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