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Dessins d’enfants on the Riemann sphere. (English) Zbl 0823.14017
Schneps, Leila (ed.), The Grothendieck theory of dessins d’enfants. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 200, 47-77 (1994).
The paper under review is divided into four parts. Part I is devoted to an almost self-contained explanation of the bijection between the set of (abstract) clean dessins and the set of isomorphic classes of clean Belyi pairs (a clean Belyi pair $$(X, \beta)$$ consists of an algebraic curve $$X$$ defined over an algebraic closure $$\overline {\mathbb{Q}}$$ of $$\mathbb{Q}$$ and a holomorphic map $$\beta: X\to \mathbb{P}^ 1 (\mathbb{C})$$ whose critical values lie in $$\{0,1, \infty\}$$ such that all ramification orders over 1 are equal to 2). In part II, the action of the Galois group $$G= \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})$$ on dessins of genus 1, and so on the profinite completion $$\widehat {\pi}_ 1$$ of the fundamental group of $$\mathbb{P}^ 1 (\mathbb{C})- \{0,1, \infty\}$$ is studied. The main result of the section, attributed by the author to H. W. Lenstra jun. says that $$G$$ acts also faithfully on trees, and so on genus 0 dessins. For the proof the author proves a couple of lemmas on the “uniqueness” of decomposition of univariate polynomials $$F= G\circ H$$ if $$\deg (H)$$ is fixed which were obtained before by J. Gutierrez and C. Ruiz de Velasco [in: Algebra and geometry, Proc. 2nd Span. Belg. Week, IISBWAG, Santiago de Compostela, Alxebra 54, 79-90 (1990; Zbl 0704.12003)].
In part III, the bijection of part I is made explicit for genus 0 dessins. Essentially, for a dessin $$D$$ the method yields the set $$O(D)$$ of all dessins in the orbit of $$D$$ under the action of $$G$$, the number field $$K_ D$$, associated to each dessin $$D'$$ in $$O(D)$$, a set of $$G$$- conjugate Belyi functions corresponding to the dessins in $$O(D)$$ and the action of $$G$$ on $$O(D)$$. The procedure involves the computation of all solutions of a system of polynomial equations whose set of solutions is finite. To this purpose, Gröbner bases are used. – Finally, in part IV, numerical examples of this procedure are displaced.
The paper is very clearly written and provides a nonexpert people in the field, as the reviewer, an easy access to be subject.
For the entire collection see [Zbl 0798.00001].

##### MSC:
 14H57 Dessins d’enfants theory 11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory 30F99 Riemann surfaces 05E20 Group actions on designs, etc. (MSC2000) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry