padicSageDays/Projects/SkewPolynomials

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 Xa=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

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)