Coverage for local/lib/python2.7/site-packages/sage/rings/polynomial/skew_polynomial_ring_constructor.py : 54%
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r""" Constructor for skew polynomial rings
This module provides the function :func:`SkewPolynomialRing`, which constructs rings of univariate skew polynomials, and implements caching to prevent the same ring being created in memory multiple times (which is wasteful and breaks the general assumption in Sage that parents are unique).
AUTHOR:
- Xavier Caruso (2012-06-29): initial version
- Arpit Merchant (2016-08-04): improved docstrings, added doctests and refactored method
- Johan Rosenkilde (2016-08-03): changes to import format
"""
############################################################################# # Copyright (C) 2012 Xavier Caruso <xavier.caruso@normalesup.org> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #****************************************************************************
from __future__ import print_function, absolute_import, division from sage import categories import cysignals import sage.rings.ring as ring from sage.structure.category_object import normalize_names from sage.rings.finite_rings.finite_field_base import is_FiniteField from sage.categories.morphism import Morphism, IdentityMorphism
def SkewPolynomialRing(base_ring, base_ring_automorphism=None, names=None, sparse=False): r""" Return the globally unique skew polynomial ring with the given properties and variable names.
Given a ring `R` and a ring automorphism `\sigma` of `R`, the ring of skew polynomials `R[X, \sigma]` is the usual abelian group polynomial `R[X]` equipped with the modification multiplication deduced from the rule `X a = \sigma(a) X`.
.. SEEALSO::
- :class:`sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_general` - :class:`sage.rings.polynomial.skew_polynomial_element.SkewPolynomial`
INPUT:
- ``base_ring`` -- a commutative ring
- ``base_ring_automorphism`` -- an automorphism of the base ring (also called twisting map)
- ``names`` -- a string or a list of strings
- ``sparse`` -- a boolean (default: ``False``). Currently not supported.
.. NOTE::
The current implementation of skew polynomial rings does not support derivations. Sparse skew polynomials and multivariate skew polynomials are also not implemented.
OUTPUT:
A univariate skew polynomial ring over ``base_ring`` twisted by ``base_ring_automorphism`` when ``names`` is a string with no commas (``,``) or a list of length 1. Otherwise we raise a ``NotImplementedError`` as multivariate skew polynomial rings are not yet implemented.
UNIQUENESS and IMMUTABILITY:
In Sage, there is exactly one skew polynomial ring for each triple (base ring, twisting map, name of the variable).
EXAMPLES of VARIABLE NAME CONTEXT::
sage: R.<t> = ZZ[] sage: sigma = R.hom([t+1]) sage: S.<x> = SkewPolynomialRing(R, sigma); S Skew Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1
The names of the variables defined above cannot be arbitrarily modified because each skew polynomial ring is unique in Sage and other objects in Sage could have pointers to that skew polynomial ring.
However, the variable can be changed within the scope of a ``with`` block using the localvars context::
sage: with localvars(S, ['y']): ....: print(S) Skew Polynomial Ring in y over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1
SQUARE BRACKETS NOTATION:
You can alternatively create a skew polynomial ring over `R` twisted by ``base_ring_automorphism`` by writing ``R['varname', base_ring_automorphism]``.
EXAMPLES:
We first define the base ring::
sage: R.<t> = ZZ[]; R Univariate Polynomial Ring in t over Integer Ring
and the twisting map::
sage: base_ring_automorphism = R.hom([t+1]); base_ring_automorphism Ring endomorphism of Univariate Polynomial Ring in t over Integer Ring Defn: t |--> t + 1
Now, we are ready to define the skew polynomial ring::
sage: S = SkewPolynomialRing(R, base_ring_automorphism, names='x'); S Skew Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1
Use the diamond brackets notation to make the variable ready for use after you define the ring::
sage: S.<x> = SkewPolynomialRing(R, base_ring_automorphism) sage: (x + t)^2 x^2 + (2*t + 1)*x + t^2
Here is an example with the square bracket notations::
sage: S.<x> = R['x', base_ring_automorphism]; S Skew Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1
Rings with different variables names are different::
sage: R['x', base_ring_automorphism] == R['y', base_ring_automorphism] False
TESTS:
You must specify a variable name::
sage: SkewPolynomialRing(R, base_ring_automorphism) Traceback (most recent call last): ... TypeError: you must specify the name of the variable
With this syntax, it is not possible to omit the name of the variable neither in LHS nor in RHS. If we omit it in LHS, the variable is not created::
sage: Sy = R['y', base_ring_automorphism]; Sy Skew Polynomial Ring in y over Univariate Polynomial Ring in t over Integer Ring twisted by t |--> t + 1 sage: y.parent() Traceback (most recent call last): ... NameError: name 'y' is not defined
If we omit it in RHS, sage tries to create a polynomial ring and fails::
sage: Sz.<z> = R[base_ring_automorphism] Traceback (most recent call last): ... ValueError: variable name 'Ring endomorphism of Univariate Polynomial Ring in t over Integer Ring\n Defn: t |--> t + 1' is not alphanumeric
Multivariate skew polynomial rings are not supported::
sage: S = SkewPolynomialRing(R, base_ring_automorphism,names=['x','y']) Traceback (most recent call last): ... NotImplementedError: multivariate skew polynomials rings not supported
Sparse skew polynomial rings are not implemented::
sage: S = SkewPolynomialRing(R, base_ring_automorphism, names='x', sparse=True) Traceback (most recent call last): ... NotImplementedError: sparse skew polynomial rings are not implemented
.. TODO::
- Sparse Skew Polynomial Ring - Multivariate Skew Polynomial Ring - Add derivations. """ raise TypeError("base_ring must be a commutative ring") base_ring_automorphism = IdentityMorphism(base_ring) else: or base_ring_automorphism.domain() != base_ring or base_ring_automorphism.codomain() != base_ring): raise TypeError("base_ring_automorphism must be a ring automorphism of base_ring (=%s)" % base_ring)
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