110 The VKCURVE package

The main function of the VKCURVE package computes the fundamental group of the complement of a complex algebraic curve in 2, using an implementation of the Van Kampen method (see for example C73 for a clear and modernized account of this method).

    gap> FundamentalGroup(x^2-y^3);
    #I  there are 2 generators and 1 relator of total length 6
    1: bab=aba
    
    gap> FundamentalGroup((x+y)*(x-y)*(x+2*y));
    #I  there are 3 generators and 2 relators of total length 12
    1: cab=abc
    2: bca=abc

The input is a polynomial in the two variables x and y, with rational coefficients. Though approximate calculations are used at various places, they are controlled and the final result is exact.

The output is a record which contains lots of information about the computation, including a presentation of the computed fundamental group, which is what is displayed when printing the record.

Our motivation for writing this package was to find explicit presentations for generalized braid groups attached to certain complex reflection groups. Though presentations were known for almost all cases, six exceptional cases were missing (in the notations of Shephard and Todd, these cases are G24, G27, G29, G31, G33 and G34). Since the a priori existence of nice presentations for braid groups was proved in B01, it was upsetting not to know them explicitly. In the absence of any good grip on the geometry of these six examples, brute force was a way to get an answer. Using VKCURVE , we have obtained presentations for all of them.

This package was developed thanks to computer resources of the Institut de Mathématiques de Jussieu in Paris. We thank the computer support team, especially Joël Marchand, for the stability and the efficiency of the working environment.

We have tried to design this package with the novice GAP3 user in mind. The only steps required to use it are

• Run GAP3 3 (the package is not compatible with GAP3 4).

• Make sure the packages CHEVIE and VKCURVE are loaded (beware that we require the development version of CHEVIE, http://www.math.jussieu.fr/\~{}jmichel/chevie.html and not the one in the GAP3.3.3.4 distribution)

• Use the function FundamentalGroup, as demonstrated in the above examples.

If you are not interested in the details of the algorithm, and if FundamentalGroup gives you satisfactory answers in a reasonable time, then you do not need to read this manual any further.

We use our own package for multivariate polynomials which is more effective, for our purposes, than the default in GAP3 3 (see Mvp). When VKCURVE is loaded, the variables x and y are pre-defined as Mvps; one can also use GAP3 polynomials (which will be converted to Mvps).

The implementation uses Decimal numbers, Complex numbers and braids as implemented in the (development version of the) package CHEVIE, so VKCURVE is dependent on this package.

To implement the algorithms, we needed to write auxiliary facilities, for instance find zeros of complex polynomials, or work with piecewise linear braids, which may be useful on their own. These various facilities are documented in this manual.

Before discussing our actual implementation, let us give an informal summary of the mathematical background. Our strategy is adapted from the one originally described in the 1930's by Van Kampen. Let C be an affine algebraic curve, given as the set of zeros in 2 of a non-zero reduced polynomial P(x,y). The problem is to compute a presentation of the fundamental group of 2 - C. Consider P as a polynomial in x, with coefficients in the ring of polynomials in y

P= α0(y)xn + α1(y) xn-1 + ... + αn-1(y) x + αn(y),
where the αi are polynomials in y. Let Δ(y) be the discriminant of P or, in other words, the resultant of P and (∂ P)/(∂ x). Since P is reduced, Δ is non-zero. For a generic value of y, the polynomial in x given by P(x,y) has n distinct roots. When y=yj, with j in 1,...,d, we are in exactly one of the following situations: either P(x,yj)=0 (we then say that yj is bad), or P(x,yj) has a number of roots in x strictly smaller than n. Fix y0 in ℂ - {y1,...,yd}. Consider the projection p: ℂ2 → ℂ, (x,y) → y. It restricts to a locally trivial fibration with base space B= ℂ - {y1,...,yd} and fibers homeomorphic to the complex plane with n points removed. We denote by E the total space p-1(B) and by F the fiber over y0. The fundamental group of F is isomorphic to the free group on n generators. Let γ1,...,γd be loops in the pointed space (B,y0) representing a generating system for π1(B,y0). By trivializing the pullback of p along γi, one gets a (well-defined up to isotopy) homeomorphism of F, and a (well-defined) automorphism φi of the fundamental group of F, identified with the free group Fn by the choice of a generating system f1,...,fn. An effective way of computing φi is by following the solutions in x of P(x,y)=0, when y moves along φi. This defines a loop in the space of configuration of n points in a plane, hence an element bi of the braid group Bn (via an identification of Bn with the fundamental group of this configuration space). Let φ be the Hurwitz action of Bn on Fn. All choices can be made in such a way that φi=φ(bi). The theorem of Van Kampen asserts that, if there are no bad roots of the discriminant, a presentation for the fundamental group of 2 - C is
< f1,...,fn | ∀ i,j, φi(fj)=fj >
A variant of the above presentation (see VKQuotient) can be used to deal with bad roots of the discriminant.

