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The Jacobi Bound Conjecture for Generically Reduced Differential Schemes

Taylor Dupuy, David Zureick-Brown

Abstract

We prove the Strong Jacobi Bound Conjecture for generically reduced components of differential schemes.

The Jacobi Bound Conjecture for Generically Reduced Differential Schemes

Abstract

We prove the Strong Jacobi Bound Conjecture for generically reduced components of differential schemes.
Paper Structure (13 sections, 15 theorems, 40 equations)

This paper contains 13 sections, 15 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Strategy of Proof
  3. Background and Notation.
  4. Differential Algebra Background
  5. Ritt's Division Algorithm and Characteristic Sets
  6. Generic Smoothness
  7. Tangent Bundles of Differential Schemes and Linearization
  8. Dimension of Tangent Bundles
  9. Linearizations of Schemes In Countably Many Variables
  10. Linearizations of $D$-Schemes
  11. Linearization and Jacobi Bounds
  12. Proof of Main Theorem
  13. Applications

Key Result

Theorem 1.1

Let $K$ be a field. Let $I =\langle f_1,\ldots,f_n\rangle \subset K[x_1,\ldots,x_n]$ be an ideal with $f_i$ of degree $d_i$ and let $X = \operatorname{Spec} K[x_1,\ldots,x_n]/I$. If $X$ has finite length, then $\operatorname{len}_K(X)\leq d_1d_2\cdots d_n$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Corollary 1.9
  • ...and 42 more