Transfinite diameter on the graph of a polynomial mapping and the DeMarco-Rumely formula

TL;DR

This paper investigates the properties of Chebyshev constants and transfinite diameters on polynomial graph mappings in complex two-space, proving a symmetry condition and providing a new proof of a related pullback formula involving resultants.

Contribution

It establishes the equality of two transfinite diameters under symmetry conditions and offers a novel proof of the DeMarco-Rumely pullback formula in ^2.

Findings

Equality of transfinite diameters under symmetry

New proof of DeMarco-Rumely pullback formula

Connection between Chebyshev constants and polynomial graphs

Abstract

We study Chebyshev constants and transfinite diameter on the graph of a polynomial mapping $f\colon\mathbb{C}^2\to\mathbb{C}^2$. We show that two transfinite diameters of a compact subset of the graph (i.e., defined with respect to two different collections of monomials) are equal when the set has a certain symmetry. As a consequence, we give a new proof in $\mathbb{C}^2$ of a pullback formula for transfinite diameter due to DeMarco and Rumely that involves a homogeneous resultant.