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Zilber dichotomy for $DCF_{0,m}$

Omar Leon Sanchez

Abstract

We prove that the theory of differentially closed fields of characteristic zero in $m\geq 1$ commuting derivations DCF$_{0,m}$ satisfies the expected form of the dichotomy. Namely, any minimal type is either locally modular or nonorthogonal to the (algebraically closed) field of constants. This dichotomy is well known for finite-dimensional types; however, a proof that includes the possible case of infinite dimension does not explicitly appear elsewhere.

Zilber dichotomy for $DCF_{0,m}$

Abstract

We prove that the theory of differentially closed fields of characteristic zero in commuting derivations DCF satisfies the expected form of the dichotomy. Namely, any minimal type is either locally modular or nonorthogonal to the (algebraically closed) field of constants. This dichotomy is well known for finite-dimensional types; however, a proof that includes the possible case of infinite dimension does not explicitly appear elsewhere.

Paper Structure

This paper contains 3 sections, 4 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The CBP for finite-dimensional types
  3. Proof of the dichotomy

Key Result

Lemma 2.3

Let $(E,\mathcal{D})$ be a $\Delta$-module over $(U,\Delta)$. Then, there is a $C_{\mathbb U}$-basis for $E^\flat$ which is also a $\mathbb U$-basis for $E$.

Theorems & Definitions (12)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.7: Canonical Base Property
  • proof
  • Corollary 2.8: Dichotomy for finite-dimensional types
  • Remark 2.9
  • Theorem 3.1
  • ...and 2 more