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Littlewood polynomials, spectral-null codes, and equipowerful partitions

Authors: Joe Buhler, Shahar Golan, Rob Pratt and Stan Wagon
Journal: Math. Comp. 90 (2021), 1435-1453
MSC (2020): Primary 11B83, 12D10; Secondary 94B05, 11Y99
Published electronically: February 26, 2021
MathSciNet review: 4232230
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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).

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Additional Information

Joe Buhler
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID: 43035

Shahar Golan
Affiliation: Department of Computer Science, Jerusalem College of Technology, Jerusalem, Israel
MR Author ID: 785537

Rob Pratt
Affiliation: SAS Institute, Inc., Cary, North Carolina 27513
MR Author ID: 1334609

Stan Wagon
Affiliation: Department of Mathematics, Macalester College, St. Paul, Minnesota 55105
MR Author ID: 179895

Keywords: Littlewood polynomials, spectral-null code, antenna array, multigrade identity, equal power sum partition, integer linear programming
Received by editor(s): December 9, 2019
Received by editor(s) in revised form: August 24, 2020, and September 29, 2020
Published electronically: February 26, 2021
Article copyright: © Copyright 2021 American Mathematical Society