On démontre que toute solution formelle $\bar{y}\left(x\right)$ d’un système d’équations analytiques réelles (resp. polynomiales réelles) $f(x,y)=0$, se relève en une solution ${\mathbf{C}}^{\infty }$ homotope à une solution analytique (resp. à une solution de Nash) aussi proche que l’on veut de $\bar{y}\left(x\right)$ pour la topologie de Krull. On utilise ce théorème pour démontrer l’algébricité (ou l’analyticité) de certains idéaux de $\mathbf{R}\left\{x\right\}$ (ou $\mathbf{R}\left[\right[x\left]\right]$), et aussi pour construire des déformations analytiques de germes d’ensembles analytiques en germes d’ensembles de Nash.

For nonquasianalytical Carleman classes conditions on the sequences $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ are investigated which guarantee the existence of a function in $C_J\left\{{\widehat{M}}_n\right\}$ such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary...

In IMUJ Preprint 2009/05 we investigated the quasianalytic perturbation of hyperbolic polynomials and symmetric matrices by applying our quasianalytic version of the Abhyankar-Jung theorem from IMUJ Preprint 2009/02, whose proof relied on a theorem by Luengo on ν-quasiordinary polynomials. But those papers of ours were suspended after we had become aware that Luengo's paper contained an essential gap. This gave rise to our subsequent article on quasianalytic perturbation theory, which developed,...

We generalize to some classes of ultradifferentiable jets or functions the classical Łojasiewicz Division Theorem and Glaeser Composition Theorem. The proof uses the desingularization results by Hironaka, Bierstone and Milman.

We prove a Weierstrass division formula for $C^{\infty }$ Whitney jets ∂̅-flat on arbitrary compact subsets of the complex plane. We also give results for Carleman classes.