Opened 3 years ago
Closed 2 years ago
#19635 closed enhancement (fixed)
Enumeration functionality for products of projective spaces over fields and finite fields
Reported by:  gjorgenson  Owned by:  

Priority:  minor  Milestone:  sage7.3 
Component:  algebraic geometry  Keywords:  
Cc:  bhutz  Merged in:  
Authors:  Grayson Jorgenson  Reviewers:  Ben Hutz 
Report Upstream:  N/A  Work issues:  
Branch:  be82af5 (Commits)  Commit:  be82af546be820168c9b464e273fb32cfc8dbe16 
Dependencies:  Stopgaps: 
Description (last modified by )
Implement enumeration functionality for products of projective spaces over fields/finite fields. Some basic class structure will need to be added to enable working specifically with such product spaces.
Also will make points of products of projective spaces hashable to increase the efficiency of the enumeration functionality. There is currently a bug with the hash implementations for points of projective spaces
P2.<x,y,z>=ProjectiveSpace(QQbar,2) H=End(P2) f=H([x^2+2*y^2,y^22*x^2,z^2]) X=P2.subscheme([x+y]) f.rational_preimages(P2([0,1,1])) # there are equal points, but with different hash values
so this will be addressed here as well. Some examples in projective_morphism need to be updated because of reordering in their outputs from the hash changes.
Change History (21)
comment:1 Changed 3 years ago by
 Branch set to u/gjorgenson/ticket/19635
comment:2 Changed 3 years ago by
 Description modified (diff)
 Milestone changed from sage6.10 to sage7.1
 Summary changed from Products of projective spaces over fields and finite fields to Enumeration functionality for products of projective spaces over fields and finite fields
comment:3 Changed 3 years ago by
 Commit set to 449d0738bfd0d9e85deea38f5ff94cdf1356e745
comment:4 Changed 3 years ago by
 Status changed from new to needs_review
comment:5 Changed 3 years ago by
 Reviewers set to Ben Hutz
 Status changed from needs_review to needs_work
A few minors things and a few larger things. I think before I do functionality testing some of these coding improvements should be made.
Minor
 You are changing the example in the ring class to give a field point. Perhaps you should change the ring instead of the output so that it is still a ring example
 <class 'sage.schemes.product_projective.point.ProductProjectiveSpaces_point_ring'> + <class 'sage.schemes.product_projective.point.ProductProjectiveSpaces_point_field'>
 no ending punctuation on errors
 in both of your brute force point searches you don't need the intermediate variable. You can just have:
for P in X.ambient_space().rational_points():
 remove references to self in doc strings. Instead say things like, this point, or this space.
 there are a number of spacing issues
 between lists of polys, such as [x^{2y}2, z^{22*w}2]
 between function parameters, such as X.affine_patch(I, True)
 between coordinates of points
 references need an underscore, [DoyleKrumm]_
points function
 you should mention that this is computing all points for dimension 0 schemes regardless of base ring.
 output: should be "all points with height at most the bound are returned"
 you might need to .sort() your doctests so that they don't depend on the machine
Major
 with 7.1.beta1 there is a depracation in one of the function you use which causes a number of doctest failures, you should update to the new function.
sage t warnlong 56.2 schemes/generic/algebraic_scheme.py # 1 doctest failed sage t warnlong 56.2 schemes/product_projective/wehlerK3.py # 1 doctest failed sage t warnlong 56.2 schemes/projective/projective_space.py # 1 doctest failed sage t warnlong 56.2 schemes/product_projective/space.py # 1 doctest failed sage t warnlong 56.2 schemes/product_projective/morphism.py # 1 doctest failed sage t warnlong 56.2 schemes/product_projective/point.py # 1 doctest failed sage t warnlong 56.2 schemes/product_projective/homset.py # 1 doctest failed
points function
 your dimension check doesn't actually work in all cases. I'd recommend using #19991 as a dependency and switching to that dimension call.
 You seem to be going to a lot of trouble to generate possible indices for affine patches. Here is something shorter
dims = [n+1 for n in X.ambient_space().dimension_relative_components()] for I in xmrange(dims):
You should be able to improve the indices creator in points_of_bounded_height and rational_points as well
 You are returning points in the ambient space instead of the scheme. You should just have
points.add(phi(PP))
 hash()  for field, finite_field are exactly the same as for ring. So you don't need these functions as they will inherit, but add their examples to ring.
