Class P_to_H_chart

An instance of this class represents a pair of approximate polynomial solutions (f0,f1) near z=z0 to a certain linear second order differential equation depending on a parameter d. If d is chosen correctly, then f0 and f1 will be closely related to the canonical hyperbolic covering of the surface PX(a_P).

The degree of polynomials used for approximate power series.

The centre around which we expand various power series.

The series ff0 and ff1 satisfy $f''+s f/2=0$, where $s$ is ss0 + d * ss1. Note that we are primarily interested in the function s0(z)=ss0(z-z0) rather than ss0 itself.

The series ff0 and ff1 satisfy $f''+s f/2=0$, where $s$ is ss0 + d * ss1. Note that we are primarily interested in the function s1(z)=ss1(z-z0) rather than ss1 itself.

The series ff0 satisfies $f''+s f/2=0$ with $f(z)=1+O(z^2)$. Note that we are primarily interested in the function f0(z)=ff0(z-z0) rather than ff0 itself.

The series ff1 satisfies $f''+s f/2=0$ with $f(z)=z+O(z^2)$. Note that we are primarily interested in the function f1(z)=ff1(z-z0) rather than ff1 itself.

This should be a $2\times 2$ invertible complex matrix, to be interpreted as follows. We have solutions $f_0$ and $f_1$ satisfying $f_i=(z-z_0)^i+O((z-z_0)^2)$, and there are also solutions $f_{00}$ and $f_{01}$ satisfying $f_{0i}=z^i+O(z^2)$. The M field should be set so that $(f_0,f_1)=M.(f_{00},f_{01})$.

Objects of the class P_to_H_map will maintain a table of charts, identified by an integer index, which is stored in this field.

This is the index of another chart, whose centre is closer to the origin. We will set the M field of this chart by comparing it with the parent chart, whose M field will have been calculated already. (The induction starts with the chart of index zero, which is centred at the origin and has M equal to the identity matrix.)

This is a function of the form $z+O(z^3)$, whose schwarzian derivative is $s$.

This sets the a_P field, and also sets the d field to a value that is approximately correct.

This method sets the fields ss0 and ss1. Note that these are independent of d, so we do not need to use this method again if we choose a new value of d.

This sets the fields ff0 and ff1

This method sets the M field of this chart based on the M field for another chart C (whose centre should be close to the centre of this chart)