Turing degree spectra of differentially closed fields

TL;DR

This paper characterizes the Turing degree spectra of countable differentially closed fields, showing they include all nonlow degrees and are precisely the jump-preimages of spectra of certain countable graphs.

Contribution

It establishes a complete characterization of the degree spectra of differentially closed fields in terms of jump-preimages of graph spectra, linking model theory and computability.

Findings

Every nonlow Turing degree is in the spectrum of some differentially closed field.

Spectra of such fields do not contain the computable degree 0.

Spectra are exactly the jump-preimages of automorphically nontrivial graph spectra.

Abstract

The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that every such field of low degree is isomorphic to a computable differential field. Relativizing the latter result and applying a theorem of Montalban, Soskova, and Soskov, we conclude that the spectra of countable differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial countable graphs.