Constructing Types in Differentially Closed Fields that are Analysable in the Constants

TL;DR

This paper investigates the analysability of types in differentially closed fields, generalizing known results and introducing canonical analysis, to construct types with prescribed properties in the theory of differential fields.

Contribution

It introduces the concept of canonical analysis and constructs types in DCF₀ with specified U-rank properties at each analysis step.

Findings

Certain equations are not analysable in constants within specific steps.

Canonical analysis may not exist for all analysable types.

Types with prescribed U-rank sequences are explicitly constructed.

Abstract

Analysability of finite $U$-rank types are explored both in general and in the theory $\mathrm{DCF}_0$. The well-known fact that the equation $\delta(\mathrm{log}\delta x)=0$ is analysable in but not almost internal to the constants is generalized to show that $\underbrace{\mathrm{log}\delta...\mathrm{log}\delta}_n x=0$ is not analysable in the constants in $(n-1)$-steps. The notion of a \emph{canonical analysis} is introduced -- namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of reductions and coreductions in theories with the canonical base property, it is constructed, for any sequence of positive integers $(n_1,...,n_\ell)$, a type in $\mathrm{DCF}_0$ that admits a canonical analysis with the property that the $i$th step has $U$-rank $n_i$.