Antonio MontesDepartament de Matemàtica Aplicada II

Universitat Politècnica de Catalunya (UPC)

Edifici Omega

Jordi Girona 1-3

08034 Barcelona

Spain

e-mail:antonio.montes_at_upc.edu

tel:+ 34 93 4131 77 04 (Secretary:+34 93 41376 80)

fax:+ 34 93 413 77 01

- Departament
de Matemàtica Aplicada II
- Facultat
d'Informàtica de Barcelona
- Facultat
de Matemàtiques i Estadística
- Universitat
Politècnica de Catalunya

Courses in the period 1981-2012.

*Computer**Algebra*(Facultat de Matemàtiques i Estadística). 5th course.*Calculus*(Facultat d'Informàtica). 1st course.*Symbolic Computation Workshop*(Facultat de Matemàtiques i Estadística). Free in UPC.*Mathematical Methods 1*(Facultat de Matemàtiques i Estadística). 1st course.*Mathematics**1*(Facultat d'Informàtica de Barcelona). 1st course.*Discrete**Mathematics*(Facultat d'Informàtica de Barcelona). 2nd course.*Algebra*(Facultat d'Informàtica de Barcelona). 1st course.*Algebra and Computing*(Facultat de Matemàtiques i Estadística). 1st course.*Numerical**Methods*(Facultat d'Informàtica de Barcelona). 3rd course.*Analysis**2*(Facultat d'Informàtica de Barcelona). 2nd course.*Fonaments**de Matemàtiques*(Facultat d’Informàtica). 1st course.*Matemàtiques**2*(Facultat d’Informàtica). 1^{st}course*.*

Àlgebra Computacional. Assignatura
optativa de la Llicenciatura de Matemàtiques
i Estadística de la Facultat de Matemàtiques
i Estadística de la UPC.

- Programa i Bibliografia.
(2012).
- Apunts d'Algebra Computacional. Antonio Montes. (Actualització: 2012 Febrer).

Main subject of interest: Computer Algebra

See
RedEACA, http://medicis.polytechnique.fr/ and Computer Algebra Internet Resources
page to find information about CA sites, mailing lists, news groups,
courses, books, journals, events, and more information about Computer Algebra.

- Comprehensive Groebner System (CGS, Kapur-Sun-Wang algorithm), see "Using Kapur-Sun-Wang Algorithm for the Gröbner Cover". A. Montes;

- Canonical Groebner Cover of a parametric ideal, see "Gröbner Bases for Polynomial Systems with parameters". A. Montes, M.Wibmer;

- Canonical representation of constructible sets, see "Computing the canonical representation of constructible sets". J.M. Brunat, A.Montes;

- Loci computation and taxonomy and applications to dynamic geometry, see "An Algorithm for Automatic Discovery of Algebraic Loci". F. Botana, A. Montes, T. Recio;

- Envelop computation and taxonomy and applications to dynamic geometry (to be published, A. Montes et al.).

To see instructions for the installation read the file GC_Readme.txt

Josep M. Brunat, Antonio Montes.

Submited to Mathematics in Computer Science (3-7-2015)

Abstract

Constructible sets are needed in many algorithms of Computer Algebra, particularly in the Gröbner Cover and other algorithms for parametric polynomial systems. In this paper we review the canonical form of constructible sets and give algorithms for computing it.

Miguel A. Abanades, Francisco Botana, Antonio Montes , Tomas Recio.

Computer-Aided Design 56 (2014) 22-33.

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. Moreover, extraneous objects are frequently appended to the true locus as side products of the locus determination process. In this paper, we propose a new method for locus computation in dynamic geometry. 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 di fferent 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 GröbnerCover algorithm. The proposed method is illustrated through a webbased 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.

A. Montes and M. Wibmer.

Proceedings of Mathematical Software ICMS 2014, Seoul, August 5-9, 2014.

