py3: fixes for sage.schemes.hyperelliptic_curves

[Erik Bray]

The main thing needed here was a __hash__. Or removing __eq__ and __ne__

Part of #24551

[Travis Scrimshaw]

You have too much data in your __hash__:

sage: P.<x> = QQ[]
sage: f0 = 4*x^5 - 30*x^3 + 45*x - 22
sage: C0 = HyperellipticCurve(f0)
sage: P.<x> = RR[]
sage: f1 = 4*x^5 - 30*x^3 + 45*x - 22
sage: C1 = HyperellipticCurve(f1)
sage: C1 == C0
True
sage: hash(C1) == hash(C0)
False

So you definitely have to remove _PP from the hash, and I would probably remove both the __class__ as subclasses are not used in the __eq__ comparison.

[Erik Bray]

I'm not 100% sure about that. I'd need to double-check what the intention is here. On Python 2 it was inheriting the flimsy default __hash__ from CategoryObject that just hashes its __repr__. So for this case I was including essentially the same data in the hash that would be needed to differentiate the reprs of two of these objects.

But maybe that's not necessary. It just wasn't clear what the intent was (if any) due to the default hash...

[Travis Scrimshaw]

I completely agree the repr hash is bad. I think the data used for equality should be what is used for the __hash__, not more. From what you're saying, it seems like we are actually fixing a bug if we only hash self._hyperelliptic_polynomials.

My understanding behind the default hash is that every object has a repr and generally two objects that compare equal have equal repr output.

I don't use/know this code, so we probably have to ask someone who understands it better to confirm if you're worried enough.

[Erik Bray]

I could certainly try a simpler repr to start with and see what happens. If that works, then I agree with you we should just use what __eq__ compares.

[Travis Scrimshaw]

Sounds good. Thanks.

[Frédéric Chapoton]

any progress here ?

[Erik Bray]

I mean, clearly not. I'd prefer if someone who knew this code were to chime in, but otherwise I'll get to it if/when I feel like taking the time to understand this code better.

[Erik Bray]

To be clear, I also agree with everything Travis wrote on this ticket; I'm just not in a rush to act on it because I don't feel qualified to make that judgment call w.r.t. this code.

[Travis Scrimshaw]

Ben, do you know about this code or know who we should cc?

[Ben Hutz]

I'm not familiar with this code. John may know.

[John Cremona]

I don't know or use the code (except perhaps occasionally) and so I do not really understand the issue here. I don't think that I have ever written such a has function so do not know what properties it is supposed to have.

What exactly was the problem which this patch is intended to fix?

[Alex J. Best]

I'm really confused about whether or not we should call these hyperelliptic curves equal in the first place. The current code returns true because the polynomials are equal which is because both variables are named x even though they are over different fields, are two curves over different fields really _equal_, even if one is just a base change of the other? For contrast

sage: P.<x,y>= RR[]
sage: C0 = Curve(y - x^2  - 1)
sage: C0
Affine Plane Curve over Real Field with 53 bits of precision defined by -x^2 + y - 1.00000000000000
sage: P.<x,y>= QQ[]
sage: C1 = Curve(y - x^2  - 1)
sage: C1
Affine Plane Curve over Rational Field defined by -x^2 + y - 1
sage: C0 == C1
False

[Erik Bray]

Replying to alexjbest:

I'm really confused about whether or not we should call these hyperelliptic curves equal in the first place. The current code returns true because the polynomials are equal which is because both variables are named x even though they are over different fields, are two curves over different fields really _equal_, even if one is just a base change of the other?

I had the same doubt before. There's some inconsistency throughout Sage about when some objects defined over different fields are considered equal under ==. In some cases it's quite deliberate, in other cases it's not clear until and unless the original author chimes in. If the author isn't even sure I would lean against calling them ==, much less having the same hash.

[John Cremona]

I agree with Alex: to be equal the fields of definition should equal and the defining polynomials identical. This is the case with elliptic curves:

sage: E = EllipticCurve([0,1])
sage: E
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
sage: K.<i> = NumberField(x^2+1)
sage: EK = E.change_ring(K)
sage: EK
Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in i with defining polynomial x^2 + 1
sage: E == EK
False
sage: hash(E)
8741953973726
sage: hash(EK)
8741951786400

I don't know what this hash function actually does: it calls hash_by_id, whatever that is. I also don't know how to find out what function is being called by E==EK either!

