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

DOI:
https://doi.org/10.1090/S0025-5718-98-01007-2

MathSciNet review:
1604387

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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 , with achieved only for polynomials of the form with in . The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in . 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 that do not arise from a perturbation of length . 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 , where 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:
https://doi.org/10.1090/S0025-5718-98-01007-2

Keywords:
Cyclotomic,
Mahler measure,
Lehmer's problem,
transfinite diameter

Received by editor(s):
February 7, 1997

Article copyright:
© Copyright 1998
American Mathematical Society