系数为回文块的随机余弦多项式的实零点
Real zeros of random cosine polynomials with palindromic blocks of coefficients
摘要：
众所周知，随机余弦多项式$ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 π) $，其中系数是独立且同分布的（i.i.d.）实值标准高斯随机变量（渐近）具有$ 2n / \sqrt{3} $期望实根。另一方面，在构造相关随机多项式的许多方法中，一种是强制系数是回文的。因此，问一个随机余弦多项式有多少个实零是有意义的。本文证明了这类多项式的实根期望个数的渐近性是$ n $，其中常数$ \mathrm{K}_\ell $（仅取决于$ \ell $）大于1，并且可以明确地由二重积分公式表示。也就是说，这样的多项式与经典的i.i.d.情况相比，具有稍微更多的期望实零点。系数
英文摘要：
It is well known that a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 π) $, with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $ 2n / \sqrt{3} $ expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree $ n $) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length $ \ell. $ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is $ \mathrm{K}_\ell \cdot 2n / \sqrt{3} $, where the constant $ \mathrm{K}_\ell $ (depending only on $ \ell $) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.
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