Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ 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 assume no other conditions on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded polynomial ring. 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 properties of their respective weighted extremal functions $\Phi_{K, q}^S$ and $V_{K, q}^S$ in connection with properties of $S$.