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

Full-text PDF Free Access

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.

- Francesco Amoroso,
*Algebraic numbers close to $1$ and variants of Mahler’s measure*, J. Number Theory**60**(1996), no. 1, 80–96. MR**1405727**, DOI https://doi.org/10.1006/jnth.1996.0114 - David W. Boyd,
*Reciprocal polynomials having small measure*, Math. Comp.**35**(1980), no. 152, 1361–1377. MR**583514**, DOI https://doi.org/10.1090/S0025-5718-1980-0583514-9 - David W. Boyd,
*Speculations concerning the range of Mahler’s measure*, Canad. Math. Bull.**24**(1981), no. 4, 453–469. MR**644535**, DOI https://doi.org/10.4153/CMB-1981-069-5 - David W. Boyd and Hugh L. Montgomery,
*Cyclotomic partitions*, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 7–25. MR**1106647** - E. Dobrowolski,
*On a question of Lehmer and the number of irreducible factors of a polynomial*, Acta Arith.**34**(1979), no. 4, 391–401. MR**543210**, DOI https://doi.org/10.4064/aa-34-4-391-401 - E. Hille,
*Analytic Function Theory, Vol. II*, Chelsea, 1987. - D. H. Lehmer,
*Factorization of certain cyclotomic functions*, Ann. of Math. (2)**34**(1933), 461–479. - E. T. Lehmer,
*A numerical function applied to cyclotomy*, Bull. Amer. Math. Soc.**36**(1930), 291–298. - M. J. Mossinghoff,
*Algorithms for the determination of polynomials with small Mahler measure*, Ph.D. Thesis, The University of Texas at Austin, 1995. - M. J. Mossinghoff,
*Polynomials with small Mahler measure*, Math. Comp. 67 (1998), 1697–1705. - G. Pólya and G. Szegő,
*Problems and theorems in analysis. Vol. II*, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR**0465631** - A. G. Postnikov,
*Introduction to analytic number theory*, Translations of Mathematical Monographs, vol. 68, American Mathematical Society, Providence, RI, 1988. Translated from the Russian by G. A. Kandall; Translation edited by Ben Silver; With an appendix by P. D. T. A. Elliott. MR**932727** - Ulrich Rausch,
*On a theorem of Dobrowolski about the product of conjugate numbers*, Colloq. Math.**50**(1985), no. 1, 137–142. MR**818097**, DOI https://doi.org/10.4064/cm-50-1-137-142 - Joseph H. Silverman,
*Exceptional units and numbers of small Mahler measure*, Experiment. Math.**4**(1995), no. 1, 69–83. MR**1359419** - C. J. Smyth,
*On the product of the conjugates outside the unit circle of an algebraic integer*, Bull. London Math. Soc.**3**(1971), 169–175. MR**289451**, DOI https://doi.org/10.1112/blms/3.2.169 - J. Stoer and R. Bulirsch,
*Introduction to numerical analysis*, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR**557543**

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
:,
26C10,
:,
12--04,
12D10,
30C15

Retrieve articles in all journals with MSC (1991): :, 26C10, :, 12--04, 12D10, 30C15

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

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

Received by editor(s):
February 7, 1997

Article copyright:
© Copyright 1998
American Mathematical Society