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On the complexity of Chow and Hurwitz forms

Mahmut Levent Doğan, Alperen Ali Ergür, Elias Tsigaridas

TL;DR

The paper investigates the bit complexity of computing Chow forms and their Hurwitz extensions in projective and multiprojective settings, proposing a deterministic, resultant-based algorithm that achieves a single exponential bound. It develops complete-intersection and general-case algorithms, leverages generalized characteristic polynomials to avoid degeneracies, and extends the framework to multiprojective spaces with detailed multihomogeneous Bézout and sparse resultant analyses. A key contribution is the first explicit bit-complexity bounds for Chow forms, plus analogous results for Hurwitz forms in both the projective and multiprojective contexts, including a clear linkage to incidence geometry and matroid/ polymatroid structures. The work also provides practical algorithms for computing multigraded Chow forms, their supports, and associated varieties, highlighting the interplay between elimination theory, invariant theory, and combinatorial structures. Overall, the results advance the algorithmic understanding of elimination in algebraic geometry and have potential implications for computational incidence geometry and related combinatorial problems.

Abstract

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model, and our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space, and explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

On the complexity of Chow and Hurwitz forms

TL;DR

The paper investigates the bit complexity of computing Chow forms and their Hurwitz extensions in projective and multiprojective settings, proposing a deterministic, resultant-based algorithm that achieves a single exponential bound. It develops complete-intersection and general-case algorithms, leverages generalized characteristic polynomials to avoid degeneracies, and extends the framework to multiprojective spaces with detailed multihomogeneous Bézout and sparse resultant analyses. A key contribution is the first explicit bit-complexity bounds for Chow forms, plus analogous results for Hurwitz forms in both the projective and multiprojective contexts, including a clear linkage to incidence geometry and matroid/ polymatroid structures. The work also provides practical algorithms for computing multigraded Chow forms, their supports, and associated varieties, highlighting the interplay between elimination theory, invariant theory, and combinatorial structures. Overall, the results advance the algorithmic understanding of elimination in algebraic geometry and have potential implications for computational incidence geometry and related combinatorial problems.

Abstract

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model, and our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space, and explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.
Paper Structure (22 sections, 25 theorems, 100 equations, 2 figures, 7 algorithms)

This paper contains 22 sections, 25 theorems, 100 equations, 2 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Previous Works and Our Results
  3. Outline of the paper
  4. Preliminaries
  5. Notation
  6. Associated hypersurfaces and Chow forms
  7. The Chow Form in $\mathbb{P}^n$
  8. The case of a complete intersection
  9. The case of an over-determined system
  10. Straightening Algorithm
  11. The Hurwitz polynomial
  12. Multigraded Chow forms
  13. The multidegree and the support of a multiprojective variety
  14. Computing the support of a multiprojective variety
  15. Associated varieties of multiprojective varieties
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $V\subset\mathbb{P}^n$ be an irreducible variety of dimension $r$. Then, the set of linear subspaces intersecting $V$, is an irreducible hypersurface of $\mathop{\mathrm{Gr}}\nolimits(n-r-1,n)$ that we call the associated hypersurface of $V$. Moreover, $\mathcal{CZ}_V$ uniquely defines $V$; that is,

Figures (2)

  • Figure 1: Example \ref{['ex:chow']}. On the left, we have $\mathop{\mathrm{supp}}\nolimits(V)$, cut out by $\alpha_1+\alpha_2=4,\alpha_1\geq 2,\alpha_2\geq 2$, as described in Theorem \ref{['thm:supp']}. On the right, we have the set of formats such that the associated variety is a hypersurface which is cut out by $\alpha_1+\alpha_2=3,\alpha_1\geq 1,\alpha_2\geq 1$.
  • Figure 2: Example \ref{['ex:hurwitz']}. The blue points are the formats for which the associated variety $\mathcal{CZ}_{\tilde{V},\alpha}$ is a hypersurface. The yellow points are the ones with $\mathcal{HZ}_{\tilde{V},\alpha}$ a hypersurface and cut out by the inequalities and the equality $\alpha_1+\alpha_2=4,\alpha_1\geq 1,\alpha_2\geq 1$.

Theorems & Definitions (59)

  • Proposition 2.1
  • Definition 1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 49 more