Lecture Notes in Computer Science LNCS 8592, p. 406-413. Springer.v

Abstract

We present the canonical GRöBNER COVER method for discussing parametric polynomial systems of equations. Its objective is to decompose the parameter space into subsets (segments) for which it exists a generalized reduced Gröbner basis in the whole segment with fixed set of leading power products on it. Wibmer's Theorem guarantees its existence. The GRöBNER COVER is designed in a joint paper of the authors, and the Singular grobcov.lib library [15] implementing it, is developed by Montes. The algorithm is canonic and groups the solutions having the same kind of properties into different disjoint segments. Even if the algorithms involved have high complexity, we show how in practice it is effective in many applications of medium difficulty. An interesting application to automatic deduction of geometric theorems is roughly described here, and another one to provide a taxonomy for exact geometrical loci computations, that is experimentally implemented in a web based application using the dynamic geometry software Geogebra, is explained in another session.

M.A. Abánades, F. Botana, A. Montes, and T. Recio.

Proceedings of Mathematical Software ICMS 2014, Seoul, August 5-9, 2014.

Lecture Notes in Computer Science LNCS 8592, p. 492-499. Springer.

Abstract

We describe here a properly recent application of the GRöBNER COVER algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm LOCUS into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra.

Antonio Montes, Tomás Recio.

Mathematics and Computers in Simulation, Vol. 104 (2014), 67-71.

Abstract

In this note we present an application of a new tool (the Gröbner 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 Gröbner 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 Gröbner 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.

**An Algorithm for Automatic Discovery
of Algebraic Loci ***F. Botana,
A. Montes, T.Recio*

Proceedings of ADG-2012, p. 53-59

We show how dynamic geometry (DG) systems can be
enhanced by using computer algebra methods. In this paper we use different

Gröbner basis methods to compute geometrical loci and
specially use the Gröbner Cover to define and compute
normal and special

parts of geometrical loci, that can then be used inside DG
programs, specially in GeoGebra.

Se detalla brevemente el desarrollo histórico de los
algoritmos para la discusión de sistemas de ecuaciones polinómicas
con parámetros y se dan los elementos esenciales para comprender y poder
utilizar el nuevo algoritmo del cubrimiento canónico de Gröbner
(GröbnerCover) de un ideal paramétrico, introducido recientemente por el
autor y por Michael Wibmer, de la Universidad de Aachen. A fin de que los no especialistas puedan comprender
bien su significado e interés, antes de abordar el tema se hace una

introducción elemental a la teoría de las bases de Gröbner,
que subyace a este nuevo algoritmo. Posteriormente se dan ejemplos donde puede
apreciarse su utilidad para resolver problemas en los que aparecen ecuaciones polinómicas con parámetros.

**ISSAC
2011. Software for computing the Gröbner cover**. *Antonio
Montes*

The objective of this software presentation is to show
the behavior and applications of the Singular library

grobcov.lib that we have recently developed (Singular web) to
compute the Gröbner cover of a parametric ideal.

* Journal of Symbolic Computation* 45 (2010) 1391-1425. doi:10.1016/j.jsc.2010.06.017

** Minimal Canonical Comprehensive Gröbner System**.

*Journal of Symbolic Computation* 44 (2009)
463-478.

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, completing so the whole MCCGS algorithm
(implemented in Maple and Singular). We show its high utility for applications,
like 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.

This paper describes the principal new improvements in the DISPGB
algorithm implemented in release 7.4 of dpgb74.mpl and in Singular.

**On the
canonical discussion of polynomial systems with parameters**. *Antonio Montes*.

Preprint arXiv: AC/0601674. Preprint MA2-IR-06-00006

Given a parametric polynomial ideal I, the algorithm DISPGB,
introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Groebner bases depending on the values of the parameters.
An improvement using a discriminant ideal to rewrite
the tree was described by Manubens
and the author in

This paper describes the principal new improvements in the DISPGB algorithm
implemented in the next release 5.0 of dpgb50.mpl. This release contains
further improvements that are described in a new preprint in preparation.

**Some Composition Determinants**** ** *Josep**
M. Brunat, Christian Krattenthaler,
Alain Lascoux, Antonio Montes*.

*Linear Algebra and Applications*. **416** (2006) 355-364. Preprint in arXiv: CO/0509157.
Preprint MA2-IR-06-00005

We compute two parametric determinants in which rows
and columns are indexed by compositions,

where in one determinant the entries are products of binomial coefficients,
while in the other the entries

are products of powers. These results generalize previous determinant
evaluations due to the first and

third author [*SIAM**
J. Matrix Anal. and Appl.* **23** (2001), 459-471] and [``A polynomial

generalization of the power-compositions determinant," {*Linear and Multilinear Algebra* (to appear)],

and they prove two conjectures of the second author [``Advanced determinant
calculus: a complement,"

preliminary version].

