Opened 9 years ago

# Skew polynomials — at Version 18

Reported by: Owned by: caruso tbd major sage-7.4 algebra skew polynomials, sd75 tfeulner, caruso, burcin, jsrn, dlucas, tscrim Xavier Caruso N/A #13214, #13303

If R is a ring equipped with an endomorphism sigma, the ring of skew polynomials over (R,sigma) is the ring of usual polynomials over R with the modified multiplication given by the rule X*a = sigma(a)*X.

Skew polynomials play an important role in several domains like coding theory or Galois representations theory in positive characteristic.

The attached patch provides:

1. a basic implementation of skew polynomials over any commutative ring (including addition, multiplication, euclidean division, gcd...)
2. a more complete implementation of skew polynomials over finite fields (including factoring)

NB: This ticket depends on tickets #13214 (which implements Frobenius endomorphisms over finite fields) and #13303 (which fixes a bug in quotient_polynomial_ring_element.pyx). For convenience, I reattach the corresponding patches here (you need to apply these two patches first).

### comment:1 Changed 9 years ago by caruso

• Status changed from new to needs_review

### comment:2 Changed 9 years ago by caruso

• Dependencies set to #13214
• Description modified (diff)

### comment:3 Changed 9 years ago by caruso

File trac_13215_skew_polynomials.2.patch replaces trac_13215_skew_polynomials.patch

### comment:4 Changed 9 years ago by tfeulner

• Cc tfeulner added
• Status changed from needs_review to needs_work

With your changes it is now possible to test some element in an Univariate Quotient Polynomial Ring to be a unit and to compute its inverse. Unfortunately, this computation gives wrong results:

sage: Z16x.<x> = Integers(16)[]
sage: GR.<y> =  Z16x.quotient(x**2 + x+1 )
sage: (2*y).is_unit()
True
sage: (2*y)^(-1)
15*y + 15
sage: (2*y)*(2*y)^(-1)
2


Thomas

### comment:5 Changed 9 years ago by caruso

Thanks for catching this!

I've updated my patch. (I've simply decided to raise a NotImplementedError? when the base ring is not a field and, by the way, I've added some doctests.)

PS: apply patch trac_13215_skew_polynomials.patch and forget trac_13215_skew_polynomials.2.patch (is it possible to remove it?)

### comment:6 Changed 9 years ago by caruso

• Status changed from needs_work to needs_review

### comment:7 Changed 9 years ago by tfeulner

Maybe you should produce a seperate trac ticket for the fix of

sage: (2*y)^(-1)
...
NotImplementedError: The base ring (=Ring of integers modulo 16) is not a field


By the way, I know that this is a problem which also occurs for usual polynomials (because of the default definition of the reduce function). Do you have any idea how to fix this:

sage: Z16x.<x> = SkewPolynomialRing(Integers(16))
sage: # Z16x.<x> = Integers(16)[] # has the same behaviour
sage: GR.<y> =  Z16x.quotient(x**2 + x+1 )
sage: I = GR.ideal(4)
sage: I.reduce(6)
6


### comment:8 Changed 9 years ago by caruso

Ok, I split my patch in two parts and I open a new ticket (cf #13303).

I'm afraid I don't understand quite well the problem with reduce. Could you explain it a bit more please?

Last edited 9 years ago by caruso (previous) (diff)

### comment:9 Changed 9 years ago by caruso

• Dependencies changed from #13214 to #13214, #13303
• Description modified (diff)

### comment:10 Changed 9 years ago by caruso

• Cc caruso added

### comment:11 Changed 9 years ago by jdemeyer

Please fill in your real name as Author.

### comment:12 Changed 9 years ago by caruso

• Authors set to Xavier Caruso

### comment:13 Changed 9 years ago by tfeulner

Hi Xavier,

you are right this is a more general problem. I started a discussion on https://groups.google.com/d/topic/sage-devel/s5y604ZPiQ8/discussion. The function reduce() does what it says returning an element of the ring. But in the definition of a QuotientRing? there is the following assumption

I.reduce(x)==I.reduce(y) \iff x-y \in I, and x-I.reduce(x) \in I, for all x \in R.

With this default behaviour, the quotient ring is just a copy of the original ring R.

Best, Thomas

### comment:14 Changed 9 years ago by burcin

• Cc burcin added
• Component changed from experimental package to algebra

It would be great to have all this in Sage. Thanks a lot for your work.

I haven't had a chance to look at the patches yet. I will try to make some time available for this in the coming weeks. For now, I have two questions,

• Did you consider using Singular's noncommutative component, Plural, as a basic data structure for skew polynomials? I expect the arithmetic to be much faster with Plural than a new implementation in Cython.
• Can the patches be broken down into smaller components for easier review?

### comment:15 follow-up: ↓ 16 Changed 9 years ago by caruso

I haven't heard about Plural before that... so, I haven't considered using it for skew polynomials yet. I will compare how fast is the arithmetic with Plural and with my own implementation in Cython and let you know.

Ok, I will split my patch in several components. (Is two enough or are you expecting more than that?).

### comment:16 in reply to: ↑ 15 Changed 9 years ago by burcin

Thanks for the prompt response! I am attending a workshop this week and I haven't had a chance to read through your patch properly yet. So please excuse me if I am completely off the mark in my response below. :)

I haven't heard about Plural before that... so, I haven't considered using it for skew polynomials yet. I will compare how fast is the arithmetic with Plural and with my own implementation in Cython and let you know.

Your patch provides more functionality than Plural, for example gcd's and factoring. The coefficient domains supported by Plural are also limited. I suppose your implementation supports more general coefficient domains, basically anything Sage already knows about.

It would be good to have a clear comparison between Plural and your implementation first. But I imagine having two skew polynomial classes in Sage, a generic implementation from your patch and one based on Plural. We could probably share the high level functionality not provided by Plural using the category framework.

I am not opposed to just including your patch and think about the Plural part later as an optimization if you say that your implementation is more capable.

Ok, I will split my patch in several components. (Is two enough or are you expecting more than that?).

I don't know yet. It depends on how many functionally independent parts there are. For instance, the short representations for morphisms can be an independent piece that is trivial to review. The hash functions for morphisms would also be a separate bug fix. Actually, I attempted to fix that a long time ago at #9016.

### comment:17 follow-up: ↓ 18 Changed 9 years ago by caruso

I had a quick look at the documentation of Plural but I didn't find how to create a skew polynomial ring (i.e. I don't know how to represent a skew polynomial ring as a G-algebra). Could you please learn me this?

### comment:18 in reply to: ↑ 17 Changed 9 years ago by burcin

• Description modified (diff)

I had a quick look at the documentation of Plural but I didn't find how to create a skew polynomial ring (i.e. I don't know how to represent a skew polynomial ring as a G-algebra). Could you please learn me this?

For example, QQ[n, S:n->n+1] :

sage: A.<n, S> = FreeAlgebra(QQ, 2)
sage: R.<n, S> = A.g_algebra({S*n: n*S + 1})
sage: R
Noncommutative Multivariate Polynomial Ring in n, S over Rational Field, nc-relations: {S*n: n*S + 1}
sage: S*n
n*S + 1
sage: S^2*n
n*S^2 + 2*S


After applying the patches on this ticket and its dependencies, I get the following error:

sage: R.<t> = ZZ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x',sigma]; S
Skew Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1
sage: x*t
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
...
sage: %debug
> /home/burcin/sage/sage-5.2/skewpolynomial_element.pyx(347)sage.rings.polynomial.skewpolynomial_element.SkewPolynomial._list_c (sage/rings/polynomial/skewpolynomial_element.c:4716)()


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