GSoC2018: Rational Point on Varieties
Hi, my name is Raghukul Raman and I have been working on implementing rational point algorithms for Schemes, in Google Summer of Code 2018. This gist describes all the work that I have done during this period.
SageMath uses the Trac issue tracking system for development, each ticket has a unique ticket number. These ticket contain all the important discussion regarding that issue, it also contain links to the commit. Apart from coding, I also reviewed some ticket of other SageMath developers. For convenience I have provides link to my commits, also current status of each ticket (including reviewed tickets) in following section.
Status of ticket which I commited to:
Status of tickets which I reviewed:
|Ticket No||Ticket Summary||Ticket Author/s||Status|
|#23627||Update points() in projective_homset.py and affine_homset.py to work over CC and CDF||Rebecca Lauren Miller, Raghukul Raman, Ben Hutz||Merged|
|#25156||multivariate power series rings don't always format latex properly||Brent Baccala, Raghukul Raman||Merged|
|#25242||is_polynomial fails when multiple roots||Ben Hutz||Merged|
|#25877||dehomogenize for projective morphism failure in number field order||Ben Hutz||Positive Reviewed|
|#25795||minor optimization in comparison between morphisms||Raghukul Raman, Travis Scrimshaw||Positive Reviewed|
First part of my project was to implement Doyle-Krumm Algorithm-4.
This algorithm provides an efficient method to compute all elements of a
number field (K) having realtive height at most B.
Initially in SageMath Algorithm-3 was implemented. So why implement algorithm-4?
A computer has a limited memory, hence there is a limit of storing data. Issues are due to the fact that in a computer we cannot work exactly with the real numbers that appear in the algorithm (heights of algebraic numbers, logarithms of real numbers, absolute values of algebraic numbers), so we must make do with rational approximations of them. Consider the following example:
sage: from sage.rings.number_field.bdd_height import bdd_height sage: K.<v> = NumberField(x^3 + x + 1) sage: bound = 3 sage: list(bdd_height(K,bound)) sage: for x in T: .... print x, e**(K(x).global_height()*K.degree()) 0 1.00000000000000 -v^2 + v - 1 2.14789903570479 v^2 - v + 1 2.14789903570479 v^2 + 1 1.46557123187677 -v^2 - 1 1.46557123187677 -1 1.00000000000000 1 1.00000000000000 -v 1.46557123187677 v 1.46557123187677 -v^2 2.14789903570479 v^2 2.14789903570479
If we change the bound to just 3.1 we get:
sage: from sage.rings.number_field.bdd_height import bdd_height sage: K.<v> = NumberField(x^3 + x + 1) sage: bound = 3.1 sage: list(bdd_height(K,bound)) sage: for x in T: .... print x, e**(K(x).global_height()*K.degree()) 0 1.00000000000000 -v^2 + v - 1 2.14789903570479 v^2 - v + 1 2.14789903570479 v^2 + 1 1.46557123187677 -v^2 - 1 1.46557123187677 -1 1.00000000000000 1 1.00000000000000 -v 1.46557123187677 v 1.46557123187677 -v^2 2.14789903570479 v^2 2.14789903570479 -2/3*v^2 + 1/3*v - 1/3 3.00000000000000 -v^2 + v 3.00000000000000 2/3*v^2 - 1/3*v + 1/3 3.00000000000000 v^2 - v 3.00000000000000 1/3*v^2 + 1/3*v + 2/3 3.00000000000000 -v + 1 3.00000000000000 -1/3*v^2 - 1/3*v - 2/3 3.00000000000000 v - 1 3.00000000000000 1/3*v^2 + 1/3*v - 1/3 3.00000000000000 -v^2 - 2 3.00000000000000 -1/3*v^2 - 1/3*v + 1/3 3.00000000000000 v^2 + 2 3.00000000000000
So with example above it is clear that we were missing points whose height were exactly 3 (due to approximation of real numbers). Algorithm-4 provides a convinent way of finding approximations that are good enough to gurantee correct results. To see my raw implementation click here.
