Newman polynomials with prescribed vanishing and integer sets with distinct subset sums

Authors:
Peter Borwein;Michael J. Mossinghoff
Affiliations:
Department of Mathematics, and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada;Department of Mathematics, University of California, Los Angeles, Los Angeles, California
Venue:
Mathematics of Computation
Year:
2003

Citing 4
Cited 0

An improved lower bound on the greatest element of a sum-distinct set of fixed order

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Abstract

We study the problem of determining the minimal degree d(m) of a polynomial that has all coefficients in {0, 1) and a zero of multiplicity m at -1. We show that a greedy solution is optimal precisely when m ≤ 5, mirroring a result of Boyd on polynomials with ±1 coefficients. We then examine polynomials of the form Πe∈E(xe +1), where E is a set of m positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that d(m) satisfies inequalities of the form 2m +c1m ≤ d(m) ≤ 103/96 . 2m+c2. Last, we consider the related problem of finding a set of m positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for m ≤ 9. extending some computations of Lunnon.