Chebyshev constants and transfinite diameter on algebraic curves in C^2

Abstract

Directional Chebyshev constants and transfinite diameter are defined for compact subsets of a complex algebraic curve in C^2 . Given such a compact subset K, a formula equating the geometric mean of the directional Chebyshev constants of K with its transfinite diameter is proved. This formula generalizes the relation between the classical transfinite diameter and Chebyshev constant in the complex plane, and is a discrete analog of Zaharjuta’s integral formula for the Fekete-Leja transfinite diameter in Cn . Further prop- erties of the Chebyshev constants and transfinite diameter are also studied.