Numerical Approximation of Coefficients of Bely̆ı Maps

In 1984, Alexander Grothendieck, inspired by a result of Gennadii Bely̆ı from 1979, constructed a finite, connected planar bipartite graph via rational functions β : P(C)→ P(C) with critical values {0, 1,∞} by looking at the inverse image of the triangle formed by these three points. He called such graphs Dessins d′Enfants. Conversely, Riemann’s Existence Theorem implies that every finite, connected planar graph arises in this way. The difficulty arises in explicitly constructing such a Bely̆ı map β from any given planar graph. We may form a valency list by considering the number of edges surrounding each vertex and each face; this forces algebraic conditions on the coefficients of the desired Bely̆ı map. Hence the construction of a Bely̆ı map can be reduced to the computation of roots of a system of nonlinear equations. In this paper, we Edray H. Goins Purdue University West Lafayette, Indiana egoins@purdue.edu Luis Melara, Corresponding author Shippensburg University Shippensburg, Pennsylvania lamelara@ship.edu Alejandra Alvarado, Corresponding author Eastern Illinois University Charleston, Illinois aalvarado2@eiu.edu reformulate the problem of finding these roots into an unconstrained optimization problem. We implement Newton’s method and a limited memory BFGS method. Convergence results for both methods are presented; including a second convergence result using the Kantorovich Theorem for Newton’s method. Numerical results are discussed and shown.