Singular

7.5.13 ncpreim_lib

Library:

ncpreim.lib

Purpose:

Non-commutative elimination and preimage computations

Author:

Daniel Andres, daniel.andres@math.rwth-aachen.de

Support: DFG Graduiertenkolleg 1632 `Experimentelle und konstruktive Algebra'

Overview:

In G-algebras, elimination of variables is more involved than in the commutative case.
One, not every subset of variables generates an algebra, which is again a G-algebra.
Two, even if the subset of variables in question generates an admissible subalgebra, there might be no admissible elimination ordering, i.e. an elimination ordering which also satisfies the ordering condition for G-algebras.

The difference between the procedure eliminateNC provided in this library and the procedure eliminate (plural) from the kernel is that eliminateNC will always find an admissible elimination if such one exists. Moreover, the use of slimgb for performing Groebner basis computations is possible.

As an application of the theory of elimination, the procedure preimageNC is provided, which computes the preimage of an ideal under a homomorphism f: A -> B between G-algebras A and B. In contrast to the kernel procedure preimage (plural), the assumption that A is commutative is not required.

References:

(BGL) J.L. Bueso, J. Gomez-Torrecillas, F.J. Lobillo: `Re-filtering and exactness of the Gelfand-Kirillov dimension', Bull. Sci. math. 125, 8, 689-715, 2001.
(GML) J.I. Garcia Garcia, J. Garcia Miranda, F.J. Lobillo: `Elimination orderings and localization in PBW algebras', Linear Algebra and its Applications 430(8-9), 2133-2148, 2009.
(Lev) V. Levandovskyy: `Intersection of ideals with non-commutative subalgebras', ISSAC'06, 212-219, ACM, 2006.

Procedures:

7.5.13.0. eliminateNC		elimination in G-algebras
7.5.13.0. preimageNC		preimage of ideals under homomorphisms of G-algebras
7.5.13.0. admissibleSub		checks whether subalgebra is admissible
7.5.13.0. isUpperTriangular		checks whether matrix is (strictly) upper triangular
7.5.13.0. appendWeight2Ord		appends weight to ordering
7.5.13.0. elimWeight		computes elimination weight
7.5.13.0. extendedTensor		tensor product of rings with additional relations