Sigma invariants for P^N, N > 1

[gh-EnderWannabe]

Currently, the sigma invariants method for dynamical systems on projective space is not implemented when the projective space has dimension greater than 1. Using the algorithm described in ​https://arxiv.org/pdf/1908.03184.pdf, this ticket implements that functionality.

[git]

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

​e9281ff	31944: added flatenning for periodic, began work on preperiodic
​1d50748	31944: working minimal periodic and preperiodic
​27888da	31954: initial commit with working code and examples
​22201d7	Merge branch 'hyperplane_transform' into periodic_points_improvement
​c324844	31944: added computation for formal periodic points
​91ebb22	31944: removed changes in tickets 31906 and 31896 from this ticket
​df038ca	31944: initial formal preperiodic points
​5d142bf	31944: examples for formal preperiodic points, fixed return scheme for formal preperiodic points
​400a5b1	Merge branch 'periodic_points_improvement' into sigma_invariants
​8b3ea6a	31994: added formal parameter

[git]

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

​39e1987	Merge branch 'u/gh-EnderWannabe/sigma_invariants' of git://trac.sagemath.org/sage into sigma_invariants
​abedd59	31994: updated documentation and list return

[git]

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

​d1476d8	31954: added sanity check with error message
​27dea94	31954: added examples for finite fields and polynomial rings
​312b538	Merge branch 'u/gh-EnderWannabe/hyperplane_transform' of trac.sagemath.org:sage into hyperplane_transform
​b31b634	Merge branch 'hyperplane_transform' into periodic_points_improvement
​50179a3	31944: cleaned up code, added example
​aaecd46	31944: cleaned up code added comments
​a6656ca	31944: fix for subschemes, added function field error message
​b4ec7c1	31944: minor fixes
​6b810a2	31944: fixed periodic typo and added test
​bd287f0	Merge branch 'periodic_points_improvement' into sigma_invariants

[bhutz]

I pushed a branch with a few comments in the main algorithm. Feel free to add some more.

Here are the issues I've found:

• I don't think putting the degrees as the first two coordinates is appropriate since unless you understand the full form of the sigma invariant polynomials it's not that helpful. If you do know the full form, then then information is not needed.

• need examples for
  • formal=True
  • finite fields
  • polynomial rings

• cycle is ignored for dimension > 1, but seems like a not implemented error should be thrown instead of simply ignoring the parameter. Also, should mention in the INPUT documentation that it is only for dim=1.

• You need to fix the signs as you pull off the coefficients in

  sigmas.append(sigma_dictionary.pop(w**i*t**j, 0))
  

• Formal doesn't seem to work as these both give the same result

  P.<x,y,z>=ProjectiveSpace(QQ,2)
  f=DynamicalSystem([x^2-z^2,y^2,z^2])
  f.sigma_invariants(2,return_polynomial=True,formal = False)
  f.sigma_invariants(2,return_polynomial=True,formal = True)
  

• random error

P.<x,y,z>=ProjectiveSpace(QQbar,2)
f=DynamicalSystem([x^2-z^2,y^2,z^2])
f.sigma_invariants(2,return_polynomial=True,formal = False)

• shouldn't these work?

  R.<c>=QQ[]
  P.<x,y>=ProjectiveSpace(R,1)
  f=DynamicalSystem([x^2+c*y^2,y^2])
  f.sigma_invariants(1,return_polynomial=True)
  
  R.<c,d>=QQ[]
  P.<x,y,z>=ProjectiveSpace(R,2)
  f=DynamicalSystem([x^2+c*z^2,y^2+d*z^2,z^2])
  f.sigma_invariants(1,return_polynomial=True)
  
  R.<c,d>=QQ[]
  P.<x,y,z>=ProjectiveSpace(FractionField(R),2)
  f=DynamicalSystem([x^2+c*z^2,y^2+d*z^2,z^2])
  f.sigma_invariants(1,return_polynomial=True)
  

• This is wrong

  P.<x,y>=ProjectiveSpace(QQ,1)
  f=DynamicalSystem([x^2+3*y^2,x*y])
  f.sigma_invariants(1,return_polynomial=True, chow=True)
  

[git]

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

​e49e34c	Merge branch 'point_transformation_improvement' into periodic_points_improvement
​3a04a8d	31944: fixed bug with formal=True over ring
​4a62742	Merge branch 'develop' into periodic_points_improvement
​af1b86b	Merge branch 'periodic_points_improvement' into sigma_invariants
​89a417d	31994 - added some comments to code
​9212723	Merge branch 'u/bhutz/sigma_invariants' of trac.sagemath.org:sage into sigma_invariants
​2be2b66	31994: added flattening
​e3693f0	31944: added error message for fraction fields of m polynomial rings
​33c6b68	Merge branch 'periodic_points_improvement' into sigma_invariants
​a410756	31994: added support for fraction fields of polynomial rings

[gh-EnderWannabe]

Pushed a change fixing the random QQbar issue, and fixing the signs.

I also added flattening and unflattening for the families with a parameter - those should work now.

