Polynomials with exponents in compact convex sets and associated weighted extremal functions: The Siciak-Zakharyuta theorem

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function \(V_{E}\) $V_E$ of a subset E of \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ , also called a pluricomplex Green function or global extremal function of E, equals the logarithm of the Siciak function \(\Phi _E\) ${\mathrm{\Phi }}_E$ if E is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on E, and the Siciak function is the upper envelope of m-th roots of the modulus of polynomials p in \(\mathcal {P}_m({\mathbb {C}}^n)\) ${\mathcal{P}}_m({\mathbb{C}}^n)$ of degree \(\le m\) $\le m$ such that \(|p|\le 1\) $|p|\le 1$ on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space \(\mathcal {P}_m({\mathbb {C}}^n)\) ${\mathcal{P}}_m({\mathbb{C}}^n)$ is replaced by \(\mathcal {P}_m^S({\mathbb {C}}^n)\) ${\mathcal{P}}_m^S({\mathbb{C}}^n)$ consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of \({\mathbb {R}}^n_+\) ${\mathbb{R}}_{+}^n$ with \(0\in S\) $0\in S$ . It states that if q is an admissible weight on a closed set E in \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ then \(V^S_{E,q}=\log \Phi ^S_{E,q}\) $V_{E,q}^S=log{\mathrm{\Phi }}_{E,q}^S$ on \({\mathbb {C}}^{*n}\) ${\mathbb{C}}^{\ast n}$ if and only if the rational points in S form a dense subset of S.