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The Equation f(X) = f(Y) in Rational Functions X = X(t), Y = Y(t)

Published online by Cambridge University Press:  04 December 2007

Roberto M. Avanzi  and
Umberto M. Zannier
Roberto M. Avanzi
Affiliation:
Institut für Experimentelle Mathematik, Universität Essen, Ellernstraße 29, 45326 Essen, Germany. e-mail: mocenigo@exp-math.uni-essen.de
Umberto M. Zannier
Affiliation:
Istituto Universitario di Architettura DCA, Santa Croce 191, 30135 Venice, Italy. e-mail: zannier@dimi.uniud.it
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Abstract

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We determine all the complex polynomials f(X) such that, for two suitable distinct, nonconstant rational functions g(t) and h(t), the equality f(g(t)) = f(h(t)) holds. This extends former results of Tverberg, and is a contribution to the more general question of determining the polynomials f(X) over a number field K such that f(X) − λ has at least two distinct K-rational roots for infinitely many λ ∈ K.

Keywords

diophantine equationsequations admitting rational functions as solutionsreducibility
Type
Research Article
Information
Compositio Mathematica , Volume 139 , Issue 3 , December 2003 , pp. 263 - 295
Copyright
© 2003 Kluwer Academic Publishers