Polynomials with coefficients from a finite set
Authors:
Peter Borwein, Tamás Erdélyi and Friedrich Littmann
Journal:
Trans. Amer. Math. Soc. 360 (2008), 5145-5154
MSC (2000):
Primary 30B30; Secondary 11C08, 30C15
DOI:
https://doi.org/10.1090/S0002-9947-08-04605-9
Published electronically:
May 27, 2008
MathSciNet review:
2415068
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of $\mathbb {C}$ is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem.
This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of $\mathbb {C}$ cannot tend to zero uniformly on an arc of the unit circle.
Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let $n$ be so large that \[ \delta _n:=33\pi \frac {\log n}{\sqrt {n}} \] satisfies $\delta _n\le 1$. We show that any polynomial in \begin{align*} K_n&:=\Big \{\sum _{k=0}^n a_k z^k : |a_0|=|a_n|=1\text { and }|a_k|\le 1\Big \} \end{align*} has at least \[ 8\sqrt {n}\log n \] zeros in any disk with center on the unit circle and radius $\delta _n$.
- Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
- P. Borwein and T. Erdélyi, Littlewood-type problems on subarcs of the unit circle, Indiana Univ. Math. J. 46 (1997), no. 4, 1323–1346. MR 1631600, DOI https://doi.org/10.1512/iumj.1997.46.1435
- B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 865–889 (English, with English and French summaries). MR 1779897
- John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449
- P. Dienes, The Taylor series: an introduction to the theory of functions of a complex variable, Dover Publications, Inc., New York, 1957. MR 0089895
- R. J. Duffin and A. C. Schaeffer, Power series with bounded coefficients, Amer. J. Math. 67 (1945), 141–154. MR 11322, DOI https://doi.org/10.2307/2371922
- Tamás Erdélyi, On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), no. 1, 97–111. MR 1827077, DOI https://doi.org/10.1307/mmj/1008719037
- P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105–119. MR 33372, DOI https://doi.org/10.2307/1969500
- Sergei Konyagin, On a question of Pichorides, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 385–388 (English, with English and French summaries). MR 1440953, DOI https://doi.org/10.1016/S0764-4442%2897%2980072-9
- John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0244463
- G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187
- A. M. Odlyzko and B. Poonen, Zeros of polynomials with $0,1$ coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317–348. MR 1252071
- Yuval Peres and Boris Solomyak, Approximation by polynomials with coefficients $\pm 1$, J. Number Theory 84 (2000), no. 2, 185–198. MR 1795789, DOI https://doi.org/10.1006/jnth.2000.2514
- Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
- H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
- G. Szegö, Tschebyscheffsche Polynome und nichtfortsetzbare Potenzreihen, Math. Ann. 87 (1922), no. 1-2, 90–111 (German). MR 1512103, DOI https://doi.org/10.1007/BF01458039
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Additional Information
Peter Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email:
pborwein@cecm.sfu.ca
Tamás Erdélyi
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email:
terdelyi@math.tamu.edu
Friedrich Littmann
Affiliation:
Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105
Email:
Friedrich.Littmann@ndsu.edu
Keywords:
Zeros,
rational functions,
Duffin–Schaeffer Theorem,
Littlewood polynomials
Received by editor(s):
June 8, 2005
Received by editor(s) in revised form:
February 15, 2006
Published electronically:
May 27, 2008
Article copyright:
© Copyright 2008
by the authors