Oriented Matroid Pairs, Theory and an Electric Application

The property that a pair of oriented matroidsM ? L , M R on E have free union and no common (non-zero) covector generalizes oriented matroid duality. This property characterizes when certain systems of equations whose only nonlinearities occur as real monotone bijections have a unique solution for all values of additive parameters. Instances include sign non-singularity of square matrices and generalizations of positive deeniteness given by Fiedler and Ptt ak. Other instances of this property include various kinds of characterizations of when an electric network problem is well-posed. Such characterizations have been given in terms of matrix pairs by Sandberg and Willson and in terms of electrical network graphs by Duun, Minty, Hasler and Neirnyck, and by Nishi and Chua. Cases of the general common covector problem are classiied. Natural matroid rank conditions are suucient for a common covector to exist. An algorithm to construct a common covector by composing certain fundamental cocircuits is given. If M L and M R have two common bases with opposite relative orientation (chirotope value) then M ? L , M R have a common covector.