Polynomials with exponents in compact convex sets and associated weighted extremal functions – fundamental results

This paper is a collection of fundamental results about the polynomial rings $\mathcal P^S(\mathbb C^n)$ where the $m$th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb R^n_+$ is compact, convex and $0\in S$. We study the relationship between $\mathcal P^S(\mathbb C^n)$ and the class $\mathcal L^S(\mathbb C^n)$ of global plurisubharmonic functions where the growth is determined by the logarithmic supporting function of $S$. We present the properties of their respective weighted extremal functions $\Phi _{K, q}^S$ and $V_{K, q}^S$ in connection with the properties of $S$. Our ambition is to give detailed proofs with minimal assumptions of all results, thus creating a self-contained exposition.