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.
Publié le : 2000-05-14
Classification:
polynomial,
zero,
1,0,1 coefficents,
pure product,
11C08,
12D10,
11B83,
11Y99
@article{1045604677,
author = {Borwein, Peter and Mossinghoff, Michael J.},
title = {Polynomials with height 1 and prescribed vanishing at 1},
journal = {Experiment. Math.},
volume = {9},
number = {3},
year = {2000},
pages = { 425-433},
language = {en},
url = {http://dml.mathdoc.fr/item/1045604677}
}
Borwein, Peter; Mossinghoff, Michael J. Polynomials with height 1 and prescribed vanishing at 1. Experiment. Math., Tome 9 (2000) no. 3, pp. 425-433. http://gdmltest.u-ga.fr/item/1045604677/