English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function withpoles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\inPSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$.When there is only one pole, or two poles in the unit ball, it turns out to beequal to the Lempert function defined from analytic disks into $\Omega$ by $L_S(z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal{O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is knownthat we always have $L_S (z) \ge G_S(z)$. In the more general case where weallow weighted poles, there is a counterexample to equality due to Carlehed andWiegerinck, with $\Omega$ equal to the bidisk.  Here we exhibit a counterexample using only four distinct equally weightedpoles in the bidisk. In order to do so, we first define a more general notionof Lempert function "with multiplicities", analogous to the generalized Greenfunctions of Lelong and Rashkovskii, then we show how in some examples this canbe realized as a limit of regular Lempert functions when the poles tend to eachother. Finally, from an example where $L_S (z) > G_S(z)$ in the case ofmultiple poles, we deduce that distinct (but close enough) equally weightedpoles will provide an example of the same inequality. Open questions arepointed out about the limits of Green and Lempert functions when poles tend toeach other.Comment: 25 page

Pluricomplex Green and Lempert functions for equally weighted poles