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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 locus algorithm and definitions will be
published in a forthcoming paper by Abanades, Botana, Montes, Recio:
"An Algebraic Taxonomy for Locus Computation in Dynamic Geometry".
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, that can also 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, locus, locusdg and killed before the output. So, if the user want to use some other internal routine, then setglobalrings() is to be called before, as most of them use these rings. |
D.2.4.4 pdivi | | Performs a pseudodivision of a parametric polynomial by a parametric ideal. Can be used without calling presiouly setglobalrings(), |
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. Can be used without calling presiouly setglobalrings(), |
D.2.4.6 Prep | | Computes the P-representation of V(N) V(M). Can be used without calling previously setglobalrings(). |
D.2.4.7 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. Can be used without calling presiouly setglobalrings(), |
D.2.4.8 locus | | : Special routine for determining the locus of points of objects. Given a parametric ideal J with parameters (a_1,..a_m) and variables (x_1,..,xn), representing the system determining the locus of points (a_1,..,a_m)) who verify certain properties, computing the grobcov G of J and applying to it locus, determines the different classes of locus components. They can be Normal, Special, Accumulation point, Degenerate. The output are the components given in P-canonical form of two constructible sets: Normal, and Non-Normal The Normal Set has Normal and Special components The Non-Normal set has Accumulation and Degenerate components. The description of the algorithm and definitions will be given in a forthcoming paper by Abanades, Botana, Montes, Recio: "An Algebraic Taxonomy for Locus Computation in Dynamic Geometry". Can be used without calling presiouly setglobalrings(), |
D.2.4.9 locusdg | | : Is a special routine for computing the locus in dinamical geometry. It detects automatically a possible point that is to be avoided by the mover, whose coordinates must be the last two coordinates in the definition of the ring. If such a point is detected, then it eliminates the segments of the grobcov depending on the point that is to be avoided. Then it calls locus. Can be used without calling presiouly setglobalrings(), |
D.2.4.10 locusto | | : Transforms the output of locus to a string that can be reed from different computational systems. Can be used without calling presiouly setglobalrings(), |
D.2.4.11 addcons | | : Given a disjoint set of locally closed subsets in P-representation, it returns the canonical P-representation of the constructible set formed by the union of them. Can be used without calling presiouly setglobalrings(), |
See also:
compregb_lib.
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