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Pure product polynomials and the Prouhet-Tarry-Escott problem


Author: Roy Maltby
Journal: Math. Comp. 66 (1997), 1323-1340
MSC (1991): Primary 11Y50, 11B75
DOI: https://doi.org/10.1090/S0025-5718-97-00865-X
MathSciNet review: 1422792
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Abstract: An $n$-factor pure product is a polynomial which can be expressed in the form $\prod _{i=1}^n(1-x^{\alpha _i})$ for some natural numbers $\alpha _1,\ldots ,\alpha _n$. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every $n$-factor pure product has norm at least $2n$. We describe three algorithms for determining the least norm an $n$-factor pure product can have. We report results of our computations using one of these algorithms which include the result that every $n$-factor pure product has norm strictly greater than $2n$ if $n$ is $7$, $9$, $10$, or $11$.


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Additional Information

Roy Maltby
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
Email: maltby@cecm.sfu.ca

DOI: https://doi.org/10.1090/S0025-5718-97-00865-X
Keywords: Prouhet-Tarry-Escott Problem, Tarry-Escott Problem, Erd\H os-Szekeres Problem
Received by editor(s): October 16, 1995
Received by editor(s) in revised form: June 19, 1996
Article copyright: © Copyright 1997 by the author