We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| 2$ on the unit circle $\partial \mathcal{D}$. This polynomial is of degree $38$ and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional $(k, n)$ with $k \in \{1, 2, 3, n-3, n-2, n-1\}$, for which no such $\{0, 1\}$--polynomial of degree $n$ exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for $\{-1, 1\}$ polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of $N(f)$ in the set of Newman and Littlewood polynomials.

中文翻译：

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关于单位圆盘内具有规定数量的零的纽曼和利特伍德多项式

我们研究了 $\{0, 1\}$ 和 $\{-1, 1\}$ 多项式 $f(z)$，称为 Newman 和 Littlewood 多项式，它们在开元盘 $\mathcal{D} = \{z \in \mathbb{C}: |z| 2$ 在单位圆 $\partial \mathcal{D}$ 上。这个多项式的阶数为 $38$，我们在我们的构造中使用了这个特殊的多项式。我们还用 $k \in \{1, 2, 3, n-3, n-2, n-1\}$ 识别（没有证明）所有异常的 $(k, n)$，其中没有这样的 $ \{0, 1\}$--多项式$n$ 存在：此类对与常规（实数和复数）皮索数相关。建立了 $\{-1, 1\}$ 多项式的类似但不太完整的结果。我们还研究了间隔纽曼多项式的乘积，并考虑了旋转的大型 Littlewood 多项式。最后，根据我们的数据，