Sirocco improvements: fix base number fields and braid monodromy of arrangements

[enriqueartal]

Now the input of the functions or methods braid_monodromy and fundamental_group is a polynomial f(x, y) (or a curve defined by this polynomial) with coefficients in a number field K with a precise embedding in QQbar.

The first main change proposed by this issue is to construct an extension K1 containing the imaginary unit; this extension comes also with an embedding in QQbar. The main reason is that several intermediate computations deal with f(x0,y) where x0 is a Gauss rational; considering these polynomials in K1[y] instead of QQbar[y] dramatically increase the speed of some of these intermediate computations.

The second main change is to allow the input to be a list of polynomials instead of just one. The method braid_monodromy starts with changes of variables of the type x: x + y till the global degree of f coincides with its y-degree. Next step is to compute the roots in x of its discriminant with respect to y using an internal funtion called disc. Both changes are useful for this function: Applying it to an example of a polynomial of degree 18 with coefficients in a number field of index 12 yields to more that 30 minutes of computation time. This polynomial has three factors. The computation of the discriminant of each factor and the resultant of the three pairs of factors is done in 15 seconds.

A new function braid_monodromy_arrangement adds a new feature, i.e., it computes the braid monodromy of the product of the polynomials in the list, with a new information, a dictionnary associating each strand to a precise polynomial.

The third main change is to provide a new way to get the presentation of the fundamental group of the complement of the curve. This new method has two advantages: the obtained presentation has information about the homotopy type of the complement of the affine curve and computations are faster. When computing the fundamental group, the relations come from the following computation; take a generator g of a free group and a braid s; the relation is obtained as (g*s)/g (the natural action of the braid group on the free group is already implemented. The problem is that these braids may be extremely long and these computations may take very long time. The new method performs less computations and cut the above method in several steps which takes into account some conjugations tricks.

Finally, some computations have been parallelized. Also, sometimes the computations stopped for precision or parallelism issues and some solutions avoid these problems.

The branch puiseux of this fork(https://github.com/enriqueartal/sage) contains the modifications.

CC: @miguelmarco

Component: algebraic geometry

Author: Enrique Artal

Branch/Commit: u/enriqueartal/sirocco_improvements__discriminant_and_optional_information @ 963d744

Issue created by migration from https://trac.sagemath.org/ticket/34415

[enriqueartal]

Branch: u/enriqueartal/sirocco_improvements__discriminant_and_optional_information

[sagetrac-git]

Commit: 42c4b3e

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

42c4b3e

Corrected info and added optional information

[enriqueartal]

comment:3

The definition of discrim takes into account factorization. The optional information has been added. A new variable change_info has been added to method braid_monodromy. The default value is False and in this case it behaves as previously, the output is a list of braids; if the value is True then a tuple with three items is returned:

• List of braids
• Number of changes of variables x : x + y to have the same degree and y-degree
• A vector with components the real and imaginary parts of the x-coordinate of the base point.

[sagetrac-git]

Changed commit from 42c4b3e to 083ff14

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

083ff14

Check tests

[enriqueartal]

Description changed:

--- 
+++ 
@@ -1,7 +1,7 @@
-Given a polynomial `f(x, y)`, the method `braid_monodromy` starts with changes of variables of the type `x:x+y` till the global degree of `f` coincides with its `y`-degree. Next step is to compute its discriminant with respect to `y`.
+Given a polynomial `f(x, y)`, the method `braid_monodromy` starts with changes of variables of the type `x: x + y` till the global degree of `f` coincides with its `y`-degree. Next step is to compute its discriminant with respect to `y`.
 
 Applying it to an example of a polynomial of degree 18 with coefficients in a number field of index 12 yields to more that 30 minutes of computation time. This polynomial has three factors. The computation of the discriminant of each factor and the resultant of the three pairs of factors is done in 15 seconds.
 
 Now `Sirocco` takes into account the factors, so computing the discriminant via the factorization would reduce the time of computation. 
 
-In the same ideas it would be useful to know to which irreducible component each string of the braids correspond. As a first approach, at least the result could contain also the number of changes of variables and the base point, which contains the information to decide about the relationship of strings and factors.
+In the same ideas, it is useful for arrangements of curves to know to which irreducible component each strand of the braids correspond. A new function is define, `braid_monodromy_arrangement` which yields both the braid monodromy and the information of the strands

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

361d6aa

Simplifying the functions

[sagetrac-git]

Changed commit from 083ff14 to 361d6aa

[enriqueartal]

comment:7

After this change, the computation of braid_monodromy and braid_monodromy_arrangement gives an error if sage runs with SAGE_NUM_THREADS>1,

File /usr/local/sage310/src/sage/schemes/curves/affine_curve.py:1769, in AffinePlaneCurve_field.braid_monodromy(self)
   1766     raise NotImplementedError("the base field must have an embedding"
   1767                               " to the algebraic field")
   1768 f = self.defining_polynomial()
-> 1769 return braid_monodromy(f)

File /usr/local/sage310/src/sage/schemes/curves/zariski_vankampen.py:980, in braid_monodromy(f, arrangement)
    977     braidscomputed = (braid_in_segment([(gfac, seg[0], seg[1])
    978                                         for seg in segs]))
    979 segsbraids = {}
--> 980 for braidcomputed in braidscomputed:
    981     seg = (braidcomputed[0][0][1], braidcomputed[0][0][2])
    982     beginseg = (QQ(seg[0].real()), QQ(seg[0].imag()))

File /usr/local/sage310/src/sage/parallel/use_fork.py:191, in p_iter_fork.__call__(self, f, inputs)
    188     alarm(max(timeout - (T - oldest), 0.1))
    190 try:
--> 191     pid = os.wait()[0]
    192     cancel_alarm()
    193     W = workers.pop(pid)

ChildProcessError: [Errno 10] No child processes

but if it is executed twice it works