Characteristic Classes: Sequences

[mjungmath]

To each additive/multiplicative characteristic class corresponds a sequence. Namely, due to the fundamental theorem of symmetric polynomials, each symmetric polynomial can be expressed in terms of the elementary symmetric polynomials. In our setup, the k-th elementary symmetric polynomials corresponds to the k-th Chern class. This allows us to express additive/multiplicative characteristic classes in terms of Chern classes (or Pontryagin classes in the real case respectively). This is the sequence.

CC: @slel @egourgoulhon @tscrim

Component: manifolds

Author: Michael Jung

Branch/Commit: 7aa0ec5

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/30211

[mjungmath]

Branch: u/gh-mjungmath/characteristic_classes_sequence

[mjungmath]

Commit: 6972030

[mjungmath]

New commits:

6972030

Trac #30211: sequence function added

[sagetrac-git]

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

5f25df5

Trac #30211: Merge branch 'develop' into characteristic_classes_sequence

[sagetrac-git]

Changed commit from 6972030 to 5f25df5

[sagetrac-git]

Changed commit from 5f25df5 to c7414bd

[sagetrac-git]

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

c7414bd

Trac #30211: docstring syntax

[mjungmath]

comment:5

This ticket is ready for review. Patchbot should turn green after this commit.

[tscrim]

comment:6

Do you have an example that gives something that is not a symmetric function over QQ?

Also, you are better off using

m = Sym.m()
m._from_dict({p: prod(ring(coeff[i]) for i in p)
              for k in range(len(coeff))
              for p in Partitions(k))

which is faster as it more directly creates the symmmetric function you want.

[mjungmath]

comment:7

Replying to @tscrim:

Do you have an example that gives something that is not a symmetric function over QQ?

Sure: x-y is no symmetric function over QQ.

Also, you are better off using

m = Sym.m()
m._from_dict({p: prod(ring(coeff[i]) for i in p)
              for k in range(len(coeff))
              for p in Partitions(k))

which is faster as it more directly creates the symmmetric function you want.

Thanks, I'll try this.

[mjungmath]

comment:8

Apparently, there is something wrong with the code since some doctests fail. I'll check on this.

[tscrim]

comment:9

Replying to @mjungmath:

Replying to @tscrim:

Do you have an example that gives something that is not a symmetric function over QQ?

Sure: x-y is no symmetric function over QQ.

I know what a symmetric function is (that is basically my main research area). What I want is a symmetric function over some other base ring besides QQ. Can you add an example where this happens? In the branch, you say don't assume QQ, so I want a doctest where that actually occurs.

[mjungmath]

comment:10

Ah, I see. I was already wondering. :D

Well yes, the standard case is QQ but at the moment, there is nothing that prevents you having a parameter in SR. This might come in useful (I don't know when exactly, but I want to keep the possibility open). Now, I wanted to make sure that the algorithm always works, even in that case, so I chose SR.

[mjungmath]

comment:11

I will try another attempt using power series only. I hope this simplifies the code significantly. Furthermore, this could improve the performance since power series compute things faster than symbolic expressions; also the use of power series is mathematically more rigorous. Finally, power series (over the symbolic ring) allow to distinct between parameters and the actual variable.

[mjungmath]

comment:13

Replying to @tscrim:

Replying to @mjungmath:

Replying to @tscrim:

Do you have an example that gives something that is not a symmetric function over QQ?

Sure: x-y is no symmetric function over QQ.

I know what a symmetric function is (that is basically my main research area). What I want is a symmetric function over some other base ring besides QQ. Can you add an example where this happens? In the branch, you say don't assume QQ, so I want a doctest where that actually occurs.

General question: do the examples have to be mathematically sensible? For instance, there are plenty of examples where the coefficients of the Taylor expansion are not rationals and SR is then a reasonable choice, but such examples are very uncommon in the application of characteristic classes.

[tscrim]

comment:14

Replying to @mjungmath:

Replying to @tscrim:

Replying to @mjungmath:

Replying to @tscrim:

Do you have an example that gives something that is not a symmetric function over QQ?

Sure: x-y is no symmetric function over QQ.

I know what a symmetric function is (that is basically my main research area). What I want is a symmetric function over some other base ring besides QQ. Can you add an example where this happens? In the branch, you say don't assume QQ, so I want a doctest where that actually occurs.

General question: do the examples have to be mathematically sensible? For instance, there are plenty of examples where the coefficients of the Taylor expansion are not rationals and SR is then a reasonable choice, but such examples are very uncommon in the application of characteristic classes.

Uncommon is different than mathematically sensible. Since it something you are supporting, it should be tested, even if it is uncommon. Something you could do is have ring being None and then have the code first try QQ, and if that doesn't work, do it again with SR. However, that would mean the code becomes more complicated. Another option would be to default to QQ with the documentation saying that if it fails over QQ, try SR.

[sagetrac-git]

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

21fee92

Trac #30211: "ring=QQ"

[sagetrac-git]

Changed commit from c7414bd to 21fee92

[sagetrac-git]

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

49d617b

Trac #30211: distinct_real

[sagetrac-git]

Changed commit from 21fee92 to 49d617b

[sagetrac-git]

Changed commit from 49d617b to 4765f29

[sagetrac-git]

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

6687d0c	Trac #30211: Merge branch 'develop' into t/30211/characteristic_classes_sequence
4765f29	Trac #30211: distinct real

[tscrim]

comment:19

Thanks. This will be good. Just one comment on the code:

             from sage.combinat.partition import Partitions
             # Get the multiplicative sequence in the monomial basis:
-            mon_pol = Sym.m().sum(prod(ring(coeff[i]) for i in p) * Sym.m()[p]
+            mon_pol = Sym.m()._from_dict({p: sum(prod(ring(coeff[i]) for i in p)
                                   for k in range(len(coeff))
-                                  for p in Partitions(k))
+                                  for p in Partitions(k)})

This will create the element faster with far less temporary objects.

[tscrim]

comment:20

You will probably also want to do something similar for the additive case.

[sagetrac-git]

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

7aa0ec5

Trac #30211: symmetric polynomial via _from_dict

[sagetrac-git]

Changed commit from 4765f29 to 7aa0ec5

[mjungmath]

comment:22

Here we go. Wait for patchbot.

@tscrim: However, I am not sure whether this ticket is still useful since I plan to reconstruct the whole architecture of characteristic classes. Perhaps it is good if you could join my talk on Friday because I plan to explain the rough idea of the current implementation and an outline of what I have in mind.

[mkoeppe]

comment:23

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

[tscrim]

comment:24

This will be good for now. If things get completely rewritten, then this can be modified/deprecated at that time.

[tscrim]

Reviewer: Travis Scrimshaw

[mjungmath]

comment:25

Thank you for the review!

[vbraun]

Changed branch from u/gh-mjungmath/characteristic_classes_sequence to 7aa0ec5