Polynomial convexity, special polynomial polyhedra and the pluricomplex Green function for a compact set in Cn

AbstractWe prove a version of the Hilbert Lemniscate Theorem in Cn. More precisely, any polynomially convex compact subset K of Cn can be approximated externally by special polynomial polyhedra P defined by proper polynomial mappings from Cn to Cn with “almost” all their zeros in P. We precise this version when K is balanced. And we also give several applications of these results: approximation of the pluricomplex Green function for a compact set with pole at infinity, a Lelong–Bremermann Theorem for functions in L+ and uniform polynomial approximation of holomorphic functions