多项式根的自由卷积幂

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2021-10-16 DOI:10.1080/10586458.2021.1980751

S. Steinerberger

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引用次数: 7

摘要

设$\mu$为实线上的紧支持概率测度。Bercovici-Voiculescu和Nica-Speicher证明了自由卷积能力$\mu^{\boxplus k}$对于任何真实的$k \geq 1$的存在。这篇短文的目的是从多项式及其导数的根的角度对$\mu^{\boxplus k}$给出一个基本的描述。这个桥梁允许我们在自由概率和多项式的渐近行为之间来回切换。

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Free Convolution Powers Via Roots of Polynomials

Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\mu^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is to give an elementary description of $\mu^{\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.