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

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

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).

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