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 of a subset E of C n , also called a pluricomplex Green function or global extremal function of E, equals the logarithm of the Siciak function Φ 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 P m (C n) of degree ≤ m such that | p | ≤ 1 on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space P m (C n) is replaced by P m S (C n) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of R + n with 0 ∈ S . It states that if q is an admissible weight on a closed set E in C n then V E , q S = log Φ E , q S on C ∗ n if and only if the rational points in S form a dense subset of S. [ABSTRACT FROM AUTHOR]