Implementing Wells' Algorithm

[rlmiller]

Implementing Wells' Algorithm for GSOC 2017. A faster way to solve for canonical height in QQ and ZZ because you don't need to factor the resultant. This algorithm is found in Elliot Wells' Paper "Computing the Canonical Height of a Point in Projective Space"

[git]

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

​3c128bb

23334 updates, added error bound

[rlmiller]

Fixed test errors so we are ready to review. Wondering if we should take out a new ticket so we can change all the prec to 53.

[chapoton]

doc does not build:

+[dochtml] OSError: [schemes  ] /local/sage-patchbot/sage/local/lib/python2.7/site-packages/sage/schemes/projective/projective_point.py:
docstring of sage.schemes.projective.projective_point.SchemeMorphism_point_projective_ring.canonical_height:39:
 WARNING: Explicit markup ends without a blank line; unexpected unindent.

[bhutz]

A took a first look and here are a few initial comments:

• also change the description of the algorithm in canonical_height in projective_morphism

• move reference to reference file and add citations here

• 1031: line too long

• a bunch of your earlier work is to get integer coefficients and coordinates. I'd recommend using the built in functions instead
  • f.normalize_coordinates() - removes gcd and clears denominators
  • Q.normalize_coordinates() - removes gcd
  • Q.clear_denominators() - clears denominators

• 'not err == None' should be 'not err is None'

• Height_I - lower case

• you do a lot of extraneous variable assignment for example
  • P = self
  • f = F
  • A_0, B_0

you don't need all these extra variables floating around

• and something not minor: The error bound calculation seems to be failing. I put in Wells example 4.4 as a test (which I recommend you do as well) and got the appropriate value when assigning N but not when assigning err.

RSA768 = 1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413
P.<x,y>=ProjectiveSpace(QQ,1)
H=End(P)
f=H([RSA768*x^2+y^2,x*y])
Q=P(RSA768,1)
Q.canonical_height(f, err=0.00000000000000001)
930.66293109982349403319550850
Q.canonical_height(f, N=50)
931.18256422718194018675677246

same was true for simple functions as well.

[bhutz]

never mind about the error bound issue. When I use the correct parameter it works as expected.

[git]

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

​09692fa

22334 simplified code, added example

[rlmiller]

Added requested example and simplified code.

---

New commits:

​c525086

Fixed call to self.clear_denominators() to check that the point is

[bhutz]

The functionality works fine in tests and the notebook. However, I have a couple minor suggestions and a part I don't understand why it works.

• add citation to projective morphism canonical_height

• shouldn't this part go under ALGORITHM?

  If function is defined over ``QQ`` uses Wells Algorithm, which allows us to
          not have to factor the resultant.
  

• Does this line really need .abs()?

  1154: H = H + R(g).abs().log()/(d**(n+1))
  

• line 1145 Res.abs() is better than abs(Res)

• Note sure if it is better, but couldn't you do

  N = ceil(R(abs(Res)).log().log(d) - R(d-1).log(d) - R(err).log(d))
  

• lines 1157-8: you really seem to be computing

  h = self.green_function(F, 0 , **kwds) - H + R(t).log()
  

I'm also slightly confused by why this is right. Doesn't Wells' paper (proposition 2.3) have

h = self.global_height() - self.green_function(F,0) - H

I assume that R(t).log() is the correction for the function coordinate normalization.

[rlmiller]

The reason our formula looks a bit different is due to the difference between what Wells calls H_infty and what greens function actually returns for the infinite place.

Left * N = ceil(R(abs(Res)).log().log(d) - R(d-1).log(d) - R(err).log(d)) as is because that way it will look more like Well's Algorithm.

---

New commits:

​06a23c1

23334 Made updates to code, added reference. Left most as is

[bhutz]

You should still condense those two lines down to one (1157-8) and perhaps you should add a comment about the difference in notation.

[git]

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

​402a906

23334 condensed code and added description

[chapoton]

doc still does not build :

sage.schemes.projective.projective_point.SchemeMorphism_point_projective_ring.canonical_height:79: WARNING: Literal block expected; none found.

probably because of the RSA768 line

Note that in the same function, you should not indent the content of the ALGORITHM section.

EDIT:

Something else: in the reference file, the link to arxiv should be written

:arxiv:`1602.04920v1`

[bhutz]

also when you combined 1157,1158 you left in the redundant term (you are both adding and subtracting this term)

R(self[1].abs()).log()

[git]

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

​684b599

23334 fixed documentaion

[chapoton]

Several typos here:

Looks diffrent than Well's Algorithm because of the diffrence
...
for the infite place

Also range(0, N) should be range(N)

This

+            h = self.green_function(F, 0 , **kwds) - H + R(t).log()
+            return h

can be made in one line (no need to store h)

[git]

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

​695b96d

23334 fixed typos and cleaned up code

[bhutz]

docs build and all tests pass. I this is ready.

[rlmiller]

Need to make sure all "Wells'", are in the correct form. Just quick typo fixes.

[git]

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

​47c1957

23334 fixed to Wells'