Non-Noetherianity of Denjoy-Carleman rings of germs

TL;DR

This paper proves that certain Denjoy-Carleman rings of germs of functions in multiple variables are not Noetherian, resolving a long-standing open problem in real algebraic geometry by linking their non-Noetherianity to failure of Weierstrass division.

Contribution

It establishes the non-Noetherian nature of Denjoy-Carleman quasi-analytic rings of germs, using failure of Weierstrass division and advanced approximation techniques.

Findings

Denjoy-Carleman rings are not Noetherian if larger than real-analytic germs.

Failure of Weierstrass division implies non-Noetherianity.

Resolves a 35-year-old open problem in real algebraic geometry.

Abstract

It is shown that Denjoy-Carleman quasi analytic rings of germs of functions in two or more variables either complex or real valued that are stable under derivation and strictly larger than the ring of real-analytic germs are not Noetherian rings. The failure of Weierstrass division on these Denjoy-Carleman classes yields a contradiction to Noetherianity via a stronger version of Artin Approximation due to Popescu as well as results on projective modules. This settles a 35-year old open problem in real algebraic geometry.