Differences between revisions 4 and 5
⇤ ← Revision 4 as of 2012-02-22 14:01:55 →
Size: 1182
Editor: caruso
Comment:
← Revision 5 as of 2012-02-22 14:12:42 → ⇥
Size: 1505
Editor: caruso
Comment:
Deletions are marked like this. Additions are marked like this.
Line 9: Line 9:
This ring is closely related to $\sigma$-modules over $k$ (which arises in $p$-adic Hodge theory). This ring is closely related to $\sigma$-modules over $k$ and, consequently, to Galois representations.
Line 11: Line 11:
The aim of the project is to implement basic arithmetic on $k[X,\sigma]$ when $k$ is a finite field The aim of the project is to implement usual functions on $k[X,\sigma]$ when $k$ is a finite field.
Line 24: Line 24:

---- /!\ '''Edit conflict - other version:''' ----
Line 25: Line 27:

---- /!\ '''Edit conflict - your version:''' ----
Do not derive from !PolynomialRing_general since this class assumes that the variable commutes with the constants (probably rather hard: need to rewrite many things)

---- /!\ '''End of edit conflict''' ----

People interested

Xavier Caruso, Jérémy Le Borgne

Description

If k is a field and \sigma a ring endomorphism of k, the ring of skew polynomials k[X,\sigma] is the usual vector space of polynomials over k equipped with the multiplication deduced from the rule a X = \sigma(a) X (a \in K)

This ring is closely related to \sigma-modules over k and, consequently, to Galois representations.

The aim of the project is to implement usual functions on k[X,\sigma] when k is a finite field.

Progress

A class has been written (for now, in python). It supports the following functions:

  • basic arithmeric (addition, multiplication, euclidean division, gcd)

  • computation of the center Z(k[X,\sigma]) -- need to add a coercion map

  • computation of the so-called map \Psi : k[X,\sigma] \to Z(k[X,\sigma])

  • computation of the associated Galois representation (via the corresponding \sigma-module)

  • factorization -- in progress

Bugs

/!\ Edit conflict - other version:

Do not derive from PolynomialRing_general since this class assumes that the variable commutes with the constants (probably rather hard: need to rewrite many things)

/!\ Edit conflict - your version:

Do not derive from PolynomialRing_general since this class assumes that the variable commutes with the constants (probably rather hard: need to rewrite many things)

/!\ End of edit conflict

padicSageDays/Projects/SkewPolynomials (last edited 2012-02-23 19:44:31 by caruso)