# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Littlewood polynomials, spectral-null codes, and equipowerful partitionsHTML articles powered by AMS MathViewer

by Joe Buhler, Shahar Golan, Rob Pratt and Stan Wagon
Math. Comp. 90 (2021), 1435-1453 Request permission

## 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
• Email: jbuhler@math.umn.edu
• Shahar Golan
• Affiliation: Department of Computer Science, Jerusalem College of Technology, Jerusalem, Israel
• MR Author ID: 785537
• Email: sgolan@jct.ac.il
• Rob Pratt
• Affiliation: SAS Institute, Inc., Cary, North Carolina 27513
• MR Author ID: 1334609
• Email: rob.pratt@sas.com
• Stan Wagon
• Affiliation: Department of Mathematics, Macalester College, St. Paul, Minnesota 55105
• MR Author ID: 179895
• Email: wagon@macalester.edu
• 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
• © Copyright 2021 American Mathematical Society
• Journal: Math. Comp. 90 (2021), 1435-1453
• MSC (2020): Primary 11B83, 12D10; Secondary 94B05, 11Y99
• DOI: https://doi.org/10.1090/mcom/3612
• MathSciNet review: 4232230