 The example changes in SchemeMorphism_polynomial_projective_space_field. why did they change? Are these are sorting problem?
 in rationalpoints()
I don't like that you are generating all the points in all the projective spaces and then generating all possible indices. We can do this more memory efficiently; something like:
P = ProductProjectiveSpaces([1,1,1,2],GF(2)) iters = [ iter(T) for T in P ] L=[] for x in iters: L.append(next(x)) # put at zero points=[P(L)] j = 0 while j < P.num_components(): try: L[j] = next(iters[j]) points.append(copy(P(L))) j = 0 except StopIteration: iters[j] = iter(P[j]) # reset L[j] = next(iters[j]) # put at zero j += 1
speaking of which. I'd like to see this as an _iter_ function in addition to a rational_points function.
comment:6 Changed 3 years ago by
The need to use copy() in the iterator was bothering me and this seems to be a more endemic problem. If you initialize a point with a list of points and change an element of that list, it changes the point. This does not seem like desired behavior as it does not behave like that for regular projective points.
P=ProjectiveSpace(1,QQ,'x') P2=ProjectiveSpace(1,QQ,'y') PP=P2.cartesian_product(P) Q=P(1,1) Q2=P2(0,1) M=[Q2,Q] QQ=PP(M);print QQ M[0]=P2([1,1]);print QQ
Consequently, I think the copy needs to be somewhere in _init_
comment:7 Changed 2 years ago by
 Commit changed from 449d0738bfd0d9e85deea38f5ff94cdf1356e745 to edfad27dd51435adf0d65e6e7e6c005c78ed7993
comment:8 Changed 2 years ago by
 Milestone changed from sage7.1 to sage7.3
 Status changed from needs_work to needs_review
Okay I think I addressed the issues from the review and this should be ready for another lookover.
I added an __iter__
function for product spaces over finite fields making them iterable, but this broke a line in the dimension function for products in algebraic_scheme.py
. In particular something like for PS in PP
will now yield points instead of the component spaces of PP.
I'm not exactly sure why the output of some of the projective morphism examples got rearranged, but it definitely happened when I changed the hash function. I think it might be related to the use of set()
.
I also tried to address the copy issue for the __iter__
and rational_points
functions. Just adding polys = copy(polys)
to the first line of the __init__
function for points of products of projective spaces seems to fix the issue. However, I think a list of projective space points is just a list of references to the actual point objects, so applying copy to such a list just copies the references but not the objects themselves. So this fix does not protect against modification of the original objects:
P1=ProjectiveSpace(1,QQ,'x') P2=ProjectiveSpace(1,QQ,'y') PP=P1.cartesian_product(P2) Q1=P1(0,1) Q2=P2(1,1) M=[Q1,Q2] pt=PP(M) print pt M[0]._coords[0] = 1 print pt
Also, copy for projective space points seems to be too shallow:
P.<x,y,z> = ProjectiveSpace(QQ,2) pt1 = P(1,2,3) pt2 = copy(pt1) print pt2 pt1._coords[0] = 7 print pt2
I'm not sure whether this really is an issue, but if it is, maybe we could override __copy__
for projective points.
comment:9 followup: ↓ 11 Changed 2 years ago by
Why did you duplicate the code in __iter__
and rational_points
? You can simply do
def rational_points(self): return list(self)
comment:10 Changed 2 years ago by
 Commit changed from edfad27dd51435adf0d65e6e7e6c005c78ed7993 to 266218f68838ce9449a4f549fb16f6990e0bd5af
Branch pushed to git repo; I updated commit sha1. New commits:
266218f  19635: improved rational_points function

comment:11 in reply to: ↑ 9 Changed 2 years ago by
Replying to vdelecroix:
Why did you duplicate the code in
__iter__
andrational_points
? You can simply dodef rational_points(self): return list(self)
I wasn't thinking; list(self)
should have the same efficiency and is much cleaner. I also added the ability to find the rational points over extensions of the base field.
comment:12 Changed 2 years ago by
 Status changed from needs_review to needs_work
Much improved. Still a couple things.
1) I agree that _copy_ is not a fault of your code. This is a more general schemes issue
E=EllipticCurve([0,0,0,0,1]) T=E(0,1) S=copy(T) T._coords[1]=1 T,S ((0 : 1 : 1), (0 : 1 : 1))
What you are doing is in line with everything else, so will be resolved when the more general generic/point copy is resolved.
2) hmmm...yes, a natural consequence of the point iterator is that iterating over the components with that simple loop no longer works. However, I think it is import to preserve a way to iterate over the components without resorting to accessing private data with ._components. I suggest a simple PP.components() function that returns ._components.
3) This documentation for rational_points() does not look right
rational points on the affine space of this space
4) base extension for rational_points() doesn't seem to work
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],QQ) X = PP.subscheme([u^3v^3, s^2t^2,]) X.dimension() X.rational_points(bound=2,F= CyclotomicField(3,'z'))
versus
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1], CyclotomicField(3,'z')) X = PP.subscheme([u^3v^3, s^2t^2,]) X.dimension() X.rational_points()
or the same for QQbar...
