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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.
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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.
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- 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		  * 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
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Do not derive from PolynomialRing_general since this class assumes that the variable commutes with the constant 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)

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

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)

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