It is well known that a random cosine polynomial $${V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} $$ V n ( x ) = ∑ j = 0 n a j cos ( j x ) , x ∈ (0, 2 π ), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $$2n/\sqrt 3 $$ 2 n / 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 ℓ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is $${{\rm{K}}_\ell} \times 2n/\sqrt 3 $$ K ℓ × 2 n / 3 , where the constant K ℓ (depending only on ℓ ) 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.

中文翻译：

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具有回文系数块的随机余弦多项式的实零点

众所周知，随机余弦多项式 $${V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} $ $ V n ( x ) = ∑ j = 0 naj cos ( jx ) , x ∈ (0, 2 π )，系数独立同分布（iid）实值标准高斯随机变量（渐近）有$$2 n/\sqrt 3 $$ 2 n / 3 预期的实数根。另一方面，在构造相关随机多项式的许多方法中，一种是强制系数为回文。因此，如果系数在固定长度 ℓ 的回文块中排序，那么询问具有相同且正态分布系数的随机余弦多项式（n 次）具有多少实零点是有意义的。我们证明了这样一个多项式的实根的期望个数的渐近线是 $${{\rm{K}}_\ell} \times 2n/\sqrt 3 $$ K ℓ × 2 n / 3 ，其中常数 K ℓ（仅依赖于 ℓ ）大于 1，可以用二重积分公式显式表示。也就是说，与具有 iid 系数的经典情况相比，此类多项式具有略多的预期实零点。