Siciak–Zahariuta extremal functions and polynomial hulls

Abstract

We use our disc formula for the Siciak–Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs. The polynomial hull K of a compact subset K of complex affine space C n is the compact set of those x ∈ C for which |P (x)| ≤ supK |P | for all complex polynomials P in n variables. The set K is said to be polynomially convex if K = K. Polynomial hulls are usually difficult to determine. By the maximum principle for holomorphic functions it is clear that x ∈ K if there is a continuous map f from the closed unit disc D into C, holomorphic on D, such that f(0) = x and f maps the unit circle into K. In the early days of polynomial convexity theory it seemed possible that K might simply be the union of all analytic discs in C with boundary in K. Stolzenberg’s counterexample of 1963 [7] showed that K \K may be nonempty—and in fact quite large (see [1])—without containing any nonconstant analytic discs. The question of whether polynomial hulls could nevertheless be somehow described in terms of analytic discs remained open for three decades, until Poletsky derived an answer from his theory of disc functionals [6] (see Theorem 2 below). In this note, we give a different characterization of the polynomial hull of a connected compact subset of C, based on our generalization in [5] of Lempert’s disc formula for the Siciak–Zahariuta extremal function from the convex case to the connected case. The gist of both results, Poletsky’s and ours, is to suitably weaken the requirement that the analytic discs in the 2000 Mathematics Subject Classification: Primary 32E20; Secondary 32U35.