A refinement for rational functions of Pólya's method to construct Voronoi diagrams

Abstract

Given a complex polynomial P with zeroes z 1 , . . . , z d , we show that the asymptotic zero-counting measure of the iterated derivatives Q (n) , n = 1, 2, . . . , where Q = R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z 1 , . . . , z d . This refines Pólya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in C m .where s is Euclidean length measure in the complex plane, and z i , z j are distinct zeroes of P . Restricting the measure to the segment of L ij that is part of the Voronoi diagram, and summing over all lines gives a measure µ S , supported on the Voronoi diagram. This will in fact be a probability measure canonically associated with the diagram.Theorem 1.1. Given a rational function Q = R/P where P has degree d ≥ 2 and distinct zeroes z 1 , . . . , z d , arXiv:1610.00921v2 [math.CA]

The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions

“…The two results in this section describe properties of the asymptotic zero-counting measure of P n /A n . Their proofs are analogous to those of Lemma 2.1 and Proposition 2.2 in [2], respectively.…”

“…In this paper, we generalize the main result of the aforementioned paper (see Theorem 1 of [2]) to the situation when f = (P/Q)e T , where P and Q are defined as previously, and T is a nonconstant polynomial. Furthermore, we assume that Q is monic and has at least two zeros, all of which are distinct.…”

“…of such a function asymptotically accumulate along the boundaries of the Voronoi diagram associated with S. This classical result is called Pólya's Shire theorem (see [3,4]). In a recent paper by Rikard Bögvad and this author (see [2]), a measure-theoretic refinement of Pólya's Shire theorem was given for the special case that f = P/Q, where P and Q are polynomials with gcd(P, Q) = 1, and P ≡ 0.…”

“…Additionally, note that the formula used to reconstruct the measure µ S in (iv) is identical to the formula used in the reconstruction of a measure from its associated logarithmic potential (see [1]). Furthermore, note that if t = 0, it follows from Theorem 1 of [2] that the logarithmic potential of the asymptotic zero-counting measure µ of the sequence (P/Q) (n) ∞ n=1 is given by…”

Pseudotropical curves

“…In particular, in algebraic geometry objects of this nature had appeared in a number of different problems, see e.g. [5], [6], [14], [17], [18], [20].…”