This document is divided in four chapters. The first chapter describes two methods for the resolution of systems of algebraic equations. They make use of elementary algebra. We begin with the definitions, stressing the problem of approximating algebraic numbers. The first method is the now classical Gröbner basis calculus. The other method computes an approximation of the solution and goes back to the (exact) algebraic properties with the help of LLL reduction.
The second chapter gives an introduction to the dessins d'enfants. We recall quickly some basic definitions of group theory, geometry or graph theory. Then we give formal definitions of the dessins, stressing the ``Grothendieck's correspondance'' between combinatoric and algebraic properties. We list the usual variants and generalisations of the dessins d'enfants.
The third chapter explains how we can build an algebraic system from the visual description of a dessin. Its solutions describe the algebraic properties of the dessin. This system can be solved with Gröbner basis, but we prefer to compute successive approximations of dessins. With that technique, we can take advantage of the geometric nature of the dessins.
The fourth chapter begins with the most simple of all dessins, as toy examples. Then we give a conjecture and computations of the number of conjugates of some specific dessin called ``Y-trees''. These computations show that the numeric approach for solving the systems is really competitive. We continue with some nice examples, we conclude with computations of dessins that represent regular extensions of Galois group some Mathieu group.