We study the regularity problem for Cauchy Riemann maps between hypersurfaces in Cn. We prove that a continuous Cauchy Riemann map between two smooth C∞ pseudoconvex decoupled hypersurfaces of finite D'Angelo type is of class C∞.

Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.

Soient $k$ un corps commutatif et $I=(p_1,\cdots ,p_m)k_n\left[X\right]$ un idéal de l’anneau des polynômes $k[X_1,\cdots ,X_n]$ (éventuellement $I=k_n\left[X\right]$). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme $p$, élément de la clôture intégrale $\overline{I}$ de l’idéal $I$, on a une représentation$$p^m=\sum _{1\le i\le m}{p}_i{q}_i,\text{avec}maxdeg\left(q_i{p}_i\right)\le mdegp+m{d}_1\cdots d_m,$$où $d_i=deg{p}_i,1\le i\le m$.

$G\subset {ℂ}^N$ is compact and convex it is known for a long time that the nonzero constant coefficients linear partial differential operators (of finite or infinite order) are surjective on the space of all analytic functions on G. We consider the question whether solutions of the inhomogeneous equation can be given in terms of a continuous linear operator. For instance we characterize those sets G for which this is always the case.