Arakelov-Zhang pairing of rational maps

[Paul Fili]

We add a function to compute the dynamical Arakelov-Zhang pairing of two rational maps defined over number fields.

[git]

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

​b3ea858

Fixed an error in use of default precision of RealField

[git]

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

​ec5eac2

21129: added a needed import statement

[Frédéric Chapoton]

please use python3-compatible syntax for print (see ​https://wiki.sagemath.org/Python3-compatible%20code)

[Frédéric Chapoton]

plus one failing doctest, see patchbot report

[Frédéric Chapoton]

I removed the bad prints and corrected the failing doctest.

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New commits:

​a3bfe9c	Merge branch 'u/paulfili/arakelov_zhang_pairing_of_rational_maps' in 8.0.rc2
​6ab8e91	trac 21129 fixing details

[Yuan Kong]

Updated to sage 9.0 codebase

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New commits:

​357a0a1

Updated our Arakelov_Zhang_pairing funciton

[git]

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

​664acad

Update Examples in Arakelov_Zhang_pairing

[Paul Fili]

There is currently a bug in prime_of_bad_reduction, reported now in #28779, which affects one of the examples here. (Apparently the bug wasn't present when the examples were first computed!). I'd rather not remove a good example, so we are marking this as dependent on that bug fix.

[Paul Fili]

The bug in #28779 should be fixed. If you apply that patch, then this one, it should now be working.

[Ben Hutz]

notice that this is listed as 13000 lines of code change. It looks like git thinks you've deleted the entire file and re-written it. It would be better not to lose the code change history. Can you see if you can get it to just have your changes?

[Paul Fili]

Replying to bhutz:

notice that this is listed as 13000 lines of code change. It looks like git thinks you've deleted the entire file and re-written it. It would be better not to lose the code change history. Can you see if you can get it to just have your changes?

Done!

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New commits:

​ba813c7	Added Arakelov Zhang pairing function to fresh copy of projective_ds.py
​0c4e9b2	Fixed error, clears LCM of denominators always, GCD as appropriate
​2778cdb	Merge branch 'u/paulfili/normalize' of git://trac.sagemath.org/sage into azpairing
​2a8b393	Updated AZ pairing and checked tests

[Ben Hutz]

Thanks!

[Erik Bray]

Ticket retargeted after milestone closed

[Matthias Köppe]

Batch modifying tickets that will likely not be ready for 9.1, based on a review of the ticket title, branch/review status, and last modification date.

[Matthias Köppe]

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

[Matthias Köppe]

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

[Matthias Köppe]

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.

[Jing Guo]

New commits:

​ba813c7	Added Arakelov Zhang pairing function to fresh copy of projective_ds.py
​2778cdb	Merge branch 'u/paulfili/normalize' of git://trac.sagemath.org/sage into azpairing
​2a8b393	Updated AZ pairing and checked tests

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​81f904b	Added Arakelov Zhang pairing function to fresh copy of projective_ds.py
​be35eb0	Updated AZ pairing and checked tests

[git]

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

​b36bf4d

21129: Format doc; TODO: Turn args to kwds properly

[git]

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

​f3e67ab

21129: Change args to kwds

[git]

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

​933a160	21129: Complete `)` in misc.py
​9b43de1	21129: Better format and doc

[git]

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

​c83217c

21129: Indent

[git]

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

​174b686

21129: Rename `RealField` and `f.domain.base_ring` vars to avoid

[git]

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

​dcce807

21129: format examples

[Alexander Galarraga]

The following throws an error:

K.<k> = CyclotomicField(3)
P.<x,y> = ProjectiveSpace(K,1)
f = DynamicalSystem_projective([x^2 + (2*k+2)*y^2, y^2])
g = DynamicalSystem_projective([x^2, y^2])
pairingval = f.arakelov_zhang_pairing(g, n=5); pairingval

The last example in the documentation is failing on my end. Does it pass for you?

Finally, some minor issues in the code. There is extra whitespace on line 1302. Line 1281 is too long, it should be wrapped into two lines, as should any line longer than that.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​e64cb5f	Added Arakelov Zhang pairing function to fresh copy of projective_ds.py
​580a8dc	Updated AZ pairing and checked tests
​7dd124b	21129: Format doc; TODO: Turn args to kwds properly
​6bf37d5	21129: Change args to kwds
​bbe6180	21129: Complete `)` in misc.py
​67e3590	21129: Better format and doc
​be6f236	21129: Indent
​0fdcfd2	21129: Rename `RealField` and `f.domain.base_ring` vars to avoid
​f1f7fb1	21129: Fix docs and a bug when `f_disc` is not rational

[git]

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

​ecf5014

21129: Wrap long lines

[Jing Guo]

Thank you, Alex. Good catch. That was caused by calling abs before norm for non-rational f_disc. Everything is fixed now.

[Alexander Galarraga]

Code itself looks good now. The test I was trying to run also doesn't fail now. Using Prop 4.3 in Birdy and Larson's paper on the pairing, I tested pairing two different dynamical systems, and the results were mathematically correct correct.

The only remaining problem is the example beginning on line 1224. According to Prop 18 of Petsche, Szpiro and Tucker, the value of the pairing should be exactly the global height of a. Thus, this is mathematically incorrect. Since it was passing before the code was modernized, it could have been something in the modernization. It is also possible that we just need to increase n, or change the noise modifier to get this example to work. Take a look and try and see what could be going wrong, and if you get stuck, this is worth emailing Paul about.

[git]

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

​22c1778

21129: Better indentation

[git]

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

​0f1e6a4

21129: Add noise_multiplier check for bad reductions

[git]

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

​a9d40fa

21129: Correct example

[git]

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

​1bc95d5	21129: Fix noise_multiplier bug
​3287b2d	Merge branch 'u/gh-guojing0/pairing' of https://github.com/sagemath/sagetrac-mirror into pairing

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​9ce8f81

21129: Correct nosie mult bug