# Blog Archives

# Topic Archive: Effective Methods

# Effective bounds for the existence of differential field extensions

We present a new upper bound for the existence of a differential field extension of a differential field

(K; D) that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in

terms of lengths of certain antichain sequences of N^m equipped with the product order. Pierce’s theory

has interesting applications to the model theory of fields with m commuting derivations, and his results

have been used when studying effective methods in differential algebra, such as the effective differential

Nullstellensatz problem. We use a new approach involving Macaulay’s theorem on the Hilbert function

to produce an improved upper bound. In particular, we see markedly improved results in the case of two

and three derivations.

This is joint work with Omar Leon Sanchez.