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. trac_12404.patch
2. trac_12404_warning.patch
3. 12404_examples.patch

to the sage library.

[saraedum]

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]

Replying to saraedum:

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

This would be "is_not_squarefree()" of course.

[zimmerma]

see also #12198.

Paul

[zimmerma]

see also #12129.

Paul

[saraedum]

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

[saraedum]

The attached patch should fix the problem.

[zimmerma]

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

[saraedum]

Replying to zimmerma:

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]

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

[saraedum]

I made a few docstring changes in the latest patch.

[saraedum]

distinguish characteristic zero and nonzero

[saraedum]

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

[zimmerma]

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]

warning about inconsistency with squarefree_decomposition

[saraedum]

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]

Replying to zimmerma:

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]

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

[was]

jdmeyer -- I positively review your patch.