is_squarefree() incorrect over imperfect fields

[saraedum]

The method is_squarefree() is incorrect for polynomials over function fields over finite fields:

sage: K.<x> = FunctionField(GF(3))
sage: R.<T> = K[]
sage: f=T^3-x
sage: f.factor(proof=False)
T^3 + 2*x
sage: f.is_squarefree()
False

---

Apply:

1. attachment: trac_12404.patch
2. attachment: trac_12404_warning.patch
3. attachment: 12404_examples.patch

to the sage library.

Depends on #9054
Depends on #13043
Depends on #12988
Depends on #10902

CC: @sagetrac-sydahmad

Component: commutative algebra

Keywords: sd40.5

Author: Julian Rueth

Reviewer: Paul Zimmermann, Jeroen Demeyer

Merged: sage-5.1.beta3

Issue created by migration from https://trac.sagemath.org/ticket/12404

[saraedum]

comment:1

The method is_squarefree() returns self.derivative().gcd(self).degree() <= 0. In the example the derivative vanishes which leads to the incorrect behaviour.

As a quick workaround one can do:

True in [d>=2 for (f,d) in f.factor(proof=False)]

[saraedum]

comment:2

Replying to @saraedum:

True in [d>=2 for (f,d) in f.factor(proof=False)]

This would be "is_not_squarefree()" of course.

[zimmermann6]

comment:3

see also #12198.

Paul

[zimmermann6]

comment:4

see also #12129.

Paul

[saraedum]

comment:5

Thanks for pointing these out. Since both are in the multivariate case, I don't think they're related.

[saraedum]

comment:6

The attached patch should fix the problem.

[saraedum]

Changed dependencies from #9054 to #9054, #12404

[saraedum]

Author: Julian Rueth

[saraedum]

Changed dependencies from #9054, #12404 to #9054, #12988

[saraedum]

Changed keywords from none to sd40.5

[zimmermann6]

Work Issues: coherence with squarefree_decomposition, fix error

[zimmermann6]

comment:9

the documentation of is_squarefree says that f is not square-free if g^2
divides f where g is a non-unit. In particular 4*x is not considered
square free by is_squarefree, but it is by squarefree_decomposition:

sage: R.<x> = ZZ[]
sage: f = 4*x
sage: f.is_squarefree()
False
sage: f.squarefree_decomposition()
(4) * x

Sage should be coherent in that matter. Personally I prefer not to decompose the coefficient content.

Moreover the following produces an error:

sage: R.<x> = ZZ[]
sage: f = 2*x^2
sage: f.is_squarefree()
...
AttributeError: 'int' object has no attribute 'is_zero'

Paul

[zimmermann6]

Reviewer: Paul Zimmermann

[saraedum]

comment:10

Replying to @zimmermann6:

the documentation of is_squarefree says that f is not square-free if g^2
divides f where g is a non-unit. In particular 4*x is not considered
square free by is_squarefree, but it is by squarefree_decomposition:

sage: R.<x> = ZZ[]
sage: f = 4*x
sage: f.is_squarefree()
False
sage: f.squarefree_decomposition()
(4) * x

Sage should be coherent in that matter. Personally I prefer not to decompose the coefficient content.

That is true. I also noted this inconsistency. To not break existing code, I'd rather add a warning section in the docstring. Generally I agree that having squarefree_decomposition() factor the content is not what one wants for most purposes.

Would you be ok with just adding a warning and an example showing this problem?

Moreover the following produces an error:

sage: R.<x> = ZZ[]
sage: f = 2*x^2
sage: f.is_squarefree()
...
AttributeError: 'int' object has no attribute 'is_zero'

This should not happen when applying the dependency #12988.

[saraedum]

comment:12

I added a patch with such a warning. I still have to check how it renders in the docs.

[saraedum]

comment:13

I made a few docstring changes in the latest patch.

[saraedum]

Attachment: trac_12404.patch.gz

distinguish characteristic zero and nonzero

[saraedum]

comment:14

Btw. I will add the respective warning for squarefree_decomposition() in #13008.

[zimmermann6]

comment:15

Julian, there is a typo in trac_12404_warning.patch: not to consider to content
should read not to consider the content.

Would it break a lot of code if is_squarefree returns False for 4*x?
Did you try?

Paul

[saraedum]

Attachment: trac_12404_warning.patch.gz

warning about inconsistency with squarefree_decomposition

[saraedum]

comment:16

Thanks, the typo should be fixed now.

No I haven't tried. I was actually thinking about external code using that method. If you insist we can change it. I don't have a very strong opinion about this. I just believe that inconsistency isn't bad as long as it's documented. IMHO breaking the interface is worse than documented inconsistency.

[jdemeyer]

comment:17

Replying to @zimmermann6:

Would it break a lot of code if is_squarefree returns False for 4*x?

Did you mean "if is_squarefree returns True for 4*x"?

[jdemeyer]

Changed dependencies from #9054, #12988 to #9054, #12988, #10902

[jdemeyer]

Attachment: 12404_examples.patch.gz

[jdemeyer]

comment:19

Positive review to the first two patches. Anybody else can review my patch?

[jdemeyer]

Changed reviewer from Paul Zimmermann to Paul Zimmermann, Jeroen Demeyer

[williamstein]

comment:20

jdmeyer -- I positively review your patch.

[jdemeyer]

Changed upstream from Not yet reported upstream; Will do shortly. to none

[jdemeyer]

Changed work issues from coherence with squarefree_decomposition, fix error to none

[jdemeyer]

Changed dependencies from #9054, #12988, #10902 to #9054,#13043, #12988, #10902

[jdemeyer]

Merged: sage-5.1.beta3