Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy-Carleman rings

TL;DR

This paper demonstrates that the Weierstrass Preparation Theorem does not hold in certain quasi-analytic Denjoy-Carleman rings of germs of functions in multiple variables, using a non-extension theorem as proof.

Contribution

It provides a counterexample showing the failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy-Carleman rings, highlighting limitations of these function rings.

Findings

Weierstrass Preparation Theorem fails in these rings.

Failure proven via a non-extension theorem.

Counterexamples in multi-variable germs.

Abstract

It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.