Most of the work in this ticket was done by Rebecca Miller and Prof. Hutz. My work was to review this ticket and the only contribution I did was to remove duplicate points. Due to inaccurate results of groebner basis calculation, same projective points we treated different. So I added a tolerance parameter, if distance between any 2 points is less that tolerance, they are considered same.
dupl_points = list(rat_points) # rat_points is list of points containing duplicates for i in range(len(dupl_points)): u = dupl_points[i] for j in range(i+1, len(dupl_points)): v = dupl_points[j] if all([(u[k]-v[k]).abs() < 2**(-tol) for k in range(len(u))]): rat_points.remove(u) break
rat_points is a set, so complexity of this algorithm is O(n^2log(n)).
After #17008, projective spaces were made unique, but affine spaces were left untouched. As a result if we generate affine patches of a projective space, they were created as same object in memory. for example:
sage: PP = ProjectiveSpace(QQ,1) sage: AA = PP.affine_patch(0) sage: BB = PP.affine_patch(1) sage: AA is BB True
There were 2 possible fixes for this:
- Remove UniqueRepresentation from projective spaces and handle #17008.
- Add UniqueRepresentation for affine spaces, taking into account patching and embedding between affine and projective spaces.
So I took the added second solution, also Peter suggested on adding embedding_index and ambient projective space as construction parameter for affine spaces. So now since two affine patches had different embedding index, they were different object in memory. I also fixed the patching and embeddinng taking into account Uniqueness of both spaces.
I also modified generator names for affine patches.
sage: P = ProjectiveSpace(QQ, 2, "xyz") sage: A = P.affine_patch(0) sage: A.gens() (y, z)
Second part of my GSoC project was to replace search enumeration for subschemes, with much faster Sieve Algorithm. Basic idea behind the algorithm is to search points modulo prime and use chinese remainder theorem to reconstruct the rational points. Suppose Y is a subscheme defined as follows:
P.<x,y,z,q>=ProjectiveSpace(QQ,3) Y=P.subscheme([x^2-3^2*y^2+z*q,x+z+4*q]) bound = 10
Then to find rational points on Y we first find point modulo primes (2, 3, 7, 11),
can be done using existing rational point finding function. Then we lift all
possible combination of these point modulo prime and check that they belong to
given subscheme (Y). Now how is this fast?
Naturally this algorithm uses the existing rational_points(), but this time the search space is significantly reduced. Second major improvement of this algorithm, is that computation of points modulo prime is done in parallel. hence the complexity is:
T(n) = max(T(pi)) for i ∊ (1,k) (since computation related to each prime is done in parallel)
Other thing that can be done, is that computation related to each several combinations
can be distributed evenly over other threads.
First of all for correctness of algorithm, sufficient primes need to be present, whose product should be greater that bound, which is given by: (reference: Preperiodic Points):
bound = 2(N/4+1)*given_bound2*√(N+1).
Now a better choice of these primes would significantly affect the efficiency of algorithm.
So to find a list of prime which would give rational points taking least amount of time,
I implemented the
Main idea behind the function is to check for all valid list of primes, and return which one takes least amount of time. Time complexity used to return best prime list is:
T = (N^2*P_max^N) + N^5*(α^dim_scheme/P_max) # α is product of all primes, P_max is largest prime in list and N is the dimension of Ambient Space
For more details of algorithm implementation visit - Sieve.
__hash__() function was not implemented for affine points in SageMath.
Adding hash function for affine points was quite easy compared to for projective
points, where (1,3,6) is same point as (3,9,18). It was simply to hash the list
of coordinates in that affine point.
sage: A.<x,y> = AffineSpace(QQ, 2) sage: X = A.subscheme(x - y) sage: hash(X([1, 1])) 1300952125 # 32-bit 3713081631935493181 # 64-bit
enum_affine_rational_field() function was missing points, for example
sage: A.<x,y> = AffineSpace(2, QQ) sage: C = Curve(x^2+y-x) sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field sage: enum_affine_rational_field(C, 10) [(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]
You can see that some points of height 9 ((-2/3, -10/9),(-1/3, -4/9),(1/3, 2/9),(2/3, 2/9)) and height 4 ((-1/2, -3/4),(1/2, 1/4),(3/2, -3/4)) which clearly lie on the subscheme are missing. This bug was due to the way they were generating rational numbers and eventually trying it on the subscheme. Actually the code was quite old and after that
range_by_height() function was implemented which could do the same thing correctly. So I used this new function and corrected this bug.
