          ### 7.5.25 ratgb_lib

Status: experimental
Library:
ratgb.lib
Purpose:
Groebner bases in Ore localizations of noncommutative G-algebras
Author:
Viktor Levandovskyy, levandov@risc.uni-linz.ac.at

Overview:
Theory: Let A be an operator algebra with `R = K[x1,.,xN]` as subring. The operators are usually denoted by `d1,..,dM`.

Assume, that A is a `G`-algebra, then the set `S=R-0` is multiplicatively closed Ore set in A. That is, for any s in S and a in A, there exist t in S and b in A, such that `sa=bt`. In other words, one can transform any left fraction into a right fraction. The algebra `A_S` is called an Ore localization of A with respect to S.

This library provides Groebner basis procedure for A_S, performing polynomial (that is fraction-free) computations only. Note, that there is ongoing development of the subsystem called Singular:Locapal, which will provide yet another approach to Groebner bases over such Ore localizations.

Assumptions: in order to treat such localizations constructively, some care need to be taken. We will assume that the variables `x1,...,xN` from above (which will become invertible in the localization) come as the first block among the variables of the basering. Moreover, the ordering on the basering must be an antiblock ordering, that is its matrix form has the left upper `NxN` block zero. Here is a recipe to create such an ordering easily: use 'a(w)' definitions of the ordering N times with intvecs `w_i` of the following form: `w_i` has first N components zero. The rest entries need to be positive and such, that `w1,..,wN` are linearly independent (see an example below).

Guide: with this library, it is possible
- to compute a Groebner basis of an ideal or a submodule in the 'rational' Ore localization D = A_S
- to compute a dimension of associated graded submodule (called D-dimension) - to compute a vector space dimension over Quot(R) of a submodule of D-dimension 0 (so called D-finite submodule)
- to compute a basis over Quot(R) of a D-finite submodule

Procedures:

 7.5.25.0. ratstd compute Groebner basis and dimensions in Ore localization of the basering Support: SpezialForschungsBereich F1301 of the Austrian FWF and Transnational Access Program of RISC Linz, Austria          