ИСТИНА

Classically Grothendieck dessin d'enfant is an embedded graph $\Gamma$ on a smooth compact oriented surface $M$ such that the complement $M\setminus \Gamma$ is homeomorphic to a disjoint union of open discs. Each Grothendieck dessin d'enfant is in a natural correspondence to a unique (up to a linear-fractional transformation) Belyi pair. Belyi pair is an algebraic curve together with a non-constant meromorphic function on this curve with at most 3 critical values. This correspondence is actual for various applications. We investigate the systems of equations relating concrete dessins with concrete Belyi pairs and some special solutions of these equations, called parasitic solutions. In the talk we provide the introduction to the theory of dessins d'enfants and formulate some new results concerning the aforesaid systems of equations. Series of examples will be provided. In particular, so-called anti-Vandermonde systems will be considered and investigated. The talk is based on the results of our joint works with N. Ya. Amburg and G.B. Shabat.