#7797: Full interface to letterplace from singular
------------------------------------------------------------------------+---
Reporter: burcin |
Owner: burcin
Type: enhancement |
Status: needs_work
Priority: major |
Milestone: sage-4.7
Component: algebra |
Keywords: singular
Author: Simon King, Michael Brickenstein, Burcin Erocal |
Upstream: N/A
Reviewer: |
Merged:
Work_issues: Unigenerated free algebra vs. univariate polynomial ring |
------------------------------------------------------------------------+---
Changes (by SimonKing):
* reviewer: split the ticket =>
* work_issues: => Unigenerated free algebra vs. univariate polynomial
ring
Old description:
> The new aim of this ticket is to add an interface to the
> [http://www.singular.uni-kl.de/Manual/latest/sing_427.htm#SEC480
> letterplace] component of Singular, namely providing
>
> * A new implementation of free algebras with fast arithmetic.
> * Degree-wise Gröbner basis computation for twosided homogeneous ideals
> of free algebras.
> * Normal form computation with respect to such ideals.
>
> and in addition
>
> * One- and twosided ideals of noncommutative rings.
> * Quotient rings of such ideals
>
> (Note that the original purpose was merely to compute Groebner bases up
> to a degree bound of two-sided ideals of free algebras, but without
> normal form computation etc.)
>
> Examples are below, in the comments.
>
> Apply trac7797-full_letterplace_wrapper.patch
>
> Depends on #10961
New description:
The new aim of this ticket is to add an interface to the
[http://www.singular.uni-kl.de/Manual/latest/sing_427.htm#SEC480
letterplace] component of Singular, namely providing
* A new implementation of free algebras with fast arithmetic.
* Degree-wise Gröbner basis computation for twosided homogeneous ideals
of free algebras.
* Normal form computation with respect to such ideals.
and in addition
* One- and twosided ideals of noncommutative rings.
* Quotient rings of such ideals
(Note that the original purpose was merely to compute Groebner bases up to
a degree bound of two-sided ideals of free algebras, but without normal
form computation etc.)
Examples are below, in the comments.
Apply trac7797-full_letterplace_wrapper_rel11068.patch
Depends on #11068
--
Comment:
I managed to split my patch. The part concerning "basic implementation of
ideals in non-commutative rings" is now at #11068. The new patch is based
on top of that.
'''__TODO__'''
Let the `FreeAlgebra` constructor always return a free algebra, not a
polynomial ring.
'''__New Feature__'''
In addition to what was described in previous comments, my letterplace
wrapper can compute ''complete'' twosided Gröbnerbases by an adaptive
algorithm. The idea is simple: If the Gröbner basis is known out to degree
`2*d-1`, but the highest degree of its generators is `d`, then the Gröbner
basis is complete.
Example:
{{{
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y-y*x,x*z-z*x,y*z-z*y,x^2*y-z^3,x*y^2+z*x^2]*F
sage: I.groebner_basis(Infinity)
Twosided Ideal (z*z*z*y*y + z*z*z*z*x, z*x*x*x + z*z*z*y, y*z - z*y, y*y*x
+ z*x*x, y*x*x - z*z*z, x*z - z*x, x*y - y*x) of Free Associative Unital
Algebra on 3 generators ('x', 'y', 'z') over Rational Field
}}}
Since the commutators are contained in the ideal, we can verify that
result with a commutative Gröbner basis, as follows:
{{{
sage: P.<c,b,a> = PolynomialRing(QQ,order='neglex')
sage: J = P*[a^2*b-c^3,a*b^2+c*a^2]
sage: J.groebner_basis()
[b*a^2 - c^3, b^2*a + c*a^2, c*a^3 + c^3*b, c^3*b^2 + c^4*a]
}}}
So, that's a good consistency test.
Apply trac7797-full_letterplace_wrapper_rel11068.patch
Depends on #11068
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7797#comment:40>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/sage-trac?hl=en.
