Zilber dichotomy for DCF_0,m

2024/11/07 by Omar León Sánchez, Sanchez, Omar Leon
Mathematics · #03C10 #03C60 #12H05 #14A99 #Advanced Algebra and Geometry #FOS: Mathematics #Logic (math.LO) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.04801

openalex publication_date 2024/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the theory of differentially closed fields of characteristic zero in m≥ 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.

Related

• Zilber's Dichotomy for Differentially Closed Fields with an Automorphism
• Model theory of fields with free operators in characteristic zero
• Restricted Trichotomy in Characteristic Zero
• Fields with several commuting derivations
• A note on the effective listing of complete types