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This ring is closely related to σ-modules over k (which arises in p-adic Hodge theory). This ring is closely related to σ-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,σ] when k is a finite field The aim of the project is to implement usual functions on k[X,σ] when k is a finite field.
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---- /!\ '''Edit conflict - other version:''' ----
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---- /!\ '''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 σ a ring endomorphism of k, the ring of skew polynomials k[X,σ] is the usual vector space of polynomials over k equipped with the multiplication deduced from the rule aX=σ(a)X (a∈K)

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

The aim of the project is to implement usual functions on k[X,σ] 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,σ]) -- need to add a coercion map

  • computation of the so-called map Ψ:k[X,σ]→Z(k[X,σ])

  • computation of the associated Galois representation (via the corresponding σ-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)