Corrigendum to the paper: Geometric Axioms for Differentially Closed Fields with Several Commuting Derivations

TL;DR

This corrigendum corrects an error in a previous paper on geometric axioms for differentially closed fields with multiple derivations, clarifying the proper axiomatization without affecting the main geometric results.

Contribution

It provides a corrected set of first-order axioms that accurately characterize the geometric properties of differentially closed fields with several commuting derivations.

Findings

The original lemma 2.6(2) was incorrect due to improper iteration of { au}.

The geometric characterization in Theorem 3.4 remains valid.

A new, simpler set of first-order axioms is proposed to replace the flawed attempt in Theorem 4.3.

Abstract

In the proof of Lemma 2.6 (2) the iteration of the map {\tau} was not performed properly and in fact the lemma is wrong; a counterexample is given by f = \bar{x}_1and k = 2. This error does not, however, affect the geometric characterization given in Theorem 3.4 but only the attempt in Theorem 4.3 to express it as a first-order set of axioms. That attempt is incorrect; the main problem being that in general {\tau}V(f_1,..., f_s) 6= V(f_1..., f_s, {\tau}f_1,..., {\tau}f_s). But a different, indeed simpler, set of first-order axioms, which we will now describe, does express the geometric characterization.