69 research outputs found
On semiconjugate rational functions
We investigate semiconjugate rational functions, that is rational functions
related by the functional equation , where is a
rational function of degree at least two. We show that if and is a pair
of such functions, then either can be obtained from by a certain
iterative process, or and can be described in terms of orbifolds of
non-negative Euler characteristic on the Riemann sphere.Comment: Final version, accepted by Geom. Funct. Ana
Prime and composite Laurent polynomials
In 1922 Ritt constructed the theory of functional decompositions of
polynomials with complex coefficients. In particular, he described explicitly
indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)).
In this paper we study the equation above in the case when f,g,p,q are
holomorphic functions on compact Riemann surfaces. We also construct a
self-contained theory of functional decompositions of rational functions with
at most two poles generalizing the Ritt theory. In particular, we give new
proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.Comment: Some of the proofs given in sections 6-8 are simplified. Some other
small alterations were mad
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