Center related classes and methods in Skew Polynomials

[arpitdm]

We propose the addition of the following methods and classes to skew polynomials:

1. class CenterSkewPolynomial_generic_dense
2. class SectionSkewPolynomialCenterInjection
3. class SkewPolynomialCenterInjection
4. class CenterSkewPolynomialRing
5. def center in class SkewPolynomialRing_general

In addition, we designed a special class SkewPolynomial_finite_order_dense for dense skew polynomial over fields when the twisting endomorphism has finite order (in which case the centre has finite index). It includes the following methods:

• reduced_trace
• reduced_norm
• is_central
• bound
• optimal_bound

Note: The original ticket #13215 first introduced this functionality (only for finite fields). That was subsequently modified to support the basic implementation of skew polynomials and the center based methods from that ticket that were removed are being reintroduced here.

Depends on #13215
Depends on #29452
Depends on #29517

CC: @tscrim @xcaruso @johanrosenkilde @sagetrac-dlucas

Component: algebra

Author: Xavier Caruso

Branch/Commit: 5929bf9

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/21262

[arpitdm]

Branch: u/arpitdm/centering_methods_skew_polynomials

[arpitdm]

Commit: 3d7cd83

[arpitdm]

comment:2

Please also note that the current code is more or less just what was in the original patch for #13215 related to Center related methods and classes. No effort has e.g. been made yet to accommodate for changes in #13215 since this addition was factored out.

---

Last 10 new commits:

9a2fad2	merged changes from Tickets 13215 and 21088
eaca253	integrated skew polynomial finite field into sage. removed some compile and doctest errors.
7664060	removed leftpow and rightpow methods from SkewPolynomial_finite_field_dense class because they require the 'bound' method which in turn requires 'center'. this will be added in another separate ticket with the rest of the center stuff.
a6e93e1	added SEEALSO and TODO blocks and made small polishes to the documentation.
130b173	improved documentation for skew_polynomials_finite_field.pyx file
15861b9	documentation builds successfully.
2d67e0e	merging updates
a2c4f06	removed unused imports, signal statements. small fixes to documentation.
5547542	added karatsuba based methods as is, from the original #13215 ticket
3d7cd83	added centering based methods and classes as is, from the original #13215 ticket

[xcaruso]

comment:3

The methods is_central and optimal_bound make sense for skew polynomials over any ring, right? So it is a bit weird (according to me) to define them only in the class SkewPolynomial_finite_field_dense (through I very probably do this first).

Similarly the reduced norm makes sense as soon as the twist map has finite order (and in addition I am very interested in using it over a finite extension of Qp). So, I would suggest to move it to the class SkewPolynomial_generic_dense and to raise ValueError or TypeError when the twist map has not finite order.

[arpitdm]

Author: Xavier Caruso

[xcaruso]

Changed branch from u/arpitdm/centering_methods_skew_polynomials to u/caruso/centering_methods_skew_polynomials

[xcaruso]

comment:6

As I said before, the features provided by this ticket do not only make sense for finite fields but more generally for fields on which the twisting endomorphism has finite order. I then renamed the class SkewPolynomial_finite_field_dense and called it SkewPolynomial_finite_order_dense.
Moreover, I polished the code and made all methods work (e.g. some imports were missing).

Ticket ready for review.

---

New commits:

891ccfa	Merge branch 't/21262/centering_methods_skew_polynomials' into integration
dd53bd1	Remove inplace methods and Karatsuba
a29393a	Replace "finite fields" by "fields with automorphism of finite order"
35f44cb	Added a method reduced_trace

[xcaruso]

Description changed:

--- 
+++ 
@@ -6,11 +6,12 @@
 
 5. def center in `class SkewPolynomialRing_general`
 
-6. Following methods to the class `SkewPolynomial_finite_field_dense`:
-- def reduced_norm
-- def reduced_norm_factor
-- def is_central
-- def optimal_bound
+In addition, we designed a special class `SkewPolynomial_finite_order_dense` for dense skew polynomial over fields when the twisting endomorphism has finite order (in which case the centre has finite index). It include the following methods:
+- reduced_trace
+- reduced_norm
+- is_central
+- bound
+- optimal_bound
 
 
-Note: The original ticket #13215 first introduced this functionality. That was subsequently modified to support the basic implementation of skew polynomials and the center based methods from that ticket that were removed are being reintroduced here. 
+Note: The original ticket #13215 first introduced this functionality (only for finite fields). That was subsequently modified to support the basic implementation of skew polynomials and the center based methods from that ticket that were removed are being reintroduced here. 

[xcaruso]

Changed commit from 3d7cd83 to 35f44cb

[xcaruso]

Changed dependencies from #13215, #21088, #21259 to #13215