On the Stability of Analytic Germs under Ultradifferentiable Perturbations

Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy-Carleman classes $\mathcal{C}_M$. For suitable divisors $P(x,t)=x^d+a_1(t)x^{d-1}+\cdots+a_d(t)$ with real-analytic coefficients $a_j$, we show that the quotient and the remainder can be chosen of class $\mathcal{C}_{M^σ}$, where $M^σ=((M_j)^σ)_{j\geq 0}$ and $σ$ is a certain Łojasiewicz exponent $σ$ related to the geometry of the roots of $P$ and verifying $1\leq σ\leq d$. We provide various examples for which $σ$ is optimal, in particular strictly less than $d$, which sharpens earlier results of…

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensio…

Topological equivalence of finitely determined real analytic plane-to-plane map-germs

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determined real analytic plane-to-plane map germs.

The Galoisian envelope of a germ of foliation: the quasi-homogeneous case

We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.

解析叶状结构的和乐与等价性

Holonomy and equivalence of analytic foliations

本文研究一类由特定向量场芽生成的奇异叶状结构，证明在适当假设下，其解析分类可归结为沿给定分界线的局部和乐的解析共轭。

Non-Noetherianity of Denjoy-Carleman rings of germs

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.