Numerical calculations for the global extremal function The global extremal function for a set $U\subset ℂ^n$ and a function $q:U\to (-\infty,+\infty]$ is defined as
$$V_{U,q}(z)=\sup\{u(z)\colon u\in ℒ, u|_U\leq q\}$$
where $ℒ$ is the Lelong class which consists of those plurisubharmonic functions $u$ on $ℂ^n$ that fulfill
$$u(z)-\log|z|\leq O(1),\quad\text{when } |z|\to\infty.$$
It is known that under certain conditions, if $\{p_j\}$ is an orthonormal basis for the space of polynomials on U weighted relative to $q$ , then
$$\lim_{n\to\infty} \frac{1}{2n}\log\left(\sum_{j=1}^n p_j(z)\overline{p_j(z)}\right)=V_{U,q}(z)$$ .
This code is intended to numerically approximate the global extremal function for subsets of $ℂ$ by this limit.
The code needs python 3, matplotlib and numpy. For those using Conda there is an environment.yml file you can use to create an environment:
conda env create -f environment.yml
conda activate greenfunction
To run the program run