Combinatorics behind discriminants of polynomial systems

Abstract

We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical results of Gelfand, Kapranov, and Zelevinsky on discriminants of one general multivariate polynomial. The developed tools are targeted at vector subspace arrangements and naturally extend to their combinatorial abstraction called polymatroids, which are the subject matter of this work. We explore relations between polymatroids and their induced matroids for bases, circuits, cycles, and rank functions. We define contractions for polymatroids corresponding to the contractions of the induced matroids. With a view towards applications to discriminants, we construct a new combinatorial structure induced by polymatroids, called BK-sets.