This algorithm is implemented in the following way.

• As input, we have a polynomial P. The polynomial is reduced if it was not.

• The discriminant Δ of P with respect to x is computed. It is a polynomial in y.

• The roots of Δ are approximated, via the following procedure. First, we reduce Δ and get Δred (generating the radical of the ideal generated by Δ). The roots {y1,...,yd} of Δred are separated by SeparateRoots (which implements Newton's method).

• Loops around these roots are computed by LoopsAroundPunctures. This function first computes some sort of honeycomb, consisting of a set S of affine segments, isolating the yi. Since it makes the computation of the monodromy more effective, each inner segment is a fragment of the mediatrix of two roots of Δ. Then a vertex of one the segments is chosen as a basepoint, and the function returns a list of lists of oriented segments in S: each list of segment encodes a piecewise linear loop γi circling one of yi.

• For each segment in S, we compute the monodromy braid obtained by following the solutions in x of P(x,y)=0 when y moves along the segment. By default, this monodromy braid is computed by FollowMonodromy. The strategy is to compute a piecewise-linear braid approximating the actual monodromy geometric braid. The approximations are controlled. The piecewise-linear braid is constructed step-by-step, by computations of linear pieces. As soon as new piece is constructed, it is converted into an element of Bn and multiplied; therefore, though the braid may consist of a huge number of pieces, the function FollowMonodromy works with constant memory. The packages also contains a variant function ApproxFollowMonodromy, which runs faster, but without guarantee on the result (see below).

• The monodromy braids bi corresponding to the loops γi are obtained by multiplying the corresponding monodromy braids of segments. The action of these elements of Bn on the free group Fn is computed by BnActsOnFn and the resulting presentation of the fundamental group is computed by VKQuotient. It happens for some large problems that the whole fundamental group process fails here, because the braids bi obtained are too long and the computation of the action on Fn requires thus too much memory. We have been able to solve such problems when they occur by calling on the bi at this stage our function ShrinkBraidGeneratingSet which finds smaller generators for the subgroup of Bn generated by the bi (see the description in the third chapter). This function is called automatically at this stage if VKCURVE.shrinkBraid is set to true (the default for this variable is false).

• Finally, the presentation is simplified by ShrinkPresentation. This function is a heuristic adaptation and refinement of the basic GAP3 functions for simplifying presentations. It is non-deterministic.

From the algorithmic point of view, memory should not be an issue, but the procedure may take a lot of CPU time (the critical part being the computation of the monodromy braids by FollowMonodromy). For instance, an empirical study with the curves x2-yn suggests that the needed time grows exponentially with n. Two solutions are offered to deal with curves for which the computation time becomes unreasonable.

A global variable VKCURVE.monodromyApprox controls which monodromy function is used. The default value of this variable is false, which means that FollowMonodromy will be used. If the variable is set by the user to true then the function ApproxFollowMonodromy will be used instead. This function runs faster than FollowMonodromy, but the approximations are no longer controlled. Therefore presentations obtained while VKCURVE.monodromyApprox is set to true are not certified. However, though it is likely that there exists examples for which ApproxFollowMonodromy actually returns incorrect answers, we still have not seen one.

The second way of dealing with difficult examples is to parallelize the computation. Since the computations of the monodromy braids for each segment are independent, they can be performed simultaneously on different computers. The functions PrepareFundamentalGroup, Segments and FinishFundamentalGroup provide basic support for parallel computing.

Subsections

  1. FundamentalGroup
  2. PrepareFundamentalGroup

110.1 FundamentalGroup

FundamentalGroup(curve [, printlevel])

curve should be an Mvp in x and y, or a GAP3 polynomial in two variables (which means a polynomial in a variable which is assumed to be y over the polynomial ring ℚ[x]) representing an equation f(x,y) for a curve in 2. The coefficients should be rationals, gaussian rationals or Complex rationals. The result is a record with a certain number of fields which record steps in the computation described in this introduction:

    gap> r:=FundamentalGroup(x^2-y^3);
    #I  there are 2 generators and 1 relator of total length 6
    1: bab=aba
    
