On integer Chebyshev polynomials
HTML articles powered by AMS MathViewer
- by Laurent Habsieger and Bruno Salvy PDF
- Math. Comp. 66 (1997), 763-770 Request permission
Abstract:
We are concerned with the problem of minimizing the supremum norm on $\lbrack 0,1\rbrack$ of a nonzero polynomial of degree at most $n$ with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to $75$ and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and improve a lower bound due to Flammang et al.References
- Peter Borwein and Tamás Erdélyi, The integer Chebyshev problem, Math. Comp. 65 (1996), no. 214, 661–681. MR 1333305, DOI 10.1090/S0025-5718-96-00702-8
- Emiliano Aparicio Bernardo, On the approximation of functions by polynomials with integer coefficients, Proc. Eighth Annual Reunion of Spanish Mathematicians (Santiago, 1967) Publ. Inst. “Jorge Juan” Mat., Madrid, 1969, pp. 21–33 (Spanish). MR 0259428
- Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal Diophantic deviation from zero, J. Approx. Theory 55 (1988), no. 3, 270–278. MR 968933, DOI 10.1016/0021-9045(88)90093-7
- Valérie Flammang, Sur le diamètre transfini entier d’un intervalle à extrémités rationnelles, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 779–793 (French, with English and French summaries). MR 1340952, DOI 10.5802/aif.1473
- V. Flammang, G. Rhin, and C. J. Smyth, The integer transfinite diameter of intervals and totally real algebraic integers, Tech. Report MS-95-033, Department of Mathematics and Statistics, University of Edinburgh, 1995.
- Naoyoshi Kanamaru, Takao Nishizeki, and Tetsuo Asano, Efficient enumeration of grid points in a convex polygon and its application to integer programming, Internat. J. Comput. Geom. Appl. 4 (1994), no. 1, 69–85. MR 1269706, DOI 10.1142/S0218195994000069
- 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, DOI 10.1090/cbms/084
- M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279, DOI 10.2307/2320934
Additional Information
- Laurent Habsieger
- Affiliation: Laboratoire d’Algorithmique Arithmétique, CNRS UMR 9936, Université Bordeaux 1, 351 cours de la Libération, F-33405 Talence Cedex, France
- Email: habsiege@math.u-bordeaux.fr
- Bruno Salvy
- Affiliation: INRIA Rocquencourt, Domaine de Voluceau, B.P. 105, F-78153 Le Chesnay Cedex, France
- Email: Bruno.Salvy@inria.fr
- Received by editor(s): October 18, 1995
- Received by editor(s) in revised form: May 3, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 763-770
- MSC (1991): Primary 11J54, 11-04, 41A10, 41-04
- DOI: https://doi.org/10.1090/S0025-5718-97-00829-6
- MathSciNet review: 1401941