Monic integer Chebyshev problem

Authors:
P. B. Borwein, C. G. Pinner and I. E. Pritsker

Journal:
Math. Comp. **72** (2003), 1901-1916

MSC (2000):
Primary 11C08; Secondary 30C10

DOI:
https://doi.org/10.1090/S0025-5718-03-01477-7

Published electronically:
January 8, 2003

MathSciNet review:
1986811

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A *monic integer Chebyshev polynomial* satisfies

and the

*monic integer Chebyshev constant*is then defined by

This is the obvious analogue of the more usual

*integer Chebyshev constant*that has been much studied.

We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

**Conjecture.** *Suppose * * is an interval whose endpoints are consecutive Farey fractions. This is characterized by * * Then*

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

**1.**Peter Borwein and Tamás Erdélyi,*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR**1367960****2.**Peter Borwein and Tamás Erdélyi,*The integer Chebyshev problem*, Math. Comp.**65**(1996), no. 214, 661–681. MR**1333305**, https://doi.org/10.1090/S0025-5718-96-00702-8**3.**G. V. Chudnovsky,*Number theoretic applications of polynomials with rational coefficients defined by extremality conditions*, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 61–105. MR**717590****4.**M. Fekete,*Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten*, Math. Zeit.**17**(1923), 228-249.**5.**M. Fekete,*Über den tranfiniten Durchmesser ebener Punktmengen II*, Math. Zeit.**32**(1930), 215-221.**6.**Le Baron O. Ferguson,*Approximation by polynomials with integral coefficients*, Mathematical Surveys, vol. 17, American Mathematical Society, Providence, R.I., 1980. MR**560902****7.**V. Flammang, G. Rhin, and C. J. Smyth,*The integer transfinite diameter of intervals and totally real algebraic integers*, J. Théor. Nombres Bordeaux**9**(1997), no. 1, 137–168 (English, with English and French summaries). MR**1469665****8.**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****9.**Laurent Habsieger and Bruno Salvy,*On integer Chebyshev polynomials*, Math. Comp.**66**(1997), no. 218, 763–770. MR**1401941**, https://doi.org/10.1090/S0025-5718-97-00829-6**10.**D. Hilbert,*Ein Beitrag zur Theorie des Legendreschen Polynoms*, Acta Math.**18**(1894), 155-159.**11.**Hugh L. Montgomery,*Ten lectures on the interface between analytic number theory and harmonic analysis*, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR**1297543****12.**Y. Okada,*On approximate polynomials with integral coefficients only*, Tohoku Math. J.**23**(1924), 26-35.**13.**I. E. Pritsker,*Small polynomials with integer coefficients*, in press.**14.**Thomas Ransford,*Potential theory in the complex plane*, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR**1334766****15.**Theodore J. Rivlin,*Chebyshev polynomials*, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR**1060735****16.**R. M. Trigub,*Approximation of functions with Diophantine conditions by polynomials with integral coefficients*, Metric questions of the theory of functions and mappings, No. 2 (Russian), Izdat. “Naukova Dumka”, Kiev, 1971, pp. 267–333 (Russian). MR**0312121****17.**M. Tsuji,*Potential theory in modern function theory*, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR**0414898**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11C08,
30C10

Retrieve articles in all journals with MSC (2000): 11C08, 30C10

Additional Information

**P. B. Borwein**

Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada

Email:
pborwein@cecm.sfu.ca

**C. G. Pinner**

Affiliation:
Department of Mathematics, 138 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

Email:
pinner@math.ksu.edu

**I. E. Pritsker**

Affiliation:
Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
igor@math.okstate.edu

DOI:
https://doi.org/10.1090/S0025-5718-03-01477-7

Keywords:
Chebyshev polynomials,
integer Chebyshev constant,
integer transfinite diameter.

Received by editor(s):
August 29, 2001

Received by editor(s) in revised form:
December 20, 2001

Published electronically:
January 8, 2003

Additional Notes:
Research of the authors was supported in part by the following grants: NSERC of Canada and MITACS (Borwein), NSF grant EPS-9874732 and matching support from the state of Kansas (Pinner), and NSF grant DMS-9996410 (Pritsker).

Article copyright:
© Copyright 2003
by the authors