[Travis Scrimshaw]

EllipticCurve is a UniqueFactory, so the input (i.e., in part, the field) uniquely identifies the curve. So the hash and == is obtained by using id as there is a unique such object in memory at a given time. Changing the hyperelliptic to suppose the UniqueRepresentation-type behavior would be a very invasive and complicated surgery I think.

So the short-term fix IMO would be deciding an appropriate notion of __eq__, and then changing the __hash__ to match. From what John is saying, we should also introduce a check that the defining fields are equal.

Also, from git blame, "Nick Alexander" and "tornaria" comes up a lot and David Kohel is listed in the copyright.

[Frédéric Chapoton]

I propose a solution : simply remove __eq__, __ne__ and __hash__ here.

Then they are all provided by the general setting of projective subschemes, which seems to be a good idea. This will compare ambient spaces and defining ideals, which amount to compare base field and polynomials.

---

New commits:

​2726198	Merge branch 'u/embray/python3/sage-schemes-hyperelliptic_curves/misc' of ssh://trac.sagemath.org:22/sage into 8.4.b0
​78ae922	py3: comparison and hash cleanup for hyperellliptic curves

[John Cremona]

That sounds very sensible to me -- I do not know why the person who implemented these special functions thought it was necessary to override those of the parent class.

As far as I am concerned this is good to merge assuming that tests all pass.

[Frédéric Chapoton]

failing doctests in hyperelliptic_padic_field.py (sigh)

[Erik Bray]

This seems like a reasonable approach, but it seems that you exposed a poor assumption that was made somewhere else. I don't know what a sage.rings.padics.padic_ZZ_pX_CR_element.pAdicZZpXCRElement is, or why it isn't hashable. Perhaps it should be?

If it definitely shouldn't be hashable (e.g. it is not immutable) then maybe the problem is in MPolynomialIdeal.__richcmp__'s assumption that the generators of an ideal are definitely hashable (and if not, it should do something else to compare gens).

[Frédéric Chapoton]

There is a _cache_key method on these objects inside src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx

But its doc claims that there is no reasonable hash..

[John Cremona]

Adding David Roe in CC since he may be able to throw light on the p-adic ring issue. Note that p-adic rings (as with the reals) are not exact, so comparison between elements is not so easy.

[Erik Bray]

Incidentally, MPolynomialIdeal.__richcmp__ was last touched by #23920. The idea to use sets to compare gens was from Travis: ​https://trac.sagemath.org/ticket/23920#comment:9

Perhaps there should be a try/except around this and attempt direct comparison if comparing by sets raises a TypeError. I'm not sure what the impact is of computing a Groebner basis, or if it can be avoided in some other way as well.

[git]

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

​2431727

adding the hash

[Frédéric Chapoton]

adding hash from the existing _cache_key seems to fix the failing doctests.. But is this a correct thing to do ?

[Erik Bray]

Replying to chapoton:

adding hash from the existing _cache_key seems to fix the failing doctests.. But is this a correct thing to do ?

No, I don't believe so, and the I think the explanation in the _cache_key docs is fairly clear why. Instead: ​https://trac.sagemath.org/ticket/25946#comment:25

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​d5573ef	py3: add a __hash__ for HyperellipticCurve_generic
​f29c001	py3: need explicit cast to int for range
​3771812	py3: comparison and hash cleanup for hyperellliptic curves
​9f9399c	tiny change in the comparison of ideals inside polynomial rings

[Frédéric Chapoton]

ok. So here is my next proposal : when comparing ideals generated by a single polynomial, just compare this generator (no need to wrap by set).

[Erik Bray]

True. Is that case common enough to merit a special case? Are there cases of ideals of rings over these types of p-adics whose ideal would be generated by more than one polynomials (and hence would still crash on the existing code)?

What I'm saying is just, why not:

try:
    same_gens = set(self.gens()) == set(other.gens())
except TypeError:
    same_gens = False

if same_gens:
    # Do whatever we do currently
else:
    # Do whatever is necessary to compare the gens (construct a GB, etc).

[git]

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

​ecc4b36

change again the comparison of ideals

[John Cremona]

Replying to embray:

True. Is that case common enough to merit a special case? Are there cases of ideals of rings over these types of p-adics whose ideal would be generated by more than one polynomials (and hence would still crash on the existing code)?