There does seem to be a big bug in the code. This one is correct:

R.<c>=QQ[]
P.<x,y>=ProjectiveSpace(R,1)
f=DynamicalSystem([x^2+c*y^2,y^2])
f.sigma_invariants(1,return_polynomial=True,chow=True)
w^3 + (-3)*w^2*t + 2*w^2 + 3*w*t^2 + (-4)*w*t + 4*c*w - t^3 + 2*t^2 + (-4*c)*t

But this fails:

R.<c>=QQ[]
P.<x,y>=ProjectiveSpace(FractionField(R),1)
f=DynamicalSystem([x^2+c*y^2,y^2])
f.sigma_invariants(1,return_polynomial=True, chow=True)
w - t

Not sure what is going on there.

As for putting the degrees in the first two coordinates, I did that so that the full form of the polynomial could be specified. I think the two degrees plus the list of coefficients with 0's is enough to fully specify the polynomial, so that doing a check of

f.sigma_invariants(1) == g.sigma_invariants(1)

actually tells us if f and g are not conjugate.

[git]

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

​97c33ff	31994: added examples, simpler code for polynomial rings
​d220330	31994: fixed n > 1

[gh-EnderWannabe]

Latest push should address all the comments except for the last.

I've been looking for the bug in

P.<x,y>=ProjectiveSpace(QQ,1)
f=DynamicalSystem([x^2+3*y^2,x*y])
f.sigma_invariants(1,return_polynomial=True, chow=True)

This map has a fixed point at infinity with multiplicity 3, which we try to account for in the second affine chart. However, it only shows up with multiplicity 1 in that affine chart. Not sure how to fix that.

[bhutz]

I think this needs to be deformed as was done in the formal periodic points.

I see four cases

• all periodic points and multipliers are distinct (typical case): runs as normal
• all periodic points distinct, but multipliers not all distinct: chow method works
• periodic points not all distinct, all other multipliers distinct: deform and use normal method
• periodic points not all distinct and multipliers not all distinct: deform + chow method

[git]

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

​b4246d2	32166: fixed identity matrix return
​eaebd33	32166: added test for identity
​57a4b39	Merge branch 'affine_preperiodic' into periodic_points_improvement
​81d7c97	32166: fixed documentation
​a63927e	Merge branch 'affine_preperiodic' into periodic_points_improvement
​77660f1	Merge branch 'periodic_points_improvement' into sigma_invariants
​008d6db	31994: added deformation
​9b5e74f	31944: fixed calls to affine preperiodic model
​14c58d3	Merge branch 'periodic_points_improvement' into sigma_invariants
​fc12a32	31994: fixed tests

[gh-EnderWannabe]

Replying to bhutz:

I think this needs to be deformed as was done in the formal periodic points.

Deformation added, that example works now.

[bhutz]

I know this is not marked as ready for review yet, but a quick comment.

The valueerror for degree counting only seems to occur for Formal=False.

Also the error should probably read '...and/or deform=True.'

[git]

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

​76249f2

31994: doc fix

[git]

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

​f7f22a8

31994: added degree check when formal=True

[gh-EnderWannabe]

Replying to bhutz:

The valueerror for degree counting only seems to occur for Formal=False.

Added

[bhutz]

The functionality is working correctly, but the documentation needs to be updated. The description is still the dimension 1 description.

[git]

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

​f1e1958	31944: fix for subscheme issue
​d3ea2be	31944: added check for morphisms
​268bc83	31944: fixed if statement
​6f2857f	31944: added example for formal preperiodic
​f385385	Merge branch 'periodic_points_improvement' into sigma_invariants
​063f159	31994: added more documentation

[bhutz]

Please look over my doc changes. Otherwise I think we ready to mark this positive.

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New commits:

​6186c58

31994: updated docs

[gh-EnderWannabe]

Docs look good to me

[vbraun]

Random failure:

**********************************************************************
File "src/sage/dynamics/arithmetic_dynamics/projective_ds.py", line 4021, in sage.dynamics.arithmetic_dynamics.projective_ds.?.preperiodic_points
Failed example:
    f.preperiodic_points(2, 1, R=K)
Expected:
    [(v : 1), (-v : 1)]
Got:
    [(-v : 1), (v : 1)]
**********************************************************************
1 item had failures:
   1 of  68 in sage.dynamics.arithmetic_dynamics.projective_ds.?.preperiodic_points
    [1406 tests, 1 failure, 49.93 s]
----------------------------------------------------------------------
sage -t --long --random-seed=0 src/sage/dynamics/arithmetic_dynamics/projective_ds.py  # 1 doctest failed
----------------------------------------------------------------------

[gh-EnderWannabe]

Added 2 lines to sort the list before returning to avoid such issues.

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New commits:

​0bb7549	Merge branch 'u/bhutz/sigma_invariants' of trac.sagemath.org:sage into sigma_invariants
​abdade3	31994: return sorted list of preperiodic points

[bhutz]

need to merge in fix for doctest failure in #32322

[git]

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

​18f87f5

Merge branch 'u/gh-EnderWannabe/sigma_invariants' of git://trac.sagemath.org/sage into sigma_invariants

[bhutz]

I removed the duplicated import that pyflakes caught. Otherwise looks fine. Patchbot hasn't picked it up since the #32322 merge, but that ticket is now closed, so no issues there.

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New commits:

​aa63aaa

31994: removed duplicated import