    gap> RecFields(r);
    [ "curve", "discy", "roots", "dispersal", "points", "segments", "loops",
      "zeros", "B", "monodromy", "basepoint", "dispersal", "braids", 
      "presentation","operations" ]
    gap> r.curve;
    x^2-y^3
    gap> r.discy;
    X(Rationals)
    gap> r.roots;
    [ 0 ]
    gap> r.points;
    [ -I, -1, 1, I ]
    gap> r.segments;
    [ [ 1, 2 ], [ 1, 3 ], [ 2, 4 ], [ 3, 4 ] ]
    gap> r.loops;
    [ [ 4, -3, -1, 2 ] ]
    gap> r.zeros;
    [ [ 707106781187/1000000000000+707106781187/1000000000000I,
       -707106781187/1000000000000-707106781187/1000000000000I ],
      [ I, -I ], [ 1, -1 ],
      [ -707106781187/1000000000000+707106781187/1000000000000I,
      707106781187/1000000000000-707106781187/1000000000000I ] ]
    gap> r.monodromy;
    [ (w0)^-1, w0, , w0 ]
    gap> r.braids;
    [ w0.w0.w0 ]
    gap> DisplayPresentation(r.presentation);
    1: bab=aba

Here r.curve records the entered equation, r.discy its discriminant with respect to x, r.roots the roots of this discriminant, r.points, r.segments and r.loops describes loops around these zeros as explained in the documentation of LoopsAroundPunctures; r.zeros records the zeros of f(x,yi) when yi runs over the various r.points; r.monodromy records the monodromy along each of r.segments, and r.braids is the resulting monodromy along the loops. Finally r.presentation records the resulting presentation (which is what is printed by default when r is printed).

The second optional argument triggers the display of information on the progress of the computation. It is recommended to set the printlevel at 1 or 2 when the computation seems to take a long time without doing anything. printlevel set at 0 is the default and prints nothing; set at 1 it shows which segment is currently active, and set at 2 it traces the computation inside each segment.

    gap> FundamentalGroup(x^2-y^3,1);
    # There are 4 segments in 1 loops
    # The following braid was computed by FollowMonodromy in 8 steps.
    monodromy[1]:=B(-1);
    # segment 1/4 Time=0sec
    # The following braid was computed by FollowMonodromy in 8 steps.
    monodromy[2]:=B(1);
    # segment 2/4 Time=0sec
    # The following braid was computed by FollowMonodromy in 8 steps.
    monodromy[3]:=B();
    # segment 3/4 Time=0sec
    # The following braid was computed by FollowMonodromy in 8 steps.
    monodromy[4]:=B(1);
    # segment 4/4 Time=0sec
    # Computing monodromy braids
    # loop[1]=w0.w0.w0
    #I  there are 2 generators and 1 relator of total length 6
    1: bab=aba

110.2 PrepareFundamentalGroup

PrepareFundamentalGroup(curve, name)

VKCURVE.Segments(name[,range])

FinishFundamentalGroup(r)

These functions provide a means of distributing a fundamental group computation over several machines. The basic strategy is to write to a file the startup-information necessary to compute the monodromy along a segment, in the form of a partially-filled version of the record returned by FundamentalGroup. Then the monodromy along each segment can be done in a separate process, writing again the result to files. These results are then gathered and processed by FinishFundamentalGroup. The whole process is illustrated in an example below. The extra argument name to PrepareFundamentalGroup is a prefix used to name intermediate files. One does first :

    gap> PrepareFundamentalGroup(x^2-y^3,"a2");
        ----------------------------------
    Data saved in a2.tmp
    You can now compute segments 1 to 4
    in different GAP sessions by doing in each of them:
        a2:=rec(name:="a2");
        VKCURVE.Segments(a2,[1..4]);
    (or some other range depending on the session)
    Then when all files a2.xx have been computed finish by
        a2:=rec(name:="a2");
        FinishFundamentalGroup(a2);

Then one can compute in separate sessions the monodromy along each segment. The second argument of Segments tells which segments to compute in the current session (the default is all). An example of such sessions may be:

    gap> a2:=rec(name:="a2");
    rec(
      name := "a2" )
    gap> VKCURVE.Segments(a2,[2]);
    # The following braid was computed by FollowMonodromy in 8 steps.
    a2.monodromy[2]:=a2.B(1);
    # segment 2/4 Time=0.1sec
    gap> a2:=rec(name:="a2");
    rec(
      name := "a2" )
    gap> VKCURVE.Segments(a2,[1,3,4]);
    # The following braid was computed by FollowMonodromy in 8 steps.
    a2.monodromy[2]:=a2.B(1);
    # segment 2/4 Time=0.1sec

When all segments have been computed the final session looks like:

    gap> a2:=rec(name:="a2");
    rec(
      name := "a2" )
    gap> FinishFundamentalGroup(a2);
    1: bab=aba

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gap3-jm
19 Feb 2018