Combinatorics behind discriminants of polynomial systems

2025/09/03 by Vladislav Pokidkin, Pokidkin, Vladislav
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2509.02963

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Representation theory for polymatroids
• Vector-valued Laurent polynomial equations, toric vector bundles and matroids
• Irreducibility of toric complete intersections
• Polymatroids are to finite groups as matroids are to finite fields
• Engineered complete intersections: slightly degenerate Bernstein--Kouchnirenko--Khovanskii
• Intersection theory of polymatroids
• The Bergman fan of a polymatroid
• Generalized virtual polytopes and quasitoric manifolds
• Iterated and mixed discriminants
• Solving Decomposable Sparse Systems
• Subspace arrangements and Cherednik algebras
• Mixed volume and an extension of intersection theory of divisors
• Determinantal singularities and Newton polyhedra
• The homotopy type of the complement of a coordinate subspace arrangement
• Link complexes of subspace arrangements
• Formality of the complements of subspace arrangements with geometric lattices
• De Concini-Procesi wonderful arrangement models - A discrete geometer's point of view

Related

• Components and codimension of mixed and \mathscrA-discriminants for square polynomial systems
• A computational approach to the discriminant of homogeneous polynomials
• Plane Curves With Minimal Discriminant
• Effective membership problem and systems of polynomial equations with multiple roots
• Engineered complete intersections: slightly degenerate Bernstein--Kouchnirenko--Khovanskii