Transfinite diameter on varieties

Transfinite diameter is a geometric notion that plays a central role in classical potential theory and complex analysis. It can also be defined in terms of univariate polynomials. In several complex variables and pluripotential theory, the Fekete-Leja transfinite diameter is a natural generalization. An important study of it was Zaharjuta's 1975 paper. His methods for studying transfinite diameter combined algebra and analysis, i.e. manipulating polynomials, and taking limits. The algebraic part in this case was relatively trivial. I am interested in studying transfinite diameter on an algebraic variety. Recently with David Cox (Amherst, MA), we were able to adapt Zaharjuta's techniques to study transfinite diameter on varieties that behave “nicely” at infinity. Almost all of the additional work is in the algebra. I will describe our work and illustrate it with concrete examples.