Relations between Transfinite Diameters on Affine Algebraic Varieties

TL;DR

This paper investigates how different polynomial bases affect the transfinite diameter of compact sets and proves their equivalence under certain conditions, with applications to algebraic varieties.

Contribution

It establishes conditions under which transfinite diameters are invariant across different polynomial bases and proves the equality of diameters defined by Cox-Ma`u and Berman-Boucksom for algebraic varieties.

Findings

Transfinite diameter remains unchanged for sufficiently similar polynomial bases.

Different definitions of transfinite diameter for algebraic varieties are equivalent.

The results unify various approaches to transfinite diameter in algebraic geometry.

Abstract

Given a compact set $K$ one may define a transfinite diameter for $K$ via a limiting process involving maximising a Vandermonde determinant over $K$ with respect to a monomial basis. Different transfinite diameters may be obtained by using different polynomial bases in the Vandermonde determinant calculation. We show that if these bases are sufficiently similar that the transfinite diameter of $K$ is unchanged. Utilising this result we show that the transfinite diameters defined by Cox-Ma`u and Berman-Boucksom for algebraic varieties are equal.