Resultants and discriminants of Chebyshev and related polynomials
HTML articles powered by AMS MathViewer
- by Karl Dilcher and Kenneth B. Stolarsky PDF
- Trans. Amer. Math. Soc. 357 (2005), 965-981 Request permission
Abstract:
We show that the resultants with respect to $x$ of certain linear forms in Chebyshev polynomials with argument $x$ are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 20402, 1966. Fifth printing, with corrections; National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents). MR 0208798
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- Tom M. Apostol, Resultants of cyclotomic polynomials, Proc. Amer. Math. Soc. 24 (1970), 457–462. MR 251010, DOI 10.1090/S0002-9939-1970-0251010-X
- Tom M. Apostol, The resultant of the cyclotomic polynomials $F_{m}(ax)$ and $F_{n}(bx)$, Math. Comp. 29 (1975), 1–6. MR 366801, DOI 10.1090/S0025-5718-1975-0366801-7
- O. I. Cygankova, Formulae for calculating the discriminants of Jacobi, Laguerre and Hermite polynomials, Izv. Vysš. Učebn. Zaved. Matematika 1962 (1962), no. 4 (29), 170–172 (Russian). MR 0138805
- Gary R. Greenfield and Daniel Drucker, On the discriminant of a trinomial, Linear Algebra Appl. 62 (1984), 105–112. MR 761061, DOI 10.1016/0024-3795(84)90089-2
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- Mourad E. H. Ismail, Discriminants and functions of the second kind of orthogonal polynomials, Results Math. 34 (1998), no. 1-2, 132–149. Dedicated to Paul Leo Butzer. MR 1635590, DOI 10.1007/BF03322044
- Scott McCallum, Factors of iterated resultants and discriminants, J. Symbolic Comput. 27 (1999), no. 4, 367–385. MR 1681345, DOI 10.1006/jsco.1998.0257
- James H. McKay and Stuart Sui Sheng Wang, A chain rule for the resultant of two polynomials, Arch. Math. (Basel) 53 (1989), no. 4, 347–351. MR 1015998, DOI 10.1007/BF01195214
- James H. McKay and Stuart Sui Sheng Wang, A chain rule for the resultant of two homogeneous polynomials, Arch. Math. (Basel) 56 (1991), no. 4, 352–361. MR 1094422, DOI 10.1007/BF01198221
- M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. MR 1033013, DOI 10.1017/CBO9780511661952
- G. Rado, Die Diskriminante der allgemeinen Kreisteilungsgleichung, J. Reine Angew. Math. 131 (1906), 49–55.
- 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
- George Shabat and Alexander Zvonkin, Plane trees and algebraic numbers, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 233–275. MR 1310587, DOI 10.1090/conm/178/01909
- Richard G. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12 (1962), 1099–1106. MR 144891, DOI 10.2140/pjm.1962.12.1099
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing Co., New York, N. Y., 1949. Translated from the second revised German edition by Fred Blum; With revisions and additions by the author. MR 0029363
Additional Information
- Karl Dilcher
- Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
- Email: dilcher@mathstat.dal.ca
- Kenneth B. Stolarsky
- Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
- Email: stolarsk@math.uiuc.edu
- Received by editor(s): November 1, 2002
- Published electronically: October 19, 2004
- Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council of Canada
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 965-981
- MSC (2000): Primary 12E10, 12E05; Secondary 13P05, 33C45
- DOI: https://doi.org/10.1090/S0002-9947-04-03687-6
- MathSciNet review: 2110427