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ON THE EQUATION f(g(x))=f(x)hm(x) FOR COMPOSITE POLYNOMIALS

Part of: Diophantine equations Algebraic number theory: global fields Real and complex fields Multiplicative number theory General field theory Sequences and sets

Published online by Cambridge University Press:  19 September 2012

HIMADRI GANGULI  and
JONAS JANKAUSKAS
HIMADRI GANGULI
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive,Burnaby, British Columbia, Canada V5A 1S6 (email: hganguli@sfu.ca)
JONAS JANKAUSKAS*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania (email: jonas.jankauskas@gmail.com)
*
✉For correspondence; e-mail: jonas.jankauskas@gmail.com
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Abstract

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In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0 in K[x] and the integer m≥2 is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2 and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x) with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

Keywords

Chebyshev polynomialcomposite polynomialsPell equationmultiplicative dependence

MSC classification

Secondary: 11B83: Special sequences and polynomials 11D57: Multiplicative and norm form equations 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11R09: Polynomials (irreducibility, etc.) 12D05: Polynomials 12E10: Special polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 155 - 161
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

A visit of the second author to IRMACS Centre, Simon Fraser University was funded by the Lithuanian Research Council (Student research support project).

References

[1]Barbeau, E. J., Pell’s Equation, Problem Books in Mathematics (Springer, New York, 2003).CrossRefGoogle Scholar
[2]Borwein, P., Choi, S. and Ganguli, H., ‘Sign changes of the Liouville function on quadratics’, Canad. Math. Bull., to appear.Google Scholar
[3]Cassaigne, J., Ferenczi, S., Mauduit, C., Rivat, J. and Sárközy, A., ‘On finite pseudorandom binary sequences, IV. The Liouville function. II’, Acta Arith. 95(4) (2000), 343–359.CrossRefGoogle Scholar
[4]Chowla, S., The Riemann Hypothesis and Hilbert’s Tenth Problem, Mathematics and Its Applications, 4 (Gordon and Breach Science Publishers, New York, 1965).Google Scholar
[5]Drungilas, P. and Dubickas, A., ‘Multiplicative dependence of shifted algebraic numbers’, Colloq. Math. 96(1) (2003), 75–81.CrossRefGoogle Scholar
[6]Dubickas, A., ‘Multiplicative dependence of quadratic polynomials’, Liet. Mat. Rink. 38(3) (1998), 295–303.Google Scholar
[7]Dubickas, A. and Steuding, J., ‘The polynomial Pell equation’, Elem. Math. 59(4) (2004), 133–143.CrossRefGoogle Scholar
[8]Lidl, R. and Niederreiter, H., Finite Fields, 2nd edn, Encyclopedia of Mathematics and its Applications, 20 (Cambridge University Press, Cambridge, 1997).Google Scholar
[9]McLaughlin, J., ‘Polynomial solutions of Pell’s equation and fundamental units in real quadratic fields’, J. London Math. Soc. (2) 67(1) (2003), 16–28.CrossRefGoogle Scholar
[10]Nathanson, M. B., ‘Polynomial Pell’s equations’, Proc. Amer. Math. Soc. 56 (1976), 89–92.Google Scholar
[11]Pastor, A. V., ‘Generalized Chebyshev polynomials and the Pell–Abel equation’, Fundam. Prikl. Mat. 7(4) (2001), 1123–1145.Google Scholar
[12]Rivlin, T. J., Chebyshev Polynomials, 2nd edn, Pure and Applied Mathematics (New York) (Wiley, New York, 1990).Google Scholar
[13]Schinzel, A., Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, 77 (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar
[14]Webb, W. A. and Yokota, H., ‘Polynomial Pell’s equation’, Proc. Amer. Math. Soc. 131(4) (2003), 993–1006 (electronic).CrossRefGoogle Scholar