
This article is cited in 4 scientific papers (total in 4 papers)
Weighted trees with primitive edge rotation groups
N. M. Adrianov^{a}, A. K. Zvonkin^{b} ^{a} Lomonosov Moscow State University, Moscow, Russia
^{b} University of Bordeaux, Bordeaux, France
Abstract:
Let $R,S\in\mathbb C[x]$ be two coprime polynomials of the same degree with prescribed multiplicities of their roots. A classical problem of number theory actively studied during the last halfcentury is, what could be the minimum degree of the difference $T=RS$. The theory of dessins d'enfants implies that such a minimum is attained if and only if the rational function $f=R/T$ is a Belyi function for a bicolored plane map all of whose faces except the outer one are of degree $1$. Such maps are called weighted trees, since they can be conveniently represented by plane trees whose edges are endowed with positive integral weights.
It is well known that the absolute Galois group (the automorphism group of the field $\bar{\mathbb Q}$ of algebraic numbers) acts on dessins. An important invariant of this action is the edge rotation group, which is also the monodromy group of a ramified covering corresponding to the Belyi function. In this paper, we classify all weighted trees with primitive edge rotation groups. There are, up to the color exchange, $184$ such trees, which are subdivided into (at least) $85$ Galois orbits and generate $34$ primitive groups (the highest degree is $32$). This result may also be considered as a contribution to the classification of covering of genus $0$ with primitive monodromy groups in the framework of the Guralnick–Thompson conjecture.
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Journal of Mathematical Sciences (New York), 2015, 209:2, 160–191
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512.542.7
Citation:
N. M. Adrianov, A. K. Zvonkin, “Weighted trees with primitive edge rotation groups”, Fundam. Prikl. Mat., 18:6 (2013), 5–50; J. Math. Sci., 209:2 (2015), 160–191
Citation in format AMSBIB
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\paper Weighted trees with primitive edge rotation groups
\jour Fundam. Prikl. Mat.
\yr 2013
\vol 18
\issue 6
\pages 550
\mathnet{http://mi.mathnet.ru/fpm1551}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3431854}
\transl
\jour J. Math. Sci.
\yr 2015
\vol 209
\issue 2
\pages 160191
\crossref{https://doi.org/10.1007/s1095801524942}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.084943362003}
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This publication is cited in the following articles:

A. K. Zvonkin, “Enumeration of weighted plane trees”, J. Math. Sci., 209:2 (2015), 258–264

J. Math. Sci. (N. Y.), 226:5 (2017), 548–560

J. Math. Sci. (N. Y.), 226:5 (2017), 623–634

J. Math. Sci. (N. Y.), 226:5 (2017), 667–693

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