multivariate factorization over QQbar

[gh-BrentBaccala]

We can currently (#8544) factor univariate polynomials over QQbar:

sage: R.<x>=QQbar[]
sage: (x^2+1).factor()
(x - I) * (x + I)

We can factor over multivariate rings, if the polynomial is actually univariate:

sage: R.<x,y>=QQbar[]
sage: (x^2+1).factor()
(x - I) * (x + I)

But we can't do full "absolute factorization" (that's what it's called in the literature):

sage: R.<x,y>=QQbar[]
sage: (x^2+y^2).factor()
...
NotImplementedError: proof = True factorization not implemented.  Call factor with proof=False.
sage: (x^2+y^2).factor(proof=False)
...
TypeError: no conversion of this ring to a Singular ring defined

This ticket implements absolute factorization by calling Singular's absfact.lib, which works for rings over Q (or transcendental extensions thereof), and uses a technique described in the attached document to handle number fields.

I flagged it major because polynomial factorization is a fundamental feature that blocks the implementation of other things, like primary decomposition of ideals.

[gh-BrentBaccala]

I added code to call Singular's absFactorize if the polynomial's coefficients are in QQ.

Here's what's now possible:

sage: R.<x,y> = QQbar[]

sage: L = (x^2+y^2).factor()
sage: L
((-1*I)*x - y) * (1*I*x - y)
sage: L.value()
x^2 + y^2

sage: p = (-7*x^2 + 2*x*y^2 + 6*x + y^4 + 14*y^2 + 47)*(5*x^2+y^2)^3*(x-y)^4
sage: F = p.factor()
sage: F
...
sage: F.value() == p
True

But something like this still doesn't work (because the coefficients aren't in QQ):

sage: (x+QQbar(sqrt(2))*y).factor()

It's something, which is better than nothing, so I think we should move it towards master, but there's still a lot of work to be done before this ticket is closed.

---

New commits:

​27f109d

Trac #25390: use Singular 'absfact' to factor multivariate polynomials over QQ

[tscrim]

I agree that small improvements are good. We can always do followup tickets to add more functionality. I do have some comments:

Can we use libsingular instead of calls to singular (which uses the pexpect interface and is much slower)?

Doc tweaks:

         ALGORITHM:
 
-        Uses Singular's `absfact` library.
+        Uses Singular's ``absfact`` library.
 
-        TODO:
+        .. TODO::
 
-        Implement absolute factorization over number fields
+            Implement absolute factorization over number fields.

Also if you could split the long output line so that it is at most ~80 chars per line.

Instead of elem_map, you could just use elem_dict.__getitem__.

I don't see the need for the internal functions polynomial_map and reverse_polynomial_map. These little trivial functions make the code harder to follow.

In Python, usually asserts are with spaces, e.g., assert minpoly.degree() == 0.

[git]

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

​e6124e2	Trac #25390: convoluted code that I want to save before cleaning
​b34f2b3	Trac #25390: correctly handle factorization over number fields
​2879cc0	Trac #25390: bug fix (and add a test case)
​747861e	Trac #25390: code cleanup suggested by tscrim's review

[gh-BrentBaccala]

Over the weekend I remembered a technique that I read years ago for factoring over a number field - factor the norm of the polynomial, which has rational coefficients and is a multiple of the original polynomial.

So, this ticket is moving towards closed more quickly that I had thought possible. I think it can now factor any polynomial over QQbar.

The algorithm is tricky enough that I'm planning to write up a short paper describing it, post it on arxiv, and link to it from the documentation. If anybody has a better suggestion for how to incorporate a three page paper explaining the algorithm of a Sage method, please let me know.

I took most of tscrim's suggestions, but I can't figure out how to access a Singular variable from libsingular. I posted a ​question on asksage, but so far no answer.

I also intend to implement factorization over AA before flipping this ticket to needs_review.

