具有独立根的随机多项式的和
Sums of random polynomials with independent roots
摘要：
我们考虑独立随机多项式的次数趋于无穷大时，其和的零点。即，令$p$和$q$是两个独立的$n$次随机多项式，其根分别独立于复平面中的概率测度$μ$和$ν$来选择。当$n$趋于无穷大时，我们计算和$p+q$的零点的极限分布。极限分布可用其对数势来描述，我们证明它是$μ$和$ν$的对数势的逐点最大值。更一般地，当$m$固定且$n$趋于无穷大时，我们考虑$m$独立度$n$随机多项式的总和。我们的结果可以被看作是描述一个版本的自由加卷积从自由概率论的零多项式。
英文摘要：
We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability measures $μ$ and $ν$ in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of $μ$ and $ν$. More generally, we consider the sum of $m$ independent degree $n$ random polynomials when $m$ is fixed and $n$ tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.
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