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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Perturbing polynomials
with all their roots on the unit circle


Authors: Michael J. Mossinghoff, Christopher G. Pinner and Jeffrey D. Vaaler
Journal: Math. Comp. 67 (1998), 1707-1726
MSC (1991): Primary :, 26C10; Secondary :, 12--04, 12D10, 30C15
MathSciNet review: 1604387
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most $4$, with $4$ achieved only for polynomials of the form $x^{2n}+cx^n+1$ with $c$ in $[-2,2]$. The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in $[-1,1]$. If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length $3$ that do not arise from a perturbation of length $4$. We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is $O(C^{\sqrt{d}})$, where $d$ is the degree, and we report on the polynomials found by this algorithm through degree 64.


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Additional Information

Michael J. Mossinghoff
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608
Email: mjm@math.appstate.edu

Christopher G. Pinner
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ontario K1N 6N5, Canada
Email: pinner@mathstat.uottawa.ca

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: vaaler@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-98-01007-2
PII: S 0025-5718(98)01007-2
Keywords: Cyclotomic, Mahler measure, Lehmer's problem, transfinite diameter
Received by editor(s): February 7, 1997
Article copyright: © Copyright 1998 American Mathematical Society