[git]

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

​b2657e2	Trac #25390: simplify factorization logic
​580c11e	Trac #25390: enhance multivariate factorization to handle AA as well as QQbar
​be078ac	Trac #25390: ensure that multivariate factorization factors are monic
​7b6c93b	Trac #25390: add a test case
​247f944	Trac #25390: slight code rearrangement

[git]

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

​efdaac1

Merge tag '8.3.beta5' into public/25390

[gh-BrentBaccala]

Still haven't figured out how to access the Singular variable from libsingular; got an answer to that ​SageMath question that made sense, but nbruin (the responder) said that it was "tricky" and he was right.

I spent two weeks trying to figure out to make this code work without doing division in polynomial rings over QQbar but it's a lot harder than I thought. I'm attaching a PDF that describes the progress I've made and the problems that I've had.

Right now, the code is fairly straightforward, but does require polynomial division over QQbar, and therefore requires Trac #25351 to be applied before it works. I've spent enough time trying to break that dependency, so I'm flipping this ticket to needs_review and leaving it the way it is. See the attached PDF for more details.

[tscrim]

I didn't realize that my suggestion would have been so complicated. Thank you for looking into it. I have two small additional nitpicks, but otherwise this ticket is a positive review assuming #25351 (which I cannot review as that is much more involved mathematically, you should cc some people who work more in that area, such as jdemeyer).

You do not need to create a list:

norm_f = prod([numfield_f.map_coefficients(h) for h in numfield.embeddings(QQbar)]).change_ring(QQ)
norm_f = prod(numfield_f.map_coefficients(h) for h in numfield.embeddings(QQbar)).change_ring(QQ)

-        REFERENCE::
+        REFERENCES:
 
-            Geddes, et. al, "Algorithms for Computer Algebra", Section 8.8
+        - Geddes, et. al, "Algorithms for Computer Algebra", Section 8.8

although I think it would be better to reference the book in the master ref file and use - [Geddes1234]_ Section 8.8 as the reference.

[nbruin]

If at all possible, you'll probably want to avoid computing norms by multiplying complex embeddings together. Especially when you have your polynomial over a number field, it's probably much better to avoid floats altogether. For instance, if your number field is given as K=QQ[t]/(h(t)) (with h(t) monic) and you have a polynomial f(x) in K[x], then you can represent your polynomial as F(x,t) in QQ[x,t], via the isomorphism K[x] ~ Q[x,t]/(h(t)). Then you have that

Norm_(K[x]/Q[x]) (f) = Resultant(F,h,t)

Code for computing resultants is usually quite optimized. You can experiment a bit to see which approach works best, but I'd expect the resultant to work quite well.

[git]

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

​3adbd36	Trac #25390: use resultant to compute polynomial norm
​9d6353a	Trac #25390: bug fix (if polynomial names conflict with number field names)
​cbc51c4	Trac #25390: update references

[gh-BrentBaccala]

I made the changes suggested by tscrim and nbruin; thanks for your feedback.

Using the resultant was more of a pain that I thought it should be. The difficulty lay mainly in needing to introduce a new variable, different from the others. If anybody can suggest a better way that what I came up with, please let me know.

[nbruin]

Variable names might be haunting us here:

sage: k=NumberField(x^2+1,"x")
sage: R=PolynomialRing(k,"x,y")
sage: P=PolynomialRing(QQ,k.gens()+R.gens())
ValueError: variable name 'x' appears more than once

Unless you can guarantee the variable names of the number field and the polynomial ring aren't clashing, it seems you cannot go about the problem this way. While it would be reasonable to assume that a user wouldn't intentionally throw this kind of awful naming at you, it's quite conceivable that automatic code would end up doing something like this. Example:

sage: matrix(k,2,2,[k.0,0,0,1]).charpoly()
x^2 + (-x - 1)*x + x

So it would seem the names of your variables for the trivariate ring should be unrelated to the names that occur for the others. Assuming you have a polynomial g in a ring P over a number field k over QQ, I think you can do your conversion with something along the lines of

R=g.parent()
k=R.base_ring()
P=PolynomialRing(QQ,len(k.gens()+R.gens()),'T')   
G=sum(P({(0,)+tuple(k):1})*v.polynomial()(P.0) for k,v in g.dict().iteritems())
NG=G.resultant(k.polynomial()(P.0),P.0)
RQ=PolynomialRing(QQ,R.gens())
NGQ=NG((0,)+RQ.gens())

Doing your conversions like this has the advantage that you don't rely on names of generators at all.

