Systems of Polynomials for Detecting Orientable Matroids

We study systems of polynomial equations that correspond to a matroid M. Each of these systems has a zero solution if and only if M is orientable. Since determining if a matroid is orientable is NP-complete with respect to the size of the input data, determining if these systems have solutions is also NP-complete. However, we show that one of the associated polynomial systems corresponding to M is linear if M is a binary matroid and thus it may be determined if binary matroids are orientable in polynomial time given the circuits and cocircuits of said matroid as the input. In the case when M is not binary, we consider the associated system of non-linear polynomials. In this case Hilbertʼs Nullstellensatz gives us that M is non-orientable if and only if a certain certificate to the given polynomials system exists. We wish to place bounds on the degree of these certificates in future research.