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Components and codimension of mixed and $\mathscr{A}$-discriminants for square polynomial systems

Vladislav Pokidkin

TL;DR

This work resolves the codimension and component structure of discriminants for square and overdetermined sparse polynomial systems by developing a combinatorial and geometric framework centered on BK-tuples (zero-defect linearly independent subtuples) and BK-multiplication. It introduces and analyzes three discriminants—$D_{\mathscr{A}}$, $D_{cay(\mathscr{A})}$, and $\mathring{D}_{\mathscr{A}}$—and provides a complete description of their irreducible components, codimensions, and degrees for all $\mathscr{A}$ with $n\le k$, organized via a poset $P_{\mathscr{A}}$ and defect $\delta(\mathscr{B})$. Central results show that linearly dependent cases reduce to sparse resultants, while BK-tuples yield a stratified decomposition into Cayley discriminants and their interactions, with explicit degree formulas derived from mixed volumes and Euler obstructions. The paper thus unifies and extends existing discriminant theory (A-, Cayley-, and mixed) and yields practical tools for computing component structure and degrees in sparse polynomial systems, including the irreducible BK-tuple scenario and the unique-circuit cases. These insights advance discriminant geometry for square and overdetermined systems and provide concrete, computable invariants for applications in algebraic geometry and computational algebra.

Abstract

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for discriminants of square and overdetermined systems of equations. This version is more involved, in the sense that the discriminant may have several components of different dimension. We enumerate all components and find their dimension and degree, for each of the three conventional ways to formalize the notion of the discriminant in this setting (namely, for mixed, Cayley and $\mathscr{A}$-discriminants).

Components and codimension of mixed and $\mathscr{A}$-discriminants for square polynomial systems

TL;DR

This work resolves the codimension and component structure of discriminants for square and overdetermined sparse polynomial systems by developing a combinatorial and geometric framework centered on BK-tuples (zero-defect linearly independent subtuples) and BK-multiplication. It introduces and analyzes three discriminants—, , and —and provides a complete description of their irreducible components, codimensions, and degrees for all with , organized via a poset and defect . Central results show that linearly dependent cases reduce to sparse resultants, while BK-tuples yield a stratified decomposition into Cayley discriminants and their interactions, with explicit degree formulas derived from mixed volumes and Euler obstructions. The paper thus unifies and extends existing discriminant theory (A-, Cayley-, and mixed) and yields practical tools for computing component structure and degrees in sparse polynomial systems, including the irreducible BK-tuple scenario and the unique-circuit cases. These insights advance discriminant geometry for square and overdetermined systems and provide concrete, computable invariants for applications in algebraic geometry and computational algebra.

Abstract

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for discriminants of square and overdetermined systems of equations. This version is more involved, in the sense that the discriminant may have several components of different dimension. We enumerate all components and find their dimension and degree, for each of the three conventional ways to formalize the notion of the discriminant in this setting (namely, for mixed, Cayley and -discriminants).
Paper Structure (8 sections, 50 theorems, 16 equations)

This paper contains 8 sections, 50 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Combinatorial Review
  3. Discriminants of polynomial systems
  4. BK-multiplication
  5. $\mathscr{A}$-discriminants
  6. Cayley discriminants
  7. Mixed discriminants
  8. Degrees of discriminants

Key Result

Theorem 1.1

For a linearly dependent tuple $\mathscr{A}$ with the minimal (by inclusion) subtuple $\mathscr{M}$ of minimal defect, 1) the $\mathscr{A}$-discriminant is the sparse resultant $R_{\mathscr{M}}$ of codimension $-\delta(\mathscr{M})$(Theorem Theorem. Discriminants for linearly dependent tuples); 2) t

Theorems & Definitions (97)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 87 more