12 research outputs found
Improving DISPGB Algorithm Using the Discriminant Ideal
In 1992, V. Weispfenning proved the existence of Comprehensive Groebner Bases
(CGB) and gave an algorithm to compute one. That algorithm was not very
efficient and not canonical. Using his suggestions, A. Montes obtained in 2002
a more efficient algorithm (DISPGB) for Discussing Parametric Groebner Bases.
Inspired in its philosophy, V. Weispfenning defined, in 2002, how to obtain a
Canonical Comprehensive Groebner Basis (CCGB) for parametric polynomial ideals,
and provided a constructive method.
In this paper we use Weispfenning's CCGB ideas to make substantial
improvements on Montes DISPGB algorithm. It now includes rewriting of the
discussion tree using the Discriminant Ideal and provides a compact and
effective discussion. We also describe the new algorithms in the DPGB library
containing the improved DISPGB as well as new routines to check whether a given
basis is a CGB or not, and to obtain a CGB. Examples and tests are also
provided.Comment: 21 pages, see also http://www-ma2.upc.edu/~montes
Minimal canonical comprehensive Gröbner systems
This is the continuation of Montes' paper "On the canonical discussion of polynomial systems with parameters''. In this paper, we define the Minimal Canonical Comprehensive Gröbner System of a parametric ideal and fix under which hypothesis it exists and is computable. An algorithm to obtain a canonical description of the segments of the Minimal Canonical CGS is given, thus completing the whole MCCGS algorithm (implemented in Maple and Singular). We show its high utility for applications, such as automatic theorem proving and discovering, and compare it with
other existing methods. A way to detect a counterexample to deny its existence is outlined, although the high number of tests done give evidence of the existence of the Minimal Canonical CGS.Postprint (published version
From graphs to tensegrity structures: Geometric and symbolic approaches
A form-finding problem for tensegrity structures is studied; given an
abstract graph, we show an algorithm to provide a necessary condition for it to
be the underlying graph of a tensegrity in (typically )
with vertices in general position. Furthermore, for a certain class of graphs
our algorithm allows to obtain necessary and sufficient conditions on the
relative position of the vertices in order to underlie a tensegrity, for what
we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio
グレブナー基底を用いた収束冪級数環での拡張イデアル所属アルゴリズムについて (数式処理とその周辺分野の研究)
Ideal membership and extended ideal membership problems are considered in rings of convergent power series. It is shown that the problems for zero-dimensional ideals in the local rings can be solved in polynomial rings. New algorithms are given to solve the problems in the local rings. The key of the algorithms is the use of ideal quotients in polynomial rings