5) shouldn't this work too
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],GF(3)) X = PP.subscheme([u^3v^3, s^2t^2,]) X.dimension() X.rational_points()
or
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],GF(3)) X = PP.subscheme([u^3v^3, s^2t^2,]) X.dimension() X.rational_points(F=GF(9,'z'))
comment:13 Changed 2 years ago by
 Commit changed from 266218f68838ce9449a4f549fb16f6990e0bd5af to 94c689bd21966bab1057f22d8dab1eeaec85314e
Branch pushed to git repo; I updated commit sha1. New commits:
94c689b  19635: Changes from second review

comment:14 Changed 2 years ago by
 Status changed from needs_work to needs_review
I think I've resolved everything from the review, but I don't know whether my changes are really the right ways to address the issues in 4) and 5).
For the base extension issues in 4), the problem actually seemed to be from rational_points
in algebraic_scheme.py
for subschemes. The same issue can be seen with projective spaces:
P.<u,v> = ProjectiveSpace(CyclotomicField(3,'z'),1) X = P.subscheme([u^3v^3]) X.rational_points()
vs
P.<u,v> = ProjectiveSpace(QQ,1) X = P.subscheme([u^3v^3]) X.rational_points(F=CyclotomicField(3,'z'))
This seemed to be happening because of line 1520 of algebraic_scheme.py
which sets X = self(F)
, but the codomain of the resulting homset is still the original subscheme, and not one over the extension. I tried changing this to X = self.base_extend(F)(F)
so that the codomain is defined over the extension. Is this what we actually want in order to find the rational points over the extension?
The issue in 5) seemed to be related to a coercion problem. There is some coercion in place when initializing projective points from lists, but not for initializing points of products of projective spaces. Things like
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],GF(3)) X = PP.subscheme([u^3v^3, s^2t^2,]) l = [QQ(1) for k in range(4)] Q = X(l)
don't work. The problem causing the issues with finite fields seemed to come from this, though I don't know why a list of rational numbers is being used to initialize a point. I think this list came about from __fast_eval
for affine morphisms. I added something similar to what's done for projective points, and this seemed to fix the problem.
comment:15 Changed 2 years ago by
I think the fix for 4 is the right thing to do. I don't fully understand what issue you are describing for (5). _fast_eval will only be called for evaluation maps and there are no maps here. I see the issue you are pointing out in the example above, and you've fixed that issue. Could you explain where this list of rational numbers is showing up in the code in my examples (4)?
comment:16 Changed 2 years ago by
The coercion problem comes up in the example
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],GF(3)) X = PP.subscheme([u^3v^3, s^2t^2,]) X.dimension() X.rational_points()
when using the affine morphisms corresponding to the affine patches used for finding points. In particular, a rational number list shows up when evaluating one of the affine morphisms at the rational points from the corresponding patch. The following traces some of the code in rational_points
for the above example to capture when the list shows up:
PP.<u,v,s,t> = ProductProjectiveSpaces([1,1],GF(3)) X = PP.subscheme([u^3v^3, s^2t^2,]) X = X(GF(3)).codomain() [Y,phi] = X.affine_patch([0,0], True) x = Y.rational_points()[0] # expanding out what happens in phi(x): print [x._coords[i].parent() for i in range(len(x._coords))] # coordinates are finite field elements P = phi._fast_eval(x._coords) print [P[i].parent() for i in range(len(P))] # resulting coordinates are rational field elements print phi.codomain().point(P, True) # this then breaks without the coercion implemented in last commit
Adding the coercion got around the issue, but I'm not sure why _fast_eval
gives a list of rational numbers here.
comment:17 Changed 2 years ago by
ok. _fast_eval is returning a list defined over a ring based on what will be most quickly computed. Then the coordinates are coerced when it is made into a point. However, for subschemes, check_satisfies_equations was being called before the coordinates were coerced, causing the error. So your fix does solve the problem, by coercing first.
The functionality checkouts to me. There are just a couple very minor things:
 functions needs one line desctiption and then more details. There are several where the docs start with a paragraph.
 wrap the super long imports (see other files for examples)
 needs space before ZZ.
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1,1],ZZ)
comment:18 Changed 2 years ago by
 Commit changed from 94c689bd21966bab1057f22d8dab1eeaec85314e to be82af546be820168c9b464e273fb32cfc8dbe16
Branch pushed to git repo; I updated commit sha1. New commits:
be82af5  19635: Minor documentation and spacing fixes.

comment:19 Changed 2 years ago by
Thanks. I also removed a couple lines in homset.py
that are no longer used with the new dimension check in place.
comment:20 Changed 2 years ago by
 Status changed from needs_review to positive_review
comment:21 Changed 2 years ago by
 Branch changed from u/gjorgenson/ticket/19635 to be82af546be820168c9b464e273fb32cfc8dbe16
 Resolution set to fixed
 Status changed from positive_review to closed
Branch pushed to git repo; I updated commit sha1. New commits:
19635: FIrst pass at implementation