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D.2.4 grobcov_lib
- Library:
- grobcov.lib
- Purpose:
- Groebner Cover for parametric ideals.
- Purpose:
- Comprehensive Groebner Systems, Groebner Cover, Canonical Forms,
Parametric Polynomial Systems.
The library contains Montes-Wibmer's algorithms to compute the
canonical Groebner cover of a parametric ideal as described in
the paper:
Montes A., Wibmer M.,
"Groebner Bases for Polynomial Systems with parameters".
Journal of Symbolic Computation 45 (2010) 1391-1425.
The central routine is grobcov. Given a parametric
ideal, grobcov outputs its Canonical Groebner Cover, consisting
of a set of pairs of (basis, segment). The basis (after
normalization) is the reduced Groebner basis for each point
of the segment. The segments are disjoint, locally closed
and correspond to constant lpp (leading power product)
of the basis, and are represented in canonical prime
representation. The segments are disjoint and cover the
whole parameter space. The output is canonical, it only
depends on the given parametric ideal and the monomial order.
This is much more than a simple Comprehensive Groebner System.
The algorithm grobcov allows options to solve partially the
problem when the whole automatic algorithm does not finish
in reasonable time.
grobcov uses a first algorithm cgsdr that outputs a disjoint
reduced Comprehensive Groebner System with constant lpp.
For this purpose, in this library, the implemented algorithm is
Kapur-Sun-Wang algorithm, because it is the most efficient
algorithm known for this purpose.
D. Kapur, Y. Sun, and D.K. Wang.
"A New Algorithm for Computing Comprehensive Groebner Systems".
Proceedings of ISSAC'2010, ACM Press, (2010), 29-36.
cgsdr can be called directly if only a disjoint reduced
Comprehensive Groebner System (CGS) is required.
- Authors:
- Antonio Montes , Hans Schoenemann.
- Overview:
- see "Groebner Bases for Polynomial Systems with parameters"
Montes A., Wibmer M.,
Journal of Symbolic Computation 45 (2010) 1391-1425.
(http://www-ma2.upc.edu/~montes/).
- Notations:
- All given and determined polynomials and ideals are in the
basering Q[a][x]; (a=parameters, x=variables)
After defining the ring, the main routines
grobcov, cgsdr,
generate the global rings
@R (Q[a][x]),
@P (Q[a]),
@RP (Q[x,a])
that are used inside and killed before the output.
If you want to use some internal routine you must
create before the above rings by calling setglobalrings();
because most of the internal routines use these rings.
The call to the basic routines grobcov, cgsdr will
kill these rings.
Procedures:
D.2.4.1 grobcov | | Is the basic routine giving the canonical Groebner cover of the parametric ideal F. This routine accepts many options, that allow to obtain results even when the canonical computation does not finish in reasonable time. |
D.2.4.2 cgsdr | | Is the procedure for obtaining a first disjoint, reduced Comprehensive Groebner System that is used in grobcov, but that can be used independently if only the CGS is required. It is a more efficient routine than buildtree (the own routine that is no more used). It uses now KSW algorithm. |
D.2.4.3 setglobalrings | | Generates the global rings @R, @P and @PR that are respectively the rings Q[a][x], Q[a], Q[x,a]. It is called inside each of the fundamental routines of the library: grobcov, cgsdr and killed before the output. If the user want to use some other internal routine, then setglobalrings() is to be called before, as the rings @R, @P and @RP are needed in most of them. globally, and more internal routines can be used, but these rings are killed by the call to any of the basic routines. |
D.2.4.4 pdivi | | Performs a pseudodivision of a parametric polynomial by a parametric ideal. |
D.2.4.5 pnormalf | | Reduces a parametric polynomial f over V(E) V(N) E is the null ideal and N the non-null ideal over the parameters. |
D.2.4.6 extend | | When the grobcov of an ideal has been computed with the default option ('ext',0) and the explicit option ('rep',2) (which is not the default), then one can call extend (GC) (and options) to obtain the full representation of the bases. With the default option ('ext',0) only the generic representation of the bases are computed, and one can obtain the full representation using extend. locus2d: Special routine for determining the locus of points of a two dimensional object. Given an ideal J with two parameters (a,b) and so many variables as needed, representing the system determining the locus of points (a,b) who verify certain geometrical properties, computing the grobcov of J and applying to it locus2d, determines the locus. locus2dto: Transforms the output of locus2d to a string that can be reed from different computational systems. |
See also:
compregb_lib.
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