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Hyperdeterminants of Polynomials

Luke Oeding

TL;DR

This work studies how the hyperdeterminant of a polynomial, viewed as a symmetric tensor, factors under symmetrization and analyzes the μ-discriminant analog. Employing a geometric approach centered on Segre-Veronese and Chow varieties and their duals, it obtains a precise factorization: the symmetrized hyperdeterminant equals a product of equations Ξ_{λ,n} for dual Chow varieties, each raised to a multiplicity M_{λ} given by multinomial coefficients. The authors prove a general refinement-compatibility result for duals and projections, derive a degree-generating framework via GKZ data, and illustrate the theory with binary-case formulas and plane-cubic phenomena, including Catalecticant-type cases. The results unify classical invariant-theoretic interpretations with modern projective-duality methods, providing both conceptual insight and practical computational tools for degrees and factor multiplicities.

Abstract

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their multiplicities. The analogous decomposition for the μ-discriminant of polynomial is found.

Hyperdeterminants of Polynomials

TL;DR

This work studies how the hyperdeterminant of a polynomial, viewed as a symmetric tensor, factors under symmetrization and analyzes the μ-discriminant analog. Employing a geometric approach centered on Segre-Veronese and Chow varieties and their duals, it obtains a precise factorization: the symmetrized hyperdeterminant equals a product of equations Ξ_{λ,n} for dual Chow varieties, each raised to a multiplicity M_{λ} given by multinomial coefficients. The authors prove a general refinement-compatibility result for duals and projections, derive a degree-generating framework via GKZ data, and illustrate the theory with binary-case formulas and plane-cubic phenomena, including Catalecticant-type cases. The results unify classical invariant-theoretic interpretations with modern projective-duality methods, providing both conceptual insight and practical computational tools for degrees and factor multiplicities.

Abstract

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their multiplicities. The analogous decomposition for the μ-discriminant of polynomial is found.

Paper Structure

This paper contains 13 sections, 19 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Combinatorial ingredients
  3. Some multi-linear algebra
  4. Geometric ingredients
  5. Segre-Veronese and Chow varieties.
  6. Dual varieties
  7. Projections and dual varieties
  8. Plane cubics again
  9. Dimension of the duals to Chow varieties
  10. Proof of main results
  11. A degree formula in the binary case
  12. A generating function for the degree of dual of a Chow variety
  13. Examples

Key Result

Theorem 1.1

The $n^{ \times d}$-hyperdeterminant of a symmetric tensor of degree $d\geq 2$ on $n\geq 2$ variables splits as the product where $\Xi_{ \lambda, n}$ is the equation of the dual variety of the Chow variety $\text{Chow}_{\lambda}\mathbb{P} ^{n-1}$ when it is a hypersurface in $\mathbb{P}^{\binom{n-1+d}{d}-1}$, $\lambda = (\lambda_{1}, \dots , \lambda_{s})$ is a partition of $d$, and the multiplicit

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • Proposition 2.2: Proposition \ref{['mark']}
  • ...and 23 more