The integer Chebyshev problem
HTML articles powered by AMS MathViewer
- by Peter Borwein and Tamás Erdélyi PDF
- Math. Comp. 65 (1996), 661-681 Request permission
Abstract:
We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval $[0,1]$ in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal “integer Chebyshev” polynomials, showing for example that on small intevals $[0, \delta ]$ and for small degrees $d$, $x^{d}$ achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the “integer transfinite diameter” of $[0,1]$ should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.References
- Francesco Amoroso, Sur le diamètre transfini entier d’un intervalle réel, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 885–911 (1991) (French, with English summary). MR 1096596, DOI 10.5802/aif.1240
- Emiliano Aparicio Bernardo, Methods for the approximate calculation of the minimum uniform Diophantine deviation from zero on a segment, Rev. Mat. Hisp.-Amer. (4) 38 (1978), no. 6, 259–270 (Spanish). MR 531469
- —, New bounds on the minimal Diophantine deviation from zero on $[0,1]$ and $[0,1/4]$, Actus Sextas Jour. Mat. Hisp.-Lusitanas (1979), 289–291.
- —, On some systems of algebraic integers of D.S. Gorshkov and their application in calculus, Rev. Mat. Hisp.-Amer. 41 (1981), 3–17 (Spanish).
- È. Aparisio, Some results in the problem of Diophantine approximations of functions by polynomials, Trudy Mat. Inst. Steklov. 163 (1984), 6–9 (Russian). International conference on analytical methods in number theory and analysis (Moscow, 1981). MR 769861
- P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities, Springer-Verlag, New York, 1995.
- P. Borwein and C. Ingalls, The Prouhet Tarry Escott problem revisited, Math. Enseign. 40 (1994), 3–27.
- 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
- M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit. 17 (1923), 228–249.
- Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, Mathematical Surveys, No. 17, American Mathematical Society, Providence, R.I., 1980. MR 560902, DOI 10.1090/surv/017
- V. Flammang, Sur la longeur des entiers algébriques totalement positifs, preprint.
- D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155–159.
- B. Kashin, On algebraic polynomials with integer coefficients with least deviation from zero on an interval, preprint.
- Tateaki Sasaki, Tomokatsu Saito, and Teruhiko Hilano, Analysis of approximate factorization algorithm. I, Japan J. Indust. Appl. Math. 9 (1992), no. 3, 351–368. MR 1189944, DOI 10.1007/BF03167271
- H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, Vol. 84, Amer. Math. Soc., Providence, R. I., 1994.
- I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganz-zahligen Koeffizienten, Math. Zeit. 1 (1918), 377–402.
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. I: Series, integral calculus, theory of functions, Die Grundlehren der mathematischen Wissenschaften, Band 193, Springer-Verlag, New York-Berlin, 1972. Translated from the German by D. Aeppli. MR 0344042
- Edward B. Saff and Richard S. Varga, On lacunary incomplete polynomials, Math. Z. 177 (1981), no. 3, 297–314. MR 618197, DOI 10.1007/BF01162064
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- C. J. Smyth, The mean values of totally real algebraic integers, Math. Comp. 42 (1984), no. 166, 663–681. MR 736460, DOI 10.1090/S0025-5718-1984-0736460-5
- Christopher Smyth, Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 1–28 (English, with French summary). MR 762691, DOI 10.5802/aif.975
Additional Information
- Peter Borwein
- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., 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
- Received by editor(s): April 25, 1994
- Received by editor(s) in revised form: September 5, 1994, and February 12, 1995
- Additional Notes: Research supported in part by NSERC of Canada.
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 661-681
- MSC (1991): Primary 11J54, 11B83
- DOI: https://doi.org/10.1090/S0025-5718-96-00702-8
- MathSciNet review: 1333305