Certainly: in ZZ_p[X] the ideal generated by p and X. But the number of gens will always be small, so why not sort the gens, however many there are?

This is an example of a common problem in Sage and other systems. If I define two ideals (or other structures) in different ways and ask whether they are equal then the test may be very expensive. But here we just test whether they have been presented in the same way (or with trivial changes such as changing the order of the generators).

What I'm saying is just, why not:

try:
    same_gens = set(self.gens()) == set(other.gens())
except TypeError:
    same_gens = False

if same_gens:
    # Do whatever we do currently
else:
    # Do whatever is necessary to compare the gens (construct a GB, etc).

---

New commits:

​ecc4b36

change again the comparison of ideals

[Frédéric Chapoton]

Here is another tentative : first compare with set() then compare with set()

[Erik Bray]

Replying to chapoton:

Here is another tentative : first compare without set() then compare with set()

That could still crash. I don't know the math well enough to construct an interestingexample case, but say you have two copies of the same ideal, but their generators just happen to be in a different order, as in:

sage: R.<x,y> = QQ[]
sage: R.ideal(x, y).gens()
[x, y]
sage: R.ideal(y, x).gens()

This is the sort of case, as John wrote, that the set() call is intended for in the first place. But if the elements x and y are not hashable, then the first test will still fail, and then the second test will be evaluated and will crash.

IMO this comparison of gens() is just a special case shortcut which is fine as-is, but it should account for the possibility that some elements just aren't hashable, that's all.

[John Cremona]

Does hashable imply not sortable? As I said, the lists of gens will be short normally, so something like testing sorted(s_gens)==sorted(o_gens) might work when the set() constructor does not? There will still be silly trivial cases such as first giving {x,y} as gens then {x,x,y} and for sure someone will then complain. But even as it is, we are not returning equality of the ideals (x,y) and (x+y,y). We are never going to catch all trivial cases (and different users' idea of a trivial case will differ) so just comparing the list of gens with no processing at all is safest and most easily explained.

[git]

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

​d47fbb3

wrap with try

[Frédéric Chapoton]

ok, I have wrapped with "try / except".

[Erik Bray]

Replying to cremona:

Does hashable imply not sortable? As I said, the lists of gens will be short normally, so something like testing sorted(s_gens)==sorted(o_gens) might work when the set() constructor does not? There will still be silly trivial cases such as first giving {x,y} as gens then {x,x,y} and for sure someone will then complain. But even as it is, we are not returning equality of the ideals (x,y) and (x+y,y). We are never going to catch all trivial cases (and different users' idea of a trivial case will differ) so just comparing the list of gens with no processing at all is safest and most easily explained.

I don't think sorting really helps here, because as I found out over in #25948, some elements can be "sorted" but the ordering is meaningless and unpredictable.

[Erik Bray]

Replying to git:

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

​d47fbb3

wrap with try

You could still get rid of (s_gens == o_gens). I don't think it's that useful.

[Frédéric Chapoton]

Replying to embray:

Replying to git:

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

​d47fbb3

wrap with try

You could still get rid of (s_gens == o_gens). I don't think it's that useful.

Yes it is. It will handle the precise case that we are dealing with in this ticket, hyperelliptic curves over p-adics.

[Erik Bray]

Ok, but I think as written you don't need the same_gens variable. I only wrote it that way in my pseudo-code because I wasn't directly looking at the real code at the time, and didn't consider using a try/except/else. I don't think we need to worry about rich_to_bool raising a TypeError.

[git]

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

​b129fc9

fix

[Frédéric Chapoton]

like that ?

---

New commits:

​b129fc9

fix

[Erik Bray]

Yeah, though it would be nice to squash all these commits before we're done here.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​f8687f7

py3: comparison cleanup for hyperelliptic curves

[Frédéric Chapoton]

done

[Erik Bray]

Thanks! Well, I'm happy, so you can set positive review if you want, or wait and see if anyone smarter than me wants to weigh on, though it seems like this approach meshes with what the other experts have said so far.

[John Cremona]

I am happy (and making no claims to be cleverer than anyone else ;))

I would never have thought of using rich_to_bool() though. Not clever enough!

[David Roe]

Seems good to me too!