A note on canonical bases and one-based types in supersimple theories

TL;DR

This paper investigates the CBP property in model theory, providing a decomposition of types of canonical bases, proving that existentially closed difference fields have CBP, and exploring applications to algebraic structures and dynamics.

Contribution

It introduces a general decomposition of types of canonical bases and proves that existentially closed difference fields possess the CBP, with applications to various algebraic theories.

Findings

Types of canonical bases can be decomposed similarly to primary decomposition.

Existentially closed difference fields of any characteristic have the CBP.

Results have implications for differential and difference varieties and algebraic dynamics.

Abstract

This paper studies the CBP, a model-theoretic property first discovered by Pillay and Ziegler. We first show a general decomposition result of types of canonical bases, which one can think of as a sort of primary decomposition. This decomposition is then used to show that existentially closed difference fields of any characteristic have the CBP. We also derive consequences of the CBP, and use these results for applications to differential and difference varieties, and algebraic dynamics.