability of matroids. First, we study systems of linear equations
over GF(2), originally alluded to by Bland and Jensen in their seminal paper on
weak orientability. The Bland-Jensen linear equations for a matroid M have a
solution if and only if M is weakly orientable. We use the Bland-Jensen system
to determine weak orientability for all matroids on at most nine elements and
all matroids between ten and twelve elements having rank three. Our experiments
indicate that for small rank, about half the time, when a simple matroid is not
orientable, it is already non-weakly orientable. Thus, about half of the small
simple non-orientable matroids of rank three are not representable over fields
having order congruent to three modulo four. For binary matroids, the
Bland-Jensen linear systems provide a practical way to check orientability.
Second, we present two extensions of the Bland-Jensen equations to slightly
larger systems of non-linear polynomial equations. Our systems of polynomial
equations have a solution if and only if the associated matroid M is
orientable. The systems come in two versions, one directly extending the
Bland-Jensen system for GF(2), and a different system working over other
fields. We study some basic algebraic properties of these systems. Finally, we
present an infinite family of non-weakly-orientable matroids, with growing rank
and co-rank. We conjecture that these matroids are minor-minimal
non-weakly-orientable matroids.
