Combinatorics behind discriminants of polynomial systems
Vladislav Pokidkin
TL;DR
This work develops a combinatorial framework based on polymatroids and their induced matroids to study discriminants of polynomial systems. It introduces BK-sets and contractions (by BK-sets and by cyclic flats), establishes rank-formulas linking polymatroid and induced-matroid notions, and reveals a rich lattice/poset structure (BK-sets and the Birkhoff poset) that governs component analyses. Realizable polymatroids are connected to partitions of vector spaces and to quotients, enabling a geometric interpretation of contractions. The approach provides concrete tools for understanding discriminants, including rank plateaus, cycles, and a BK-partition scheme that underpins a potential component-wise solving strategy and relates to conjectures on irreducibility of discriminants. Overall, the paper lays a combinatorial foundation thatgeneralizes classical discriminant results and informs sequel work on component classification and algorithmic resolution 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.