Summary

In answer to a question of M. Aschenbrenner and L. van den Dries, we show that no differentially closed field possesses a differential valuation.

Introduction

In connection with their work on H-fields [1], M. Aschenbrenner and L. van den Dries asked whether a differentially closed field can admit a nontrivial (Rosenlicht) differential valuation.

If K is a field and v is a Krull valuation on K and L/K is an extensions field, then there is at least one extension of v to a valuation on L. It is known that the analogous statement for differential specializations on differential fields is false. Indeed, anomalous properties of specializations of differential rings were observed already by Ritt [11] and examples of nonextendible specializations are known (see Exercise 6(c) of Section 6 of Chapter IV of [7] and [4, 5, 9] for a fuller account).

In this short note, we answer their question negatively by exhibiting a class of equations which cannot be solved in any differentially valued field even though they have solutions in differentially closed fields. In a forthcoming work of Aschenbrenner, van den Dries and van der Hoeven [2], the main results of this note are explained via direct computations.

I thank M. Aschenbrenner and L. van den Dries for bringing this question to my attention and discussing the matter with me and the Isaac Newton Institute for providing a mathematically rich setting for those discussions.