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2000 Polynomials with height 1 and prescribed vanishing at 1
Peter Borwein, Michael J. Mossinghoff
Experiment. Math. 9(3): 425-433 (2000).


We study the minimal degree d(m) of a polynomial with all coefficients in $\{-1,0,1\}$ and a zero of order m at 1. We determine d(m) for $m\leq10$ and compute all the extremal polynomials. We also determine the minimal degree for $m=11$ and $m=12$ among certain symmetric polynomials, and we find explicit examples with small degree for $m\leq21$. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several roots of unity.


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Peter Borwein. Michael J. Mossinghoff. "Polynomials with height 1 and prescribed vanishing at 1." Experiment. Math. 9 (3) 425 - 433, 2000.


Published: 2000
First available in Project Euclid: 18 February 2003

zbMATH: 0999.12001
MathSciNet: MR1795875

Primary: 11C08 , 12D10
Secondary: 11B83 , 11Y99

Keywords: 1,0,1 coefficents , polynomial , pure product , zero

Rights: Copyright © 2000 A K Peters, Ltd.


Vol.9 • No. 3 • 2000
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