Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call "regenerative pairs," which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).

中文翻译：

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Littlewood 多项式、谱空码和等幂分区

让 $[n]$ 表示 $\{0,1, ... , n-1\}$。多项式 $f(x) = \sum a_i x^i$ 是长度为 $n$ 的 Littlewood 多项式 (LP)，如果 $a_i$ 对于 $i \in [n]$ 是 $\pm 1$，并且 $ a_i = 0$ 对于 $i \ge n$。如果这样的 LP 可被 $(x-1)^m$ 整除，则称其为 $m$ 阶。找到$m$阶LP长度的集合$L_m$的问题等价于找到$m$阶的谱空码的长度，并且找到$n$使得$[n]$承认一个划分为两个子集，它们的前 $m$ 个矩相等。扩展Boyd等人的技术和结果，我们完全确定$L_7$和$L_8$，并证明192是$L_9$的最小元素。我们的主要工具是使用整数线性规划进行仔细有针对性的搜索（既可以找到 LP，也可以针对特定的 $n$ 和 $m$ 反驳它们的存在），以及一个意想不到的新概念（由观察到的 LP 的对称特性产生），我们称之为“再生对”，它在 $L_m$ 中产生无限的等差数列。我们证明对于 $m \le$ 8，只要有一个长度为 $n$ 且阶为 $m$ 的 LP，就有一个长度为 $n$ 且阶为 $m$ 是对称的（resp.~antisymmetric），如果m 是偶数（resp.~odd）。