[with patch, needs work] Action of MatrixGroup on a MPolynomialRing

Sorry, in the above code I forgot to copy/paste the line

sage: R.<x,y> = GF(3)[]

Moreover, for the reasons above, the ticket should belong to *commutative* algebra, not just algebra (I was clicking on the wrong button).

The patch matrixgroupCall.patch is without doctests, and I am not sure if it couldn't be done better. So, it needs more work.

For example, Martin mentioned the possibility (off list) to create a pyx file with a Cython function, and then the call method would use that function. It might pay off here, since there are tight loops and since the method has to deal with tuples or lists. So Cdefining might speed things up.

Opinions?

At least, the following now works:

sage: R.<x,y>=GF(3)[]
sage: M=Matrix(GF(3),[[1,2],[1,1]])
sage: G=MatrixGroup([M])
sage: g=G.0
sage: p=x*y^2
sage: g(p)
x^3 + x^2*y - x*y^2 - y^3

call method for MatrixGroupelement? (this time with doc test) and left_matrix_action for MPolynomial

Some changes:

Nicolas Thiéry suggested to implement the action of matrix groups by a method for polynomials (I call the method left_matrix_action), and, for convenience, also provide a __call__ method for MatrixGroupElement? that refers to left_matrix_action.

This has several advantages:

• The __call__ works for any class that has a left_matrix_action method, hence, it is not a-priori restricted to polynomials.
• I've put left_matrix_action into multi_polynomial.pyx, hence, I can use Cython.

I have two Cython concerns left:

1. The innerst loop is

   prod([Im[k]**X[k] for k in xrange(n)])
   

   where k is c'defined as int. Should this better be done in a for-loop, rather then creating a list and calling prod?
2. The variable X is of type polydict.ETuple, so I can not directly c'define X. One could do

       cdef tuple X
       for i from 0<i<l:
           X = tuple(Expo[i])
   

   But would this be faster?

Another version of left_matrix_action

In matrixgroupCallNew.patch (to be applied after the first patch), I modified the method according to my above concerns. In the example from my original post, the average running time improves from ~240 microseconds to 164 microseconds, and in a larger example it improved from 6.5s to 5.4s

Nevertheless, I made two separate patches, so that the reviewer (if there is any...) can compare by him- or herself.

Cheers

Simon

One observation: Reverse the outer loop

        for i from l>i>=0:
            X = tuple(Expo[i])
            c = Coef[i]
            for k from 0<=k<n:
                if X[k]:
                    c *= Im[k]**X[k]
            q += c

It results in a further improvement of computation time. Is this coincidence? Or is it since summation of polynomials should better start with the smallest summands?

Anyway, I didn't change the patch yet.

Slight improvement; extended functionality

Replying to SimonKing:

One observation: Reverse the outer loop {{{ for i from l>i>=0: X = tuple(Expo[i]) c = Coef[i] for k from 0<=k<n: if X[k]: c *= Im[k]**X[k] q += c }}} It results in a further improvement of computation time. Is this coincidence? Or is it since summation of polynomials should better start with the smallest summands?

I made a couple of tests, and there was a small but consistent improvement. So, in the third patch (to be applied after the other two) I did it in that way.

The left_matrix_action shall eventually be used for computing the Reynolds operator of a group action; moreover, the Reynolds operator should be applicable on a list of polynomials. Then, the function would repeatedly compute the image of the ring variables under the action of some group element. But then it would be better to compute that image only once and pass it to left_matrix_action. The new patch provides this functionality. Example (continuing the original example):

sage: L=[X.left_matrix_action(g) for X in R.gens()]
sage: p.left_matrix_action(L)
x^3 + x^2*y - x*y^2 - y^3

I did confirm that the patches apply cleanly, that

sage: M = Matrix(GF(3),[[1,2],[1,1]])
sage: G = MatrixGroup([M])
sage: g = G.0
sage: g

[1 2]
[1 1]
sage: P.<x,y> = PolynomialRing(GF(3),2)
sage: p = x*y^2
sage: g(p)
x^3 + x^2*y - x*y^2 - y^3
sage: (x+2*y)*(x+y)^2
x^3 + x^2*y - x*y^2 - y^3

works, and that the code seems well-documented.

However, I can't do testing on this machine (intrepid ubuntu) and some of the code is written in Cython, which I can't really 100% vouch for. Seems okay though and simple enough. Since speed was a topic of the comments above, my only question is that the segment

 	396	        for i from 0<=i<l: 
 	397	            X = Expo[i] 
 	398	            c = Coef[i] 
 	399	            q += c*prod([Im[k]**X[k] for k in xrange(n)]) 

could probably be rewritten as a one-line sum, which might (or might not) be faster.

Maybe Martin Albrecht could comment on the Cython code?

If Martin (for example) passes the Cython code, and the docstrings pass sage -testall, I would give it a positive review.

cdef list Im 
if isinstance(M,list): 
  Im = M 

shouldn't Im = M take care of the type checking anyway, so that a try-except block is sufficient? Also, I think maybe the type of p should be checked in the __call__ method and a friendly error message raised? Not sure though.

Cython code looks good (just read it).

REFEREE REPORT:

Check this out:

sage: R.<x,y> = GF(3)[]
sage: M=Matrix(GF(3),[[1,2],[1,1]])
sage: M2=Matrix(GF(3),[[1,2],[1,0]])
sage: G=MatrixGroup([M, M2])
sage: (G.0*G.1)(p)
-x^2*y + x*y^2 - y^3
sage: G.0(G.1(p))
x^2*y + x*y^2 + y^3

Oops, your *left action* -- which it better be if you use that notation -- ain't a left action! Oops

-- William

Really Oops. Sorry.

I implemented it analogous to what is done in Singular. Perhaps I am mistaken in the sense that it is supposed to be a right action (which then would deserve another notation).

sage: (G.0(G.1((p))))
-x^2*y + x*y^2 - y^3
sage: (G.1*G.0)(p)
-x^2*y + x*y^2 - y^3

However, I think it doesn't matter what Singular does. I will look up the literature whether one really wants a right or left action, and how the left action is supposed to be.

Replying to SimonKing:

However, I think it doesn't matter what Singular does. I will look up the literature whether one really wants a right or left action...

I mean, something like "one has a left action on a variety, which gives rise to a right action on the coordinate ring". I have to sort it out.

If this is the case, then it should be better implemented in the __mul__ method of polynomials, isn't it? Such as

sage: p*G.1*G.0==p*(G.1*G.0)
True

Left actions should use call, right actions should use *exponentiation*.

Substitution is a right action. Substitution of the *inverse* is a left action.