Enable polynomial content over padic fields

[saraedum]

Currently, one cannot call content() for a polynomial defined over Qp:

sage: K = Qp(3)
sage: R.<t> = K[]
sage: t.content()
TypeError: ground ring is a field.  Answer is only defined up to units.

The intention is apparently to protect the user from calling content() if he mistakenly got from Zp into Qp, since the content there would be always zero or one. However, this can be annoying when writing algorithms which work over Zp and Qp but which have to take the content into account over Zp.

Additionally, the content shows a somewhat strange behaviour for zero polynomials::

sage: R = Zp(3)
sage: S.<t> = R[]
sage: f = S(R(0,3)); f
(O(3^3))
sage: f.is_zero()
True
sage: f.content()
3 + O(3^23)
sage: _.is_zero()
False

I don't think that the current behaviour is mathematically incorrect, but I believe it's not very intuitive.

Component: padics

Keywords: sd59, days71

Author: Julian Rüth

Branch: 880984a

Reviewer: Jeroen Demeyer, Aly Deines

Issue created by migration from https://trac.sagemath.org/ticket/13619

[saraedum]

Author: Julian Rueth

[roed314]

comment:4

Attachment: trac_13619.patch.gz

I'm happy with this. Let me know when you want a review.

[tscrim]

comment:5

As a quick informal review, the OUTPUT: block is indented one too many times.

[nilesjohnson]

Branch: u/niles/ticket/13619

[nilesjohnson]

comment:8

rebased and converted to git branch; changed indentation of OUTPUT: block, but no other changes

---

New commits:

3ace36a

Trac #13619: Enable content for polynomials defined over Qp.

[nilesjohnson]

Commit: 3ace36a

[saraedum]

Changed branch from u/niles/ticket/13619 to u/saraedum/ticket/13619

[saraedum]

Changed keywords from none to sd59

[jdemeyer]

Changed commit from 3ace36a to e57bdb5

[jdemeyer]

comment:14

Is there are reason you are only doing this for capped_relative_dense rings and not in the generic Polynomial_padic?

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New commits:

e57bdb5

Merge branch 'develop' into ticket/13619

[jdemeyer]

comment:15

This should be done for generic padics polynomials. I would prefer to get the same answer for Zp and Qp, i.e. p^k where k is the minimal valuation. That would be more useful than always getting 1 as answer over Qp.

[jdemeyer]

Reviewer: Jeroen Demeyer

[jdemeyer]

comment:20

Replying to @saraedum:

What should be the answer if a coefficient has negative valuation?

p^k where k is the minimal valuation of the coefficients.

I would rather stick to the usual definition of content.

The "usual" definition doesn't care about the unit part, so answering p^k is a more useful and equally correct answer over Qp.

[saraedum]

comment:21

Replying to @jdemeyer:

I think it's certainly not a good idea that content() sometimes returns an element and sometimes returns an ideal. Ideally, there would be two methods:

• content() returning an ideal in the mathematical sense
• content_gen() returning an element, preferably a generator of the fractional ideal generated by the coefficients.

What is the value of content_gen()? It is almost always equivalent to content().gen() unless in some cases where both should raise an error?

[saraedum]

comment:22

Replying to @jdemeyer:

Replying to @saraedum:

What should be the answer if a coefficient has negative valuation?

p^k where k is the minimal valuation of the coefficients.

I would rather stick to the usual definition of content.

The "usual" definition doesn't care about the unit part, so answering p^k is a more useful and equally correct answer over Qp.

Ok. You are right. I'll change it.

[sagetrac-git]

Changed commit from e57bdb5 to 55e1b96

[sagetrac-git]

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

dc1dcee	Merge branch 'develop' into t/13619/ticket/13619
55e1b96	changed content of p-adic polynomials over Qp

[jdemeyer]

comment:25

See [comment:14]

[sagetrac-git]

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

355c144

Merge branch 'develop' into t/13619/ticket/13619

[sagetrac-git]

Changed commit from 55e1b96 to 355c144

[sagetrac-git]

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

880984a

Move capped relative content() implementation to the generic p-adic polynomial class.

[sagetrac-git]

Changed commit from 355c144 to 880984a

[adeines]

Changed reviewer from Jeroen Demeyer to Jeroen Demeyer, Aly Deines

[adeines]

comment:29

Looks good to me.

[vbraun]

Changed branch from u/saraedum/ticket/13619 to 880984a

[adeines]

Changed commit from 880984a to none

[adeines]

Changed keywords from sd59 to sd59, sd71

[adeines]

Changed keywords from sd59, sd71 to sd59, days71

[jdemeyer]

Changed author from Julian Rueth to Julian Rüth