Zilber dichotomy for $DCF_{0,m}$

TL;DR

This paper proves a key model-theoretic dichotomy for differentially closed fields with multiple derivations, extending known results to include infinite-dimensional types.

Contribution

It establishes the Zilber dichotomy for DCF_{0,m} in all dimensions, filling a gap in the existing literature.

Findings

Minimal types are either locally modular or nonorthogonal to constants.

The dichotomy holds for both finite and infinite-dimensional types.

Provides a comprehensive proof including the infinite-dimensional case.

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.