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The Dimension Conjecture Implies The Jacobi Bound Conjecture

Taylor Dupuy, David Zureick-Brown

Abstract

We prove that the Dimension Conjecture implies the Jacobi Bound Conjecture.

The Dimension Conjecture Implies The Jacobi Bound Conjecture

Abstract

We prove that the Dimension Conjecture implies the Jacobi Bound Conjecture.
Paper Structure (18 sections, 24 theorems, 107 equations, 1 figure)

This paper contains 18 sections, 24 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strategy of proof
  3. The Rosenfeld--Gröbner algorithm
  4. Acknowledgements
  5. Background
  6. Notation and Conventions For Schemes
  7. Notation and Conventions for Differential Schemes
  8. The Jacobi Bound Conjecture
  9. Characteristic Sets
  10. The Dimension Conjecture
  11. Ritt's Cycle Trick
  12. Ritt's Reductions: Divisions That Preserve Jacobi numbers
  13. Existence of the First Form
  14. Filling Ritt's Gap: Existence Of The Second Form
  15. Ritt's Proof of The Linear Case
  16. ...and 3 more sections

Key Result

Theorem 1.3

Suppose that $K$ is a differentially closed field. Suppose that the Dimension Conjecture is true. Then the Jacobi Bound Conjecture is true.

Figures (1)

  • Figure 1: A Ritt pencil. We have written $\Sigma$ as a union of irreducible component $\Sigma_1,\Sigma_2,\Sigma_3,\Sigma_4$. The fibers $\Gamma_{\mu}$ and $\Gamma_{\nu}$ of the pencil $\Gamma$ have an intersection which contains a base locus. The base locus is contained in $\Sigma$ and contains $\Sigma_1$.

Theorems & Definitions (98)

  • Conjecture 1.1: Jacobi Bound Conjecture Kolchin1968
  • Conjecture 1.2: Dimension Conjecture Cohn1983 Ritt1950
  • Theorem 1.3
  • Example 1.4
  • Example 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 88 more