sage: A.<x,y> = AffineSpace(2, QQ) sage: C = Curve(x^2+y-x) sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field sage: enum_affine_rational_field(C, 10) [(-2, -6), (-1, -2), (-2/3, -10/9), (-1/2, -3/4), (-1/3, -4/9), (0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0), (3/2, -3/4),(2, -2), (3, -6)]
Earlier for finding rational point on product projective spaces defined over rational field,
they were using
points_of_bounded_height() function, and trying each point if it belongs to
the given subscheme. Firstly I added a new file for all rational point finding algorithms,
just like in projective and affine spaces, to make things universal. Secondly I made other
modification, similar to ones present in projective_rational_field.py.
Lastly I implemented
enum_product_projective_rational_field() , in which rational points for each component
of ambient space were found, and then all points from the cartesian product of result were
checked if they belong to given subscheme. I have also corrected
function for product projective spaces. This function returns an iterator of the rational
points on given space, but the implementation was memory inefficient, so I changed it
to pure iterator based implementation. The following examples works now:
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2,1], QQ) sage: X = PP.subscheme([x^2 + x*y + y*z, u*u-v*u]) sage: from sage.schemes.product_projective.rational_point import \ enum_product_projective_rational_field sage: enum_product_projective_rational_field(X,4) [(-2 : 4 : 1 , 0 : 1), (-2 : 4 : 1 , 1 : 1), (-1 : 1 : 0 , 0 : 1), (-1 : 1 : 0 , 1 : 1), (-2/3 : -4/3 : 1 , 0 : 1), (-2/3 : -4/3 : 1 , 1 : 1), (-1/2 : -1/2 : 1 , 0 : 1), (-1/2 : -1/2 : 1 , 1 : 1), (0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 1), (0 : 1 : 0 , 0 : 1), (0 : 1 : 0 , 1 : 1), (1 : -1/2 : 1 , 0 : 1), (1 : -1/2 : 1 , 1 : 1)]
Sieve algorithm for product is almost similar to that for projective spaces, but while constructing points
from modulo points, we use
LLL reduction on each component and then combine all components. Apart from that
complexity is also different (used in
good_primes function). Consider:
sage: from sage.schemes.product_projective.rational_point import sieve sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2,1], QQ) sage: X = PP.subscheme([x^2 + y^2 - x*z, u*u-v*u]) sage: sieve(X,2) [(0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 1), (1/2 : -1/2 : 1 , 0 : 1), (1/2 : -1/2 : 1 , 1 : 1), (1/2 : 1/2 : 1 , 0 : 1), (1/2 : 1/2 : 1 , 1 : 1), (1 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 1 : 1)]
points_of_bounded_height() function for projective spaces gave different results
enum_projective_rational_point(). Consider this example:
sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: print sorted(list(P.points_of_bounded_height(bound=3))) [(-2 : 1), (-1 : 1), (-1/2 : 1), (0 : 1), (1/2 : 1), (1 : 0), (1 : 1), (2 : 1)]
Now you can see that some points of height 3 are not present. Apart from missing point, it returned some point whose height were greater than bound. To correct this I replaced the algorithm for QQ with this:
zero = (0,) * (n+1) for c in cartesian_product_iterator([srange(-B,B+1) for _ in range(n+1)]): if gcd(c) == 1 and c > zero: yield self(c)
How do these condition come from?? Well
gcd(c) = 1 is necessary to prevent
k*c (since they all are same projective points) adding to list, and
c > zero
ensures the case when
k = 1,-1.
There was no comparison operator between morphisms, for product projective spaces.
Now how do we check if 2 morphism are equal for projective spaces?
