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Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants

Vladislav Pokidkin

TL;DR

The paper resolves Esterov's conjecture on the irreducibility of $\mathscr{A}$-discriminants by developing a combinatorial–geometric framework based on realizable polymatroids, dual realizations, and a stratification of the dual space. It proves that for an irreducible BK-tuple, the intersection of the determinant hypersurface with the corresponding subspace is irreducible, via a detailed analysis of flats and defects. This irreducibility result is then transferred to $\mathscr{A}$-discriminants, showing irreducibility of $D_{\mathscr{A}}$ over any characteristic-zero field and providing a complete codimension description distinguishing lir and nir cases over $\mathbb{C}$. The work links polymatroid theory with algebraic geometry, yielding a pathway to describe components, codimensions, and degeneracies for discriminants of square polynomial systems, with implications for their algebraic and geometric structure.

Abstract

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose discriminant is an irreducible hypersurface. This resolves a conjecture of Esterov, and, in a sequel paper, leads to a complete description of components and codimensions for discriminants of square systems of equations.

Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants

TL;DR

The paper resolves Esterov's conjecture on the irreducibility of -discriminants by developing a combinatorial–geometric framework based on realizable polymatroids, dual realizations, and a stratification of the dual space. It proves that for an irreducible BK-tuple, the intersection of the determinant hypersurface with the corresponding subspace is irreducible, via a detailed analysis of flats and defects. This irreducibility result is then transferred to -discriminants, showing irreducibility of over any characteristic-zero field and providing a complete codimension description distinguishing lir and nir cases over . The work links polymatroid theory with algebraic geometry, yielding a pathway to describe components, codimensions, and degeneracies for discriminants of square polynomial systems, with implications for their algebraic and geometric structure.

Abstract

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose discriminant is an irreducible hypersurface. This resolves a conjecture of Esterov, and, in a sequel paper, leads to a complete description of components and codimensions for discriminants of square systems of equations.
Paper Structure (6 sections, 13 theorems, 10 equations)

This paper contains 6 sections, 13 theorems, 10 equations.

Table of Contents

  1. Combinatorial Review
  2. Realizable polymatroids
  3. Dual realization
  4. Polymatroid partition of the dual vector space
  5. Irreducible intersections of subspaces with determinant
  6. Esterov's conjecture on irreducibility of discriminants

Key Result

Theorem 1

A. For an irreducible BK-tuple $(L_{1},...,L_{n})$ of vector subspaces from an $n$-dimensional vector space over a field of characteristic zero, the intersection of the determinant hypersurface $Det$ with the subspace $L$ is irreducible in the space of square matrices of the size $n$.

Theorems & Definitions (21)

  • Theorem
  • Theorem
  • Proposition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Lemma 2.1
  • proof
  • ...and 11 more