**A polynomial
generalization of the power-compositions determinant****.**** ** *Josep**
M. Brunat, Antonio Montes*.

*Linear and Multilinear Algebra* **55**-4 (2007) 303-313. arXiv: math.CO/0601756. Preprint MA2-IR-06-00005

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

alpha

(

det (

In 1992, V. Weispfenning proved the existence of Comprehensive Gröbner 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

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.

Extended Abstract in

We present important improvements and a thorough redesign of the algorithm DisPGB in the new release 2.1 of the DPGB

Let

Let

Let *F*
be a set of polynomials in the variables *x_1,...,x_n* with
coefficients in *R*[*a_1,..a_n*], where *R*
is a

UFD and *a**_1,..,a_m* a set of
parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all
the cases over the parameters leading to different reduced Gröbner
basis, when the parameters in *F* are substituted in an extension field *K*
of *R*. This new algorithm improves Weispfenning CGB algorithm, obtaining a reduced set of
compatible and disjoint cases. A final improvement determines the minimal
singular variety outside of which the Gröbner basis
specializes properly (generic case). These constructive methods provide a very
satisfactory discussion with rich geometrical interpretation in the
applications.

**The power-composition determinant
and its application to global optimization.** *Josep** M. Brunat*, *Antonio
Montes*.

*SIAM** J. Matrix Anal.** Appl.*
**23-**2 (2001) 459-471.

Preprint in *Actas** de EACA-2000*,

Let *C*(*n,p*) be the set of
*p*-compositions of an integer *n*, i.e., the set of *p*-tuples *alpha = *(*alpha_1,..,alpha_p*) of
nonnegative integers such that *alpha_1 *+..+ a*lpha_p**
*= *n*. The main result of this paper is an explicit formula for the
determinant of the matrix whose entries are *alpha*^{*beta*}
= a*lpha_1*^{*beta_1*} .. *alpha_p*^{*beta_p*} where alpha,beta belong to *C*(*n,p*).
The formula shows that the determinant is positive and has a nice
factorization. As an application, it is

shown that the polynomials *p_alpha*(*x*) =
(*alpha_1 x_1+ .. + alpha_p
x_p*)^*n* with alpha in *C*(*n,p*) form a basis of the vector space *H_n*[*x_1, .. ,x_p*]
of homogeneous polynomials of degree *n* in *p* variables.
The result is of interest in the context of global optimization because it
allows an explicit representation of polynomials as a difference of convex
functions.

**Basic Algorithms for Specialization
in Groebner Bases.** ** ** *Antonio
Montes .*
(1999).

*Actas** de EACA-99*.

Some
basic algorithms for dealing with Groebner bases with
parametres are given. They allow to
construct a general algorithm for a general discussion of polynomial
systems with parametrs that improves the
Comprehensive Groebner Basis Algorithm of Weisfenning. The complete algorithm will be given in a
forthcoming paper.

**Algebraic solution of the load-flow
problem for a 4-nodes electrical network.**** ***Antonio Montes*.

*Mathematics & Computer in Simulation*, **45 **(1998) 163-174.

Using
algebraic techniques to triangulate polynomial systems of equations we are able
to do it for the system describing a complete four-nodes
electrical network with all the paramters. The
advantages of the algebraic solution compared to the numerical one are
discussed.

**Solving the Load Flow Problem Using the
Groebner Basis.** ** ***Antonio Montes, Jordi Castro.*

*Sigsam** Bulletin*, **29-**1 (1995)
1-13.

The
load-flow problem must be solved in Electrical Engeenering
very often with different values of the parameters. Theoretically, it is interessting to obtain a general solution for all values of
the parameters using Gröbner basis. In this paper we
compute it for a 3-nodes general network and compare the results and
conditioning with usual numerical computations.

**Numerical Conditioning of a System
of Algebraic Equations with a Finite Number of Solutions Using Groebner Bases.** ** **
*Antonio Montes*.

*Sigsam** Bulletin*, **27-**1 (1993)
12-19.

Numerical
conditioning for the problem and for the algorithm in computing the roots of a
0-dimensional polynomial system are given and applied to examples triangularized using Groebner
bases.

(*Last update of this page: September 16, 2014*)