A few weeks ago, Dong Quan Ngoc Nguyen (University of Notre Dame) came to visit here at Purdue. We spoke a little about the computer package Bertini (created by Daniel Bates, Jonathan Hauenstein, Andrew Sommese and Charles Wampler) and whether the homotopy continuation method can be used to compute Belyi maps and Shabat Polynomials. I’ve been working on setting up a system of polynomial equations whose solutions give the coefficients of the Belyi maps, so it really comes down to actually finding the solutions to these equations. The hope is that a polynomial homotopy continuation method will be much more efficient than say, using Groebner bases, to find all solutions!

Let me try and set up how this would work by working through some explicit examples.

## Belyi Maps and Dessins d’Enfants

Say that we have a connected planar bipartite graph

Problem.Given a connected planar bipartite graph, find a rational function which satisfies the following:

- Its only critical values are
. are the “black” vertices. are the “white” vertices. are the “midpoints” of the faces. as the edges is the inverse image of the interval from to .

We say

Computationally, I find working with connected planar bipartite graphs very difficult, so instead let me work with its valency list. Denote

and denote the positive integer

for

each as elements of

Proposition 1.Continue notation as above.

- The Euler Characteristic and the Degree Sum Formula together imply
. - The following algebraic variety has dimension 0:
In particular, Bezout’s theorem implies . - Every point
in gives rise to a Belyi map such that is its Dessin d’Enfant, and we have the explicit points Rather explicitly,

The basic idea behind this result is that we have the identity

## Finding All Points using Bertini

Given a degree sequence

Algorithm. Denoteas that extension of which is the field generated by the coordinates of the points .

- Using both the Homotopy Continuation and Newton’s Method, numerically find all of the points
in . - Using LLL, find a monic polynomial
such that . Note that the roots of are in one-to-one correspondence with the points in . - Also using LLL, find polynomials
in such that for . Then for each root of .

I would like to use Bertini to complete Step #1. My graduate student Jacob Bond has code which can perform Steps #2 and #3 relatively quickly. I’ll say a few more words below when I discuss some examples.

## Shabat Polynomials and Trees

Let me discuss a class of examples in some detail. We say that a connected planar graph is a tree if it has only one face.

I will view it as a bipartite graph by coloring its vertices “black” and placing “white” vertices at the midpoints of the edges. Assuming there are

where we must have

Proposition 2.Continue notation as above. If we can findnontrivial complex numbers such that is identically zero as a polynomial in , then the tree of interest is the Dessin d’Enfant of the Belyi map .

This Belyi map is not just a rational function of degree

## Worked Examples of Shabat Polynomials

Given a collection of

### Example 1

Say that

### Example 2

Say that

where

Setting

### Example 3

Consider the star graph

We have already seen this when

We are interested in an identity of the form

then we recover the identity

### Example 4

Now consider any tree with

Since

in terms of

Question. Can Bertini find all solutions to?

I only know that one solution corresponds to

Using the substitution

However, this isn’t the only Shabat polynomial because this isn’t the only tree with this degree sequence! By using a slight permutation of the degree sequence above, say

it is easy to verify that a completely different set of identities is as follows:

Using the substitution