Model Theory Seminar

Ruizhang Jin, Department of Pure Mathematics, University of Waterloo

"Model-theoretic Analysability in Differentially Closed Fields"

We are interested in types that are internal to or analysable in the constants in the theory of $DCF_0$. Using as a building block the well-known fact that the equation $\delta(\mathrm {log}\delta x)=0$ is analyzable in but not internal to the constants, we are able to construct analysable types with a unique canonical analysis such that each step of the analysis is of given $U$-rank. In addition, we look for criteria for when a 2-step analysable type is indeed internal. We develop a partial result when the binding group for the second step of the analysis is $\mathbb G_m(\mathcal C)$, the constant points of the multiplicative group: the type is internal if and only if the 2-step analysable type is (almost) the direct product of two types that are internal to the constants. However, the case is more complicated when the binding group for the second step is additive, as the previously mentioned criterion no longer holds.

MC 5403