Dynamical Systems On Berkovich Space

[gh-EnderWannabe]

The goal of this ticket is to implement basic dynamical system functionality on Berkovich space over Cp. The immediate goal is to be able to compute the image of points. Type I points can be computed with the existing functionality in Sage. Type II, III and IV points will require additional functionality, to be implemented in this ticket.

This ticket relies heavily on the implementation of Berkovich space in #29844.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​7ed429f	29844: trivial fix
​46259d4	Merge remote-tracking branch 'trac/develop' into t/29844/berkovich
​af306db	Merge branch 'u/saraedum/berkovich' of git://trac.sagemath.org/sage into 29844
​2dc9a45	Merge branch '29844' into berkovich_dynamical
​1633f58	29949: seperated systems over projective and affine space
​2e4a100	29844: can pass padic field to projective space
​40f6519	Merge branch '29844' into berkovich_dynamical
​3231a76	29844: base_ring for projective now returns a ring
​0a24e2c	Merge branch '29844' into berkovich_dynamical
​e1e7732	29949: image of Type II and III points under polynomial

[git]

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

​5434d54

29949: preliminary image of Type II points + constructor funcationality

[git]

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

​016f058

29949: added image of Type II for all cases

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​e77c1a1	29844: fixed git merge issues
​5130500	29949: restored some functionality after git crash
​b3ea2e8	29949: reformated classcalls, docs
​71c3fd3	29949: added image of type III points for number fields
​af1b339	29949: better check for poles in type III disk in padic case
​1327b8a	29844 doc clean-up
​eaa636c	29844: deleted duplicate functions. added element file to docs
​b72d178	29844: fixed parent error in involution map
​c1becdb	29844: minor doc updates
​d54619e	29949: Merge branch 'berkovich_revisions' into berkovich_dynamical

[git]

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

​67feb5d

29949: better check for poles in type III call

[git]

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

​2261e04

29499: added as_scheme_dynamical_system and fixed radius for image of type III

[git]

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

​db60890	29949: added 'ideal' parameter and fixed dehomogenize + homogenize
​b8f1a56	29949: better check for poles in image of type III

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​f640e21	29844: fixed minor spacing
​c83784a	29844: polynomial ring can be passed to projective Berkovich
​780b1d7	29844: added affine space as option for affine berk space
​2b5d6ed	29844: deleted trailing whitespace
​c5ced64	29844: added note in scheme overview
​32b42aa	29949: Merge branch 'berkovich_revisions' into berkovich_dynamical
​3887c4d	29949: Merge branch 'develop' into berkovich_dynamical
​691031d	29949: fixed broken constructor
​a15755f	29949: fixed conjugate
​72f9c7d	29949: more examples and new_ideal for conjugate

[git]

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

​18cddd9

29949: added resultant and base_ring methods

[bhutz]

btw, you should mark this as "needs-review" and mark yourself as author.

I'm still waiting for compilation to finish, but here is some comments from a code review. Nothing really major here.

need to include latest 29844

do you really needs the ._polys data as this is already contained in ._system._polys?

14: date needed

58: should give the class dynamical system

60: p-adic

95: affine dynamical system

98: z^{2 + 1}

129: extra words

172: You seem to be trapping the error in creating the dynamical system, which would give information about what failed in trying to create the dynamical system. It seems like that information would be helpful rather than the more generic error you display

175: What if you have a dynamical system defined on a subscheme?

_eq_, _neq_: you have no tests that check that two systems are unequal

256: ending .

274: need ' - '

319: shouldn't this be relative dimension? Although I'd be inclined to just say 'dimension' as the base will be a field anyway.

318: class of dynamical system

348: trailing whitespace

370: The error is really that you cannot convert not that you know it is affine. Again you masking the more specific initialization error. I can somewhat see why that might be desirable here.

374: the should be projective not affine

375: dimension_absolute. So this will fail for the base being say ZZ

379: p-adic

405: Perhaps name this as_projective_dynamical_system. There is no a cooresponding function for affine. Looking at the documentation, it seems like this function is meant to be in the class:DynamicalSystem_Berkovich. In which case the name is fine.

455: What is you scale by p?

473: extra line

505,512,536,538: ending .

524: , adjugate

The _call_ documentation isn't visible to the user, so these description need to be moved. Perhaps to the class documentation.

650: isn't that error that the point is not in the domain? again, why is the error message from the coercion not better?

