proof=False unnecessary in factor()

[zimmermann6]

There are currently no known counter examples where Singular would return an incorrect factorisation. It might be very very slow but does not return wrong answers as far as we know. Hence, we should drop proof=False.

Depends on #10903

CC: @malb @simon-king-jena

Component: commutative algebra

Keywords: sd34

Author: Martin Albrecht

Reviewer: Paul Zimmermann

Merged: sage-5.1.beta0

Issue created by migration from https://trac.sagemath.org/ticket/10902

[malb]

comment:1

I reported this upstream: http://www.singular.uni-kl.de:8002/trac/ticket/320

[zimmermann6]

comment:2

Dear Martin,

Replying to @malb:

I reported this upstream: http://www.singular.uni-kl.de:8002/trac/ticket/320

this ticket is empty on my browser (firefox). Also, is there a workaround to factor bivariate
polynomials over a prime field from within Sage, in particular over GF(2)?

Paul

[malb]

comment:3

Paul, the Singular developers replied on that ticket and state that this bug is fixed in Singular 3-1-2. Updating to 3-1-2 is now #10903.

As for a workaround, I cannot immediately think of any. It's a pretty grim situation, but at least now there is a Singular developer working on factoring.

[zimmermann6]

comment:4

Replying to @malb:

Paul, the Singular developers replied on that ticket and state that this bug is fixed in Singular 3-1-2. Updating to 3-1-2 is now #10903.

As for a workaround, I cannot immediately think of any. It's a pretty grim situation, but at least now there is a Singular developer working on factoring.

thanks Martin. Once Singular is updated to 3-1-2, I will try to find examples where the factorization returned by Singular is incomplete. It would help to have concrete examples
in the documentation.

Paul

[malb]

comment:5

3-1-2 should also improve this, but there are some issues that are only resolved in whatever is the next version of Singular, cf. http://www.singular.uni-kl.de:8002/trac/ticket/291#comment:4

[sagetrac-dsm]

comment:6

Is this really so bad?

def check(f0, f1):
    CP = CartesianProduct(*[f0.base_ring()]*len(f0.variables()))
    for xvec in CP:
        f0_val = f0.subs(dict((v, xv) for v, xv in zip(f0.variables(), xvec)))
        f1_val = f1.subs(dict((v, xv) for v, xv in zip(f0.variables(), xvec)))
        print xvec, f0_val, f1_val, f0_val == f1_val

sage: f0 = x^5 * y^8 * (x*y^4 + y^3 + 1)
sage: f1 = x^5 * y^6 * (x*y^4 + y^3 + 1)
sage: check(f0, f1)
[0, 0] 0 0 True
[0, 1] 0 0 True
[1, 0] 0 0 True
[1, 1] 1 1 True

The factor of y^2 has no effect, so aren't the two polynomials equivalent in GF(2)? This happens pretty frequently:

sage: f0 = x**8+y**8+1
sage: f1 = f0.factor(proof=False)
sage: f1
(x + y + 1)^6
sage: f1.expand()
x^6 + x^4*y^2 + x^2*y^4 + y^6 + x^4 + y^4 + x^2 + y^2 + 1
sage: check(f0, f1.expand())
[0, 0] 1 1 True
[0, 1] 0 0 True
[1, 0] 0 0 True
[1, 1] 1 1 True

Is there a case where the two expressions aren't equivalent as functions? Or am I missing something obvious like usual?

[malb]

comment:7

Replying to @sagetrac-dsm:

Is there a case where the two expressions aren't equivalent as functions? Or am I missing something obvious like usual?

But polynomials over GF(2) are not merely boolean functions. There's a one-to-one mapping between square-free polynomials over GF(2) and boolean functions, but polynomials are more than that. So I'd say it is very bad.

[zimmermann6]

comment:8

here is another example over GF(2):

sage: R.<x,y> = GF(2)[]
sage: p=x^8 + y^8; q=x^2*y^4 + x; f=p*q
sage: lf = f.factor(proof=False)
sage: f-lf
x^10*y^4 + x^2*y^12 + x^8*y^4 + x^6*y^6 + x^4*y^8 + x^2*y^10 + x^9 + x*y^8 + x^7 + x^5*y^2 + x^3*y^4 + x*y^6

An example over GF(3):

sage: R.<x,y> = GF(3)[]
sage: p = -x*y^9 + x
sage: q = -x^8*y^2
sage: f = p*q
sage: f
x^9*y^11 - x^9*y^2
sage: f.factor(proof=False)
y^2 * (y - 1)^6 * x^6

[zimmermann6]

comment:9

and an example over GF(5):

sage: R.<x,y> = GF(5)[]
sage: p=x^27*y^9 + x^32*y^3 + 2*x^20*y^10 - x^4*y^24 - 2*x^17*y
sage: q=-2*x^10*y^24 + x^9*y^24 - 2*x^3*y^30
sage: f=p*q; f-f.factor(proof=False)
-2*x^37*y^33 - 2*x^42*y^27 + x^36*y^33 - 2*x^30*y^39 + x^41*y^27 - 2*x^35*y^33 + x^30*y^34 + 2*x^29*y^34 + x^23*y^40 + 2*x^14*y^48 - x^13*y^48 + 2*x^7*y^54 + 2*x^37*y^18 + 2*x^42*y^12 - x^36*y^18 + 2*x^30*y^24 - x^41*y^12 + 2*x^35*y^18 - x^27*y^25 - 2*x^26*y^25 - x^20*y^31 - x^30*y^19 - 2*x^29*y^19 - x^23*y^25 - 2*x^14*y^33 + x^13*y^33 - 2*x^7*y^39 + x^27*y^10 + 2*x^26*y^10 + x^20*y^16

and one over GF(7):

sage: R.<x,y> = GF(7)[]
sage: p=-3*x^47*y^24
sage: q=-3*x^47*y^37 - 3*x^24*y^49 + 2*x^56*y^8 + 3*x^29*y^15 - x^2*y^33
sage: f=p*q
sage: f-f.factor(proof=False)
2*x^94*y^61 + 2*x^71*y^73 + x^103*y^32 - 2*x^59*y^61 - 2*x^76*y^39 - 2*x^36*y^73 + 3*x^49*y^57 - x^68*y^32 + 2*x^41*y^39 - 3*x^14*y^57

Examples can be constructed at will with the following code (change GF(7) by whatever you want):

R.<x,y> = GF(7)[]
nterms=2
for d in range(1,10^6):
   print d
   sys.stdout.flush()
   p = R.random_element(degree=d,terms=nterms)
   while p.degree()<=0:
      p = R.random_element(degree=d,terms=nterms)
   q = R.random_element(degree=d,terms=nterms)
   while q.degree()<=0:
      q = R.random_element(degree=d,terms=nterms)
   f = p*q
   lp = p.factor(proof=False)
   lq = q.factor(proof=False)
   lf = f.factor(proof=False)
   if lp*lq <> lf:
      print p, q
      raise ValueError

[zimmermann6]

Changed keywords from none to sd34