[git]

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

​5b23d3f

Trac #25390: improve norm calculation code

[gh-BrentBaccala]

OK, I re-coded it the way nbruin suggested.

[nbruin]

+            norm_f = sum(norm_ring({tuple(k)[1:]:v})
+                         for k,v in norm_flat.dict().iteritems())

Please don't think that unpacking a polynomial into a dictionary is the best way of handling it! To "lift" number field elements, I didn't see another way, but in this case

norm_f = norm_flat((0,)+norm_ring.gens())

would work just fine, as far as I can see. If you have a good reason to use the dict upacking (is it faster?) then it's fine, of course; but otherwise using the simpler code is probably preferable.

[git]

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

​a3eb8ef

Trac #25390: simplify norm calculation code

[gh-BrentBaccala]

I thought there had to a better way to do it!

[git]

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

​ac30e3a

Merge tag '8.4' into public/25390

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​730a40d	Trac #25351: remove @handle_AA_and_QQbar decorator from reduce method
​a9fde71	Trac #25351: remove @handle_AA_and_QQbar decorator from _normal_basis_libsingular
​4f547af	Trac #25351: Put @handle_AA_and_QQbar decorator on groebner_basis(), and remove
​d187260	Trac #25351: improve @handle_AA_and_QQbar to handle keyword arguments
​f83f2bd	Trac #25351: add test cases to methods decorated with @handle_AA_and_QQbar
​fb145a9	Trac #25351: modify a test case to check keywords arguments to @handle_AA_and_QQbar
​758e4ea	Merge tag '8.7.rc0' into public/25351
​7ba975e	Merge tag '8.7' into public/25351
​bd8ab1b	Merge tag '8.8.beta2' into public/25351
​94a6c6c	Merge branch 'public/25351' into public/25390

[git]

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

​1a3b28c	Merge tag '8.8.beta7' into public/25390
​7830854	Trac #25351: ensure Python 3 compatibility
​ebfbf2d	Merge tag '8.8.beta5' into public/25351
​b602cb1	Merge tag '8.8.beta7' into public/25351
​ba45dd6	Merge branch 'public/25351' into public/25390

[git]

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

​10554b0	Trac #25390: fix some doctests
​f36b116	Merge tag '8.8.rc0' into public/25390

[git]

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

​64dd01e	Merge tag '8.9.rc1' into public/25390
​4a955fb	Trac #25390: fix failing doctest
​3fd38a8	Trac #25390: remove a duplicate check for _factor_multivariate_polynomial
​dd4840c	Trac #25390: add a TODO item - maybe use univariate factorization code

[git]

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

​a9c1e8b

Trac #25390: replace iteritems() with items() for Python 3 compatibility

[git]

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

​6305cc1

Merge tag '9.0.beta3' into public/25390

[tscrim]

Two minor things: The first is the .. TODO:: block should be indented. The second is more stylistic, but I would avoid the blankline after the start of the for loops.

[git]

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

​91d467e

Trac #25390: indent TODO block and remove some blank lines

[gh-BrentBaccala]

Replying to tscrim:

Two minor things: The first is the .. TODO:: block should be indented. The second is more stylistic, but I would avoid the blankline after the start of the for loops.

OK, done.

I didn't remove the blank lines on the second for loop because it's so long, I feel the blank lines improve it readability.

[tscrim]

Thanks. I disagree about the space because I have never really seen that elsewhere in Python code and it looks strange to be because of that. Yet, I don't think it is important enough to hold up the ticket.