Antonio Montes

A New Algorithm for Discussing Gröbner Bases with Parameters

Let F be a set of polynomials in the variables __x=x1, . . . , xnwith coefficients in R __ a , where R is a UFD and __ a=a1, . . . ,am a set of parameters. In this paper we present a new algorithm for discussing Grobner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Grobner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning?s comprehensive Grobner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Grobner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications.

Gröbner bases for polynomial systems with parameters

Grobner bases are the computational method par excellence for studying polynomial systems. In the case of parametric polynomial systems one has to determine the reduced Grobner basis in dependence of the values of the parameters. In this article, we present the algorithm GrobnerCover which has as inputs a finite set of parametric polynomials, and outputs a finite partition of the parameter space into locally closed subsets together with polynomial data, from which the reduced Grobner basis for a given parameter point can immediately be determined. The partition of the parameter space is intrinsic and particularly simple if the system is homogeneous.

Improving the DISPGB algorithm using the discriminant ideal

In 1992, V. Weispfenning proved the existence of Comprehensive Grobner Bases (CGB) and gave an algorithm for computing 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 Grobner Bases. Inspired by its philosophy, V. Weispfenning defined, in 2002, how to obtain a Canonical Comprehensive Grobner 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 for checking whether a given basis is a CGB or not, and for obtaining a CGB. Examples and tests are also provided.

Algebraic solution of the load-flow problem for a 4-nodes electrical network

The load-flow problem for an electrical network is formulated as a system of polynomial equations in several variables. In present Electrical Engineering it is solved using numerical methods. But the system contains parameters and must very often be solved in real time in order to simulate events. It would be, in principle, useful to reduce the system to echelon or triangular form, using algebraic techniques, as, for instance, Grobner bases, in order to obtain the solution once for all and then use it in any simulation. In this paper, algorithms to triangulate the load-flow equations of a 4-nodes electrical network are presented. These algorithms have been implemented in Maple, and simulations using them are given. The advantages of the algebraic solution compared to the numerical one are discussed. In particular, the algebraic solution allows us to compute, in a very simple way, derivatives and the related numerical conditioning of the problem and also the numerical conditioning of the algorithm.

An algebraic taxonomy for locus computation in dynamic geometry

Abstract The automatic determination of geometric loci is an important issue in Dynamic Geometry. In Dynamic Geometry systems, it is often the case that locus determination is purely graphical, producing an output that is not robust enough and not reusable by the given software. Parts of the true locus may be missing, and extraneous objects can be appended to it as side products of the locus determination process. In this paper, we propose a new method for the computation, in dynamic geometry, of a locus defined by algebraic conditions. It provides an analytic, exact description of the sought locus, making possible a subsequent precise manipulation of this object by the system. Moreover, a complete taxonomy, cataloging the potentially different kinds of geometric objects arising from the locus computation procedure, is introduced, allowing to easily discriminate these objects as either extraneous or as pertaining to the sought locus. Our technique takes profit of the recently developed GrobnerCover algorithm. The taxonomy introduced can be generalized to higher dimensions, but we focus on 2-dimensional loci for classical reasons. The proposed method is illustrated through a web-based application prototype, showing that it has reached enough maturity as to be considered a practical option to be included in the next generation of dynamic geometry environments.

Generalizing the Steiner–Lehmus theorem using the Gröbner cover

In this note we present an application of a new tool (the Grobner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Grobner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, Steiner–Lehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Grobner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind.

The Power-Compositions Determinant and Its Application to Global Optimization

Let C(n,p) be the set of p-compositions of an integer n, i.e., the set of p-tuples

A polynomial generalization of the power-compositions determinant

\alpha=(\alpha_1,\ldots,\alpha_p)

Software using the Gröbner cover for geometrical loci computation and classification

of nonnegative integers such that

Erratum to “A new algorithm for discussing Gröbner bases with parameters” J. Symbolic Comput. 33 (1-2) (2002) 183-208]

\alpha_1+\cdots+\alpha_p=n