Non-Noetherianity of Denjoy-Carleman rings of germs

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.