منابع مشابه
Transfinite Diameter and the Resultant
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 .
متن کاملTransfinite 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 کاملThe monic integer transfinite diameter
We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter tM(I) is defined as the infimum of all such supremums. We show that if I has length 1, then tM(I) = 1 2 . We make three general conjectures relating to the value of tM(I) for intervals I of length less than 4. We also con...
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 کاملTrace of totally positive algebraic integers and integer transfinite diameter
Explicit auxiliary functions can be used in the “Schur-SiegelSmyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm, and we improve the known lower bounds for the absolute trace of totally positive algebraic integers. This improve...
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ژورنال
عنوان ژورنال: Journal für die reine und angewandte Mathematik (Crelles Journal)
سال: 2007
ISSN: 0075-4102,1435-5345
DOI: 10.1515/crelle.2007.077