Constructing Belyĭ Maps from Valencies

Recently Kevin Pilgrim asked how one could construct a Belyĭ Map from a given set of valencies. I’ll cover this idea in more detail in my course, but I thought I’d put out a general answer now for those who are interested.

Definitions

Say that \Gamma = \bigl( B \cup W, \, E \bigr) is a loopless, connected, planar bipartite graph which we wish to express as the Dessin d’Enfant of some Belyĭ map \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C). We explain how to construct such a Belyĭ map. We are motivated by the discussions found on Mathematica Stack Exchange at the web pages here and here. My 2013 REU worked on implementing these ideas over the summer as part of a Mathematica notebook.

Express the “black” vertices as B = \bigl \{ P_\alpha \, \bigl| \, \alpha \in I \bigr \}, the “white” vertices as W = \bigl \{ Q_\beta \, \bigl| \, \beta \in J \bigr \}, and the midpoints of the faces as F = \bigl \{ R_\chi \, \bigl| \, \chi \in K \bigr \}. Since B, \, W, \, F \subseteq \mathbb P^1(\mathbb C), we will express P_\alpha = \bigl( a_\alpha \, : \, b_\alpha \bigr), Q_\beta = \bigl( c_\beta \, : \, d_\beta \bigr) and R_\chi = \bigl( e_\chi \, : \, f_\chi \bigr). Say that each “black” vertex P_\alpha has m_\alpha edges incident, each “white” vertex Q_\beta has n_\beta edges incident i.e. “black” vertices adjacent, and each face R_\chi has k_\chi “white” vertices adjacent. The sets \bigl \{ m_\alpha \, \bigl| \, \alpha \in I \bigr \}, \bigl \{ n_\beta \, \bigl| \, \beta \in J \bigr \}, and \bigl \{ k_\chi \, \bigl| \, \chi \in K \bigr \} are the valencies of the vertices, edges, and faces, respectively. (By the way, a “clean” Dessin d’Enfant would correspond to n_\beta = 2.)

Fundamental Equation

Since we would like to write B = f^{-1}(0), W = f^{-1}(1), and F = f^{-1}(\infty) for some rational function f(z) = p(z) / q(z) of degree \deg(f) = |I| + |J| + |K| - 2 = \displaystyle \sum_{\alpha \in I} m_\alpha = \sum_{\beta \in J} n_\beta = \sum_{\chi \in K} k_\chi, we must have the factorizations


\begin{aligned} p(z) & = + p_0 \prod_{\alpha \in I} \biggl[ b_\alpha \, z - a_\alpha \biggr]^{m_\alpha} \\[5pt] p(z) - q(z) = & = -q_0 \prod_{\beta \in J} \biggl[ d_\beta \, z - c_\beta \biggr]^{n_\beta} \\[5pt] q(z) & = -r_0 \prod_{\chi \in K} \biggl[ f_\chi \, z - e_\chi \biggr]^{k_\chi} \\[5pt] \end{aligned} \quad \implies \quad \beta(z) = \dfrac {p(z)}{q(z)} = - \dfrac {p_0}{r_0} \ \dfrac {\displaystyle \prod_{\alpha \in I} \biggl[ b_\alpha \, z - a_\alpha \biggr]^{m_\alpha}}{\displaystyle \prod_{\chi \in K} \biggl[ f_\chi \, z - e_\chi \biggr]^{k_\chi}}

for some nonzero constants p_0, q_0, and r_0. Hence, given the sets \bigl \{ m_\alpha \, \bigl| \, \alpha \in I \bigr \}, \bigl \{ n_\beta \, \bigl| \, \beta \in J \bigr \}, and \bigl \{ k_\chi \, \bigl| \, \chi \in K \bigr \} we wish to find a_\alpha, \, b_\alpha, \, c_\beta, \, d_\beta, \, e_\chi, \, f_\chi, \, p_0, \, q_0, \, r_0 \in \mathbb C such that


p_0 \prod_{\alpha \in I} \biggl[ b_\alpha \, z - a_\alpha \biggr]^{m_\alpha} + q_0 \prod_{\beta \in J} \biggl[ d_\beta \, z - c_\beta \biggr]^{n_\beta} + r_0 \prod_{\chi \in K} \biggl[ f_\chi \, z - e_\chi \biggr]^{k_\chi} = 0

identically as a polynomial in z. If you can find coefficients which make this polynomial identity true for all z, then you have the desired Belyĭ map f(z) = p(z) / q(z).

Remarks

Since there are \deg(f) + 1 coefficients in this polynomial equation, we find a system of |I| + |J| + |K| - 1 homogeneous equations in |I| + |J| + |K| + 3 unknowns — so there will be infinitely many f(z) to choose from. In practice, it is useful to use a Möbius transformation in the form


\gamma(z) = \dfrac {b_0 \, e_0 - a_0 \, f_0}{d_0 \, e_0 - c_0 \, f_0} \cdot \dfrac {d_0 \, z - c_0}{b_0 \, z - a_0} \qquad \implies \qquad \begin{aligned} \gamma(P_0) & = \infty \\[5pt] \gamma(Q_0) & = 0 \\[5pt] \gamma(R_0) & = 1 \end{aligned}

The corresponding rational map \beta = f \circ \gamma^{-1} satisfies \beta(0) = 1, \beta(1) = \infty, and \beta(\infty) = 0; while the corresponding system has |I| + |J| + |K| - 1 homogeneous equations in |I| + |J| + |K| unknowns. We expect to find only finitely many \beta(z) to choose from.

As I mentioned before, my 2013 REU worked on implementing these ideas over the summer as part of a Mathematica notebook. We had trouble getting Mathematica to solve this polynomial equation when the sizes of the valencies got too large, such as 6 or 7. I’m working on getting the code to run much faster so that I can post something online soon.

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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