We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1\left(ℂ\right)$, with $k$ marked critical points $c_1,...,c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda$, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1,...,c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\mathrm{Supp}(T_1\wedge \cdots \wedge T_k)$.

We calculate the transfinite diameter for the real unit ball $B_d:=x\in {ℝ}^d:\left|x\right|\le 1$ and the real unit simplex $T_d:=x\in {ℝ}_{+}^d:{\sum }_{j=1}^d{x}_j\le 1.$

On définit sur un espace vectoriel $E$ une classe de topologies qui rendent la multiplication continue, mais ne sont pas vectorielles en général. Sur un espace complexe $E$ elles permettent d’obtenir encore les principales propriétés des fonctions plurisousharmoniques. De telles topologies séparées sont localement pseudo-convexes (mais non localement convexes en général) : cette notion intervient dans les extensions données récemment par l’auteur du théorème de Banach-Steinhaus aux familles de polynômes...

The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter d(K), characterizing the asymptotic size of K, the Chebyshev constant τ(K), characterizing the minimal uniform deviation of a monic polynomial on K, and the capacity c(K), describing the asymptotic behavior of the Green function $g_K\left(z\right)$ at infinity, coincide. In this paper we give a survey of results on multidimensional notions of transfinite diameter, Chebyshev constants and capacities, related...

We discuss problems on Hankel determinants and the classical moment problem related to and inspired by certain Vandermonde determinants for polynomial interpolation on (quadratic) algebraic curves in ℂ².

It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\epsilon }$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.