Move invariant_generators to libsingular

[mmarco]

This patch moves the .invariant_generators() method of finite matrix groups to libsingular, which simplifies much the code.

It also adds the possibility of defining the ring in which the result should be:

sage: m1 = matrix(QQ, [[0, -1], [1, 0]])
sage: G = MatrixGroup([m1])
sage: G.invariant_generators()
[x0^2 + x1^2, x0^4 + x1^4, x0^3*x1 - x0*x1^3]
sage: R.<x,y> = QQ[]
sage: G.invariant_generators(R)
[x^2 + y^2, x^4 + y^4, x^3*y - x*y^3]

[SimonKing]

The code now is simpler. But there was a reason to have it like that: Singular isn't good at listing the elements of a group (which is needed for the Reynolds operator).

If I recall correctly, there were examples for which the computation of the Reynolds operator in Singular was too slow. Apparently these examples didn't go into the doctests. But perhaps they are available at the trac ticket for the original version of the code?

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New commits:

​24817f2

Moved invariant_generators to libsingular, added option to fix the ring

[mmarco]

Ok, i read in the header that you moved the computation of the reynolds operator to gap, but i didn't see any call to gap in the code. Now i see it, you mean this part, right?:

ReyName = 't'+singular._next_var_name()
singular.eval('matrix %s[%d][%d]'%(ReyName,self.cardinality(),n))
for i in range(1,self.cardinality()+1):
     M = Matrix(elements[i-1],F)
     D = [{} for foobar in range(self.degree())]
     for x,y in M.dict().items():
     D[x[0]][x[1]] = y
      for row in range(self.degree()):
          for t in D[row].items():
          singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)'
          %(ReyName,i,row+1,ReyName,i,row+1, repr(t[1]),t[0]+1))

What i don't understand is this part:

else:
    ReyName = 't'+singular._next_var_name()
    singular.eval('list %s=group_reynolds((%s))'%(ReyName,Lgens))
    IRName = 't'+singular._next_var_name()
    singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])'%(IRName,ReyName))

If i am getting it right, it is supposed to cover the case where there are no elements in the group. In that case we should just return the ring itself.

[git]

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

​6a025ba

Compute reynolds operator before passing it to singular

[mmarco]

Ok, now it computes the reynolds operator before passing it to singular.

I am now working on the modular case. I having trouble getting the output of invariant_ring from libsingular. The singular command is supposed to return three matrices, but calling it through libsingular only gets the first one:

sage: from sage.libs.singular.function import singular_function
sage: import sage.libs.singular.function_factory
sage: sage.libs.singular.function_factory.lib('finvar.lib')
sage: inring = singular_function('invariant_ring')
sage: F=FiniteField(2)
sage: R.<x,y> = F[]
sage: m1 = matrix(R, 2, [0,1,1,0])
sage: inring(m1)
[x + y   x*y]

Any clue about how to get around this?

[git]

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

​2fa234a

Modular case

[mmarco]

Ok, i think it should be ok now. I added the modular case.

I think that this code should be faster than the previous one, since it does the same, but using the faster libsingular interface rather than the string based one. If you have some interesting examples to test, please benchmark them.

[cheuberg]

merge conflict.