Transfinite Diameter and the Resultant
نویسندگان
چکیده
We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ C by a regular polynomial map F , expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F .
منابع مشابه
Transfinite Diameter
The transfinite diameter is a way of quantifying the size of compact sets in Euclidean space. This quantity is related to the Hausdorff dimension and the Lebesgue measure, but gives a slightly different perspective on the set than either of those do. In this paper, we introduce the transfinite diameter, and outline some attempts to calculate this quantity for three sets in R. For z1, z2, . . . ...
متن کاملA pr 2 00 7 Transfinite diameter , Chebyshev constant and energy on locally compact spaces
We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the ...
متن کاملOn an Equivalent Definition of the Transfinite Diameter
PHILIP DAVIS AND HENRY POLLAK Let (1) /(f) = af + a0 + ai/l + ■ ■ ■ , a > 0, map the exterior of the unit circle C onto the exterior of a simplyconnected domain B. It is part of a general theory (see e.g. [3]) that the quantity 1/a can be identified with the transfinite diameter of B. In particular, let Tn(z) be the polynomial of degree n with leading coefficient 1 whose L°° norm over B, i.e. t...
متن کاملDiameter Problems for Univalent Functions with Quasiconformal Extension Paul
This paper utilizes the method of extremal length to study several diameter problems for functions conformal outside of a disc centered at the origin, with a standard normalization, which possess a quasiconformal extension to a ring subdomain of this disc. Known results on the diameter of a complementary component of the image domain of a univalent function are extended. Applications to the tra...
متن کاملA Robin formula for the Fekete-Leja transfinite diameter
We generalize the classical Robin formula to higher dimensions. Mathematics Subject Classification (2000) Primary 32U35 · 32U20 · 14G40; Secondary 31B15 · 31C10 The purpose of this note is to give a formula for the Fekete-Leja transfinite diameter on CN , generalizing the classical Robin formula d∞(E) = e−V(E) for the usual transfinite diameter. We will disengage this from a formula for the sec...
متن کامل