Littlewood polynomials, spectral-null codes, and equipowerful partitions

Abstract:

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 $i \ge n$. An LP has 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 initiated by Boyd, we completely determine $L_7$ and $L_8$, and prove that $\min L_9=192$ and $\min L_{10}=240$. Our primary tools are (a) symmetry, and (b) the use of carefully targeted searches using integer linear programming both to find LPs and to disprove their existence for specific $n$ and $m$. Symmetry plays an unexpected role and leads to the concept of regenerative pairs, which produce infinite arithmetic progressions in $L_m$. We prove for $m \le$ 8 (and conjecture later for all $m$) that 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).