On Distribution of Zeros of Random Polynomials in Complex Plane
On Distribution of Zeros of Random Polynomials in Complex Plane
复制标题
复平面上随机多项式零点的分布
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Zaporozhets
中科院分区：
文献类型：
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作者：
I. Ibragimov;D. Zaporozhets
设\(G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}\)是一个随机多项式，具有i.i.d.系数（实数或复数）。我们证明了gn (z)的根的参数在[0,2 π]中渐近均匀分布为\(n\,\rightarrow \,\infty \)。我们还证明了\(\mathbf{E}\,\ln (1 + \vert \xi _{0}\vert )\,<\,\infty \)条件是根在单位周长附近渐近集中的充分必要条件。
Let \(G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}\) be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of G n (z) are uniformly distributed in [0, 2π] asymptotically as \(n\,\rightarrow \,\infty \). We also prove that the condition \(\mathbf{E}\,\ln (1 + \vert \xi _{0}\vert )\,<\,\infty \) is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.