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The Resultant of Chebyshev Polynomials

Published online by Cambridge University Press:  20 November 2018

David P. Jacobs ,
Mohamed O. Rayes  and
Vilmar Trevisan
David P. Jacobs
Affiliation:
School of Computing, Clemson University, Clemson, SC 29634–0974, U.S.A.e-mail: [email protected]
Mohamed O. Rayes
Affiliation:
Dept. of Comp. Sci. and Eng., Southern Methodist University, U.S.A.e-mail: [email protected]
Vilmar Trevisan
Affiliation:
Instituto de Matemática, UFRGS, Porto Alegre, Brazile-mail: [email protected]
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Abstract

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Let ${{T}_{n}}$ denote the $n$-th Chebyshev polynomial of the first kind, and let ${{U}_{n}}$ denote the $n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res$({{T}_{m}},\,{{T}_{n}})$. Similarly, we give a formula for res$({{U}_{m}},\,{{U}_{n}})$.

Keywords

11Y1168W20resultantChebyshev polynomial
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 288 - 296
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Apostol, T. M., Resultants of cyclotomic polynomials. Proc. of the Amer. Math. Soc. 24(1970), 457–462.Google Scholar
[2] Cohn, H. and Kumar, A., Universally optimal distribution of points on spheres. J. Amer. Math. 20(2007), no. 1, 99–148. doi:10.1090/S0894-0347-06-00546-7Google Scholar
[3] Dilcher, K. and Stolarsky, K. B., Resultants and discriminants of Chebyshev and related polynomials. Trans. Amer. Math. Soc. 357(2005), no. 3, 965–981. doi:10.1090/S0002-9947-04-03687-6Google Scholar
[4] Gel’fand, I. M., Kapranov, M. M., and Zelevinksky, A. V., Discriminants, resultants and multidimensional determinants. Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994.Google Scholar
[5] Gishe, J. and Ismail, M. E. H., Resultants of Chebyshev polynomials. Z. Anal. Anwend. 27(2008), no. 4, 491–500. doi:10.4171/ZAA/1368Google Scholar
[6] Rayes, M. O., Trevisan, V., and Wang, P. S., Factorization properties of Chebyshev polynomials. Comput. Math. Appl. 50(2005), no. 8–9, 1231–1240. doi:10.1016/j.camwa.2005.07.003Google Scholar
[7] Roberts, D. P., Discriminants of some Painlevé polynomials. In: Number theory for the millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 205–221.Google Scholar
[8] Rivlin, T. J., The Chebyshev polynomials. From approximation theory to algebra and number theory. Second ed., Pure and Applied Mathematics, John Wiley & Sons, New York, 1990.Google Scholar
[9] van der Waerden, B. L., Modern algebra. v. 1, Ungar, New York, 1950.Google Scholar