Implement an example of a graded algebra with basis, and improve the later

[jhpalmieri]

The summary says it all. See also the patch on the Sage-Combinat patch server:

​http://combinat.sagemath.org/hgwebdir.cgi/patches/file/tip/trac_9280-graded-algebras-example.patch

Thanks to Jason Bandlow and Franco Saliola who participated to the improvement of the example. It now depends on #10193.

[nthiery]

Hi John,

For the record: we went through your patches with Franco and Jason, and discussed quite a bit around it. We will post here shortly an updated patch with some little suggestions.

[jhpalmieri]

Replying to nthiery:

Hi John,

For the record: we went through your patches with Franco and Jason, and discussed quite a bit around it. We will post here shortly an updated patch with some little suggestions.

Is it "shortly" yet? :)

[jhpalmieri]

In the sage-combinat patch, there are a few typos and some other issues:

• in sage/categories/graded_algebras_with_basis.py, the docstring for "degree" says "The degree of this element in the graded polynomial algebra." Delete "polynomial".

• in sage/categories/examples/graded_algebras_with_basis.py, the docstring for "one_basis" contains '(0,...,0`), and I think this should be changed to ``(0,...,0)``.

• in sage/categories/examples/graded_algebras_with_basis.py, the docstring for the main class is now outdated: it still refers to "basis_function" and "_basis_fcn", which don't exist any more, and also to "homogeneous_component", which is now part of the default implementation, not something specific to this example.

I'm attaching a referee patch which fixes these.

There are also some doctests for "basis" in sage/categories/graded_algebras_with_basis.py which are marked as "todo: not implemented". Do we need to wait for these to be fixed, or should we consider this ready for review? It may not be ideal, but we could also change

sage: A.basis(6) # todo: not implemented (output)
Family (y^{2}, x^{3}

to

sage: A.basis(6) # todo: not implemented (output)
Family (y^{2}, x^{3}
sage: list(A.basis(6))
[y^{2}, x^{3}]

By the way, all tests pass with this patch and with the one from #10193. So perhaps we could also delete the commented-out part at the beginning of the example, where it says

# TODO: double check that we can now discard this function

[jhpalmieri]

apply on top of sage-combinat patch

[saliola]

There are a couple of patches on the sage-combinat queue experimenting with moving some of the generic methods into the category GradedAlgebraWithBasis:

• trac_9280-graded-algebras-example-review-fs.patch
• trac_9280-graded-algebras-example.patch

[vdelecroix]

Sorry for the long delay for the ticket but #10193 is now ready !!

[chapoton]

Franco, Nicolas, what can we do with this ticket ? Should we use the patch from the combinat queue or the patch here ?

[chapoton]

let me take the patch of sage-combinat as a starting point.

for the bot: apply only trac_9280-graded-algebras-example-fs.patch

[jhpalmieri]

I don't know why I'm listed as an author in the file "sage/categories/examples/graded_modules_with_basis.py"; I don't think I had anything to do with that.

[chapoton]

The part of this patch concerning modules has been separated into ticket #11688 : the ticket #11688 should go first, then this one will need to be rebased on it.

I upload here the "algebra only patch" that will be the new starting point.

[chapoton]

this needs to be rebased

[tscrim]

Since the graded algebras with basis example is using (weighted) integer vectors, we need #12453. I'd like to attach the branch "public/categories/graded_examples-9280", but trac is giving me an error when I try...

[tscrim]

New commits:

​32dc807	Fixed failing doctests.
​c467bdb	Merge branch 'public/refactor_integer_vectors-12453' into public/categories/graded_examples-9280
​ef2ddce	#12453: Refactored IntegerVectors to use ClonableIntArray.
​c82765f	Initial review changes.
​e2c3ae8	Merge branch 'public/categories/examples-9280' into public/categoires/graded_example-9280
​6fffda7	Merge branch 'master' into public/categories/examples-9280
​3491e78	imported patch trac_9280_nomodule.patch

[git]

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

​161cedb	Merge branch 'develop' into public/categories/graded_examples-9280
​badae2a	Merge branch 'develop' into public/categories/graded_examples-9280
​eb793cc	Merge branch 'develop' into public/refactor_integer_vectors-12453
​b9f278e	Merge branch 'develop' into public/refactor_integer_vectors-12453
​2346b02	Merge branch 'public/refactor_integer_vectors-12453' into public/categories/graded_examples-9280

[git]

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

​12bac19	Merge branch 'develop' into public/refactor_integer_vectors-12453
​fb05e6c	Merge branch 'public/refactor_integer_vectors-12453' into public/categories/graded_examples-9280

[chapoton]

needs rebase

[jhpalmieri]

Yet another instance of someone asking a question related to this. Six (!) years ago, when I opened this ticket, I thought it would be good to have an example in the Sage library and in the documentation, and I really can't understand why this hasn't been taken care of yet. I am not interested in working on it myself any more, but I find it incredibly frustrating that this ticket has languished for so long.

​http://ask.sagemath.org/question/34577/can-sage-compute-a-basis-for-the-graded-parts-of-a-graded-ring/

[tscrim]

It is because of the dependency on integer vectors, which led to #12453. We can either give a new example based on another object or we review #12453 (which I just did a [non-trivial] rebase to the latest beta).