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

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.2010 Mathematics Subject Classification. Primary 26E10; 32B05; Secondary 46E25.

A Note on the Weierstrass Preparation Theorem in Quasianalytic Local Rings

“…A similar property has been announced in [1] for quasianalytic Denjoy-Carleman classes. More precisely, the statement made in [1] claims that if a Denjoy-Carleman class contains strictly the analytic system, then Weierstrass preparation does not hold, even if we allow the unit and the distinguished polynomial to be in any wider quasianalytic Denjoy-Carleman class.…”

“…A similar property has been announced in [1] for quasianalytic Denjoy-Carleman classes. More precisely, the statement made in [1] claims that if a Denjoy-Carleman class contains strictly the analytic system, then Weierstrass preparation does not hold, even if we allow the unit and the distinguished polynomial to be in any wider quasianalytic Denjoy-Carleman class. The approach there, pretty different from ours, leads to a precise investigation of the following extension problem: does a function belonging to a quasianalytic Denjoy-Carleman class defined on the positive real axis extend to a function belonging to a wider Denjoy-Carleman class defined on the real axis?…”

Pathological Phenomena in Denjoy–Carleman Classes

“…which is (1). (2) is obvious either again from Faà di Bruno's formula, or by looking at the formal power series of g at 0.…”

“…However, despite quasianalytic classes satisfying "quasianalytic continuation", their theory remains not well-understood. This is in a large because many standard techniques for analytic functions, namely the Weierstrass division and preparation theorems, fail in general for quasianalytic Denjoy-Carleman classes (see [1,8,9,13,15]). This makes deciding whether these classes are Noetherian very difficult.…”

On Division of Quasianalytic Function Germs

“…[6]) and even the Weierstrass preparation theorem (cf. [1]). Neither do more general quasianalytic local rings defined by imposing certain natural axioms (stability by composition, implicit function theorem and monomial division); see [9] for Weierstrass division and [16] for Weierstrass preparation.…”