# Pluripotential Numerics

## TL;DR

This paper develops numerical methods to approximate key quantities in Pluripotential Theory, including extremal functions, transfinite diameter, and equilibrium measures, using polynomial meshes and orthonormalization.

## Contribution

It introduces new algorithms for approximating pluripotential quantities with proven convergence and demonstrates their effectiveness through numerical tests.

## Key findings

- Convergence of transfinite diameter approximation.
- Uniform convergence of extremal function approximation.
- Successful numerical tests in simple cases.

## Abstract

We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the \emph{extremal plurisubharmonic function} $V_E^*$ of a compact $\mathcal L$-regular set $E\subset \C^n$, its \emph{transfinite diameter} $\delta(E),$ and the \emph{pluripotential equilibrium measure} $\mu_E:=\ddcn{V_E^*}.$ The methods rely on the computation of a \emph{polynomial mesh} for $E$ and numerical orthonormalization of a suitable basis of polynomials. We prove the convergence of the approximation of $\delta(E)$ and the uniform convergence of our approximation to $V_E^*$ on all $\C^n;$ the convergence of the proposed approximation to $\mu_E$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein Markov measures. Numerical tests are presented for some simple cases with $E\subset \R^2$ to illustrate the performances of the proposed methods.