Note on a differential algebra bound

TL;DR

This paper proves that the previously established bound on Morley powers for nonorthogonal types in differentially closed fields is tight, using Galois theory and descent arguments to demonstrate the existence of such types.

Contribution

It establishes the exact bound for Morley powers where nonorthogonal types remain weakly orthogonal, refining prior results with new methods.

Findings

The bound n+3 and m+3 is tight for weak orthogonality.

Constructs examples of types with degree of nonminimality 2 exhibiting this property.

Employs Galois theory and descent arguments to prove the result.

Abstract

In a recent article, Freitag, Moosa and the author showed that in differentially closed fields of characteristic zero, if two types are nonorthogonal, then their n+3 and m+3 Morley powers are not weakly orthogonal, where n and m are their respective Lascar ranks. In this short note, we prove that the bound is tight: there are such types with weakly orthogonal n+2 and m+2 Morley powers. The types in question were constructed by Freitag and Moosa as examples of types with degree of nonminimality 2. As interesting as our result are our methods: we rely mostly on Galois theory and some descent argument for types, combined with the failure of the inverse Galois problem over constant parameters.