652: backed

654: are defined

667: some spaces for readability

719: space - space

798: this input seems more descriptive than for the other classes

849: Wouldn't homogenization be faster that calling the constructor. Also, what if you get something other than affine/projective?

847: dimensional absolute again

874: projective Berkovich space?

898: , space

918: same as before

need blank line at the end

[bhutz]

A couple more things:

• 808+. An action is defined for matrices and dynamical systems: F = F*M

• for 808-816. Why don't you do

dehomogenize_hom = list((F*M).dehomogenize(1))

I didn't try it, but doesn't that do the same thing?

• 813: R.gen(0) also works

• 872 #add comment that this is the start of Type III

• Seems like this should work, since the equivalent call works for projective

  sage: Q.<z> = QQ[]
  sage: A.<a> = NumberField(z^3 + 20)
  sage: ideal = A.prime_above(3)
  sage: P.<z> = AffineSpace(A, 1)
  sage: B = Berkovich_Cp_Affine(A, ideal)
  sage: f = DynamicalSystem_Berkovich([z^2], domain=B)
  

• This is what I mean by the subscheme examples

  P.<x,y>=ProjectiveSpace(QQ,1)
  X=P.subscheme([x-y])
  f=DynamicalSystem_projective([x^2,y^2],domain=X)
  DynamicalSystem_Berkovich(f)
  

Your error message does not make any sense for this one.

• I also ran a few of Benedetto's image examples from his exercises and got all correct answers.

[git]

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

​1b06163	29949: fixed prime_above in conjugate
​ad55ab7	29949: added matrix imports
​ae3400a	Merge branch 'u/gh-EnderWannabe/berkovich' of git://trac.sagemath.org/sage into berkovich_number_field
​614bed9	29844: fixed equality for Berkovich Space, added hashing for Berkovich Space, and added test suites
​fbfef60	Merge branch 'berkovich_number_field' into berkovich_dynamical
​2f7742b	Merge branch 'develop' into berkovich_dynamical
​1523bd4	29949: fixed spacing issues, better error messages

[gh-EnderWannabe]

Replying to bhutz:

• This is what I mean by the subscheme examples

  P.<x,y>=ProjectiveSpace(QQ,1)
  X=P.subscheme([x-y])
  f=DynamicalSystem_projective([x^2,y^2],domain=X)
  DynamicalSystem_Berkovich(f)
  

Your error message does not make any sense for this one.

Changed the error message slightly. The key point I'm trying to get across is the is_ProjectiveSpace() must return True when called on the domain.

Replying to bhutz:

Deleted the ._polys data, switched to dimension relative. As for the error messages in the try catch blocks, in the notebook all the error messages are displayed, so the failed conversion error messages are still shown.

[bhutz]

A few things here:

• look at the pyflakes plug-in for unused imports

• you're missing a few doctests for coverage (the _init_ functions)

• typo in docs:

The image of type III points can be computed has long as the

• add reference in comment

 # point is now type III, so we compute using Proposition 7.6 [of Benedetto]

• I'm also getting no tab completion for Berkovich dynamical systems. I have no idea why the tab completion isn't working. Might have to ask sage-devel that one.

[git]

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

​65fd85b	Merge branch 'develop' into berkovich_number_field
​15a3b09	29844: fixed hash example
​228e2fe	29844: removed unused variables and added examples
​e7b86f7	29844: added examples
​6502126	29844: added more examples
​cb0b8c6	Merge branch 'berkovich_number_field' into berkovich_dynamical
​3878f2e	29949: added tests, removed unused imports
​42ed3ab	Merge branch 'u/gh-EnderWannabe/berkovich_dynamical' of git://trac.sagemath.org/sage into berkovich_dynamical

[gh-EnderWannabe]

Added the necessary examples for 100% coverage, and fixed the unused imports. I asked on the sage-devel google group about how to fix the auto complete.

[git]

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

​c274cbe	Merge branch 'develop' into berkovich_dynamical
​0cdf6e4	29949: fixed patchbot errors

[bhutz]

I went to check tab completion on the newest beta, and I'm sill getting no tab completion here.

Note that I had a doctest failure:

sage -t --warn-long 57.6 --random-seed=0 berkovich_ds.py  # 1 doctest failed

[git]

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

​94b3c8d

29949: fixed test failure caused by 30165

[mkoeppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

Setting a new milestone for this ticket based on a cursory review.

[bhutz]

Everything builds and passes for me.

The tab complete does seem to be purely a jupyter issue. For me, disabling jedi with

%config Completer.use_jedi = False

caused tab completion to function properly.