H1: (x0, x1,...,xn) → (y0, y1,...,yn) and H2: (x0, x1,...,xn) → (z0, z1,...,zn) be 2 morphism defined from Projective space of dimension n to itself.
So they are same projective morphisms if:
y0/z0 = x1/z1 = ... = yn/zn # (since P, c*P are same projective points)
Now for product projective space, we just need to check this condition for each projective component. So now you can run this example:
sage: PP.<x,y,z,a,b> = ProductProjectiveSpaces([2,1], ZZ) sage: H = End(PP) sage: f = H([x^2*y*z, x*y^2*z, x*y*z^2, a^2, b^2]) sage: g = H([x, y, z, a^3, a*b^2]) sage: f == g True sage: f != g False
Purpose of dehomogenize() function is to remove the kth coordinate of given
projective point and convert it into a point of kth affine_patch. But product
projective point is formed by some projective points. For products this function
take a list(L) as input where L[i] is the coordinate we need to dehomogenize from
ith projective point component. So we append result of each projective point
component and then finally convert them back into
sage: PP = ProductProjectiveSpaces([2, 2, 2], QQ, 'x') sage: A = PP([2, 4, 6, 23, 46, 23, 9, 3, 1]) sage: A.dehomogenize([0, 1, 2]) (2, 3, 1/2, 1/2, 9, 3)
Comparison Function between morphism was creating list of coordinates and then doing comparison. Instead of creating list we could direclty compare using the tuple of coordinates.
sage: P = ProjectiveSpace(QQ,2000,'x') sage: H = End(P); L = P.gens(); LL =  sage: for i in L: ..... LL.append(i*i) sage: f = H(L); g = H(LL) %timeit sage: f == g False CPU time: 10.83 s, Wall time: 10.83 s %timeit sage: updated__eq__(f,g) False CPU time: 0.00 s, Wall time: 0.00 s
local_height function for product projective points. We basically find max of global_height/local_height over each component. For computing
local_height we also need to pass in any ideal of the base ring. This works in SageMath now:
sage: PP = ProductProjectiveSpaces(QQ, [2,2], 'x') sage: Q = PP([1, 7, 5, 18, 2, 3]) sage: Q.global_height() 1.94591014905531 sage: P = ProductProjectiveSpaces(QQ, [1,2], 'x') sage: Q = P([1, 4, 1/2, 2, 32]) sage: Q.local_height(2) 4.15888308335967
Height functions for morphisms are defined as follows:
local_height: maximum of the local height of the coefficients in any of the coordinate functions of this map.
global_height: maximum of the absolute logarithmic heights of the coefficients in any of the coordinate functions of this map.
So we simply enumerate over coefficent of all the polynomials and compute heights, and finally return max. However,
if base ring is
QQbar, height functions are not present for coefficients. In that we have to convert this map, to a
map defined over Number Field. This has been left as an TODO. An example of this fix is:
sage: u = QQ['u'].0 sage: R = NumberField(u^2 - 2, 'v') sage: PP.<x,y,a,b> = ProductProjectiveSpaces([1, 1], R) sage: H = End(PP) sage: O = R.maximal_order() sage: g = H([3*O(u)*x^2, 13*x*y, 7*a*y, 5*b*x + O(u)*a*y]) sage: g.global_height() 2.56494935746154 sage: R.<z> = PolynomialRing(QQ) sage: K.<w> = NumberField(z^2-5) sage: P.<x,y,a,b> = ProductProjectiveSpaces([1, 1], K) sage: H = Hom(P,P) sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2, a^2, 6*b^2 + 1/9*a*b]) sage: f.local_height(K.ideal(3)) 2.19722457733622
We can define mappings from projective space to affine space using
To generate a projective_embedding, we pass in the embedding index, this index represents the coordinate,
which we add in to generate projective space. So this index, should be greater than 0 and should be less than or equal to,
the dimension of affine scheme. This check was buggy in projective_embedding function, which I fixed:
sage: A.<x,y> = AffineSpace(ZZ, 2) sage: A.projective_embedding(4) Traceback (most recent call last): ... ValueError: argument i (=4) must be between 0 and 2, inclusive