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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Perturbing polynomials with all their roots on the unit circle
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by Michael J. Mossinghoff, Christopher G. Pinner and Jeffrey D. Vaaler PDF
Math. Comp. 67 (1998), 1707-1726 Request permission

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
  • MR Author ID: 630072
  • ORCID: 0000-0002-7983-5427
  • 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
  • MR Author ID: 319822
  • Email: pinner@mathstat.uottawa.ca
  • Jeffrey D. Vaaler
  • Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
  • MR Author ID: 176405
  • Email: vaaler@math.utexas.edu
  • Received by editor(s): February 7, 1997
  • © Copyright 1998 American Mathematical Society
  • 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