The design of controllers for dynamical systems will be addressed in this presentation. The process is based on an algebraic representation of the system, and it relies heavily on the use of Groebner bases. The main design tool is a division algorithm for multivariable polynomials. The control law is obtained by forcing the vanishing of the polynomial remainder. The design procedure will be shown using a classical example: the inverted pendulum.
This presentation will be given in Spanish.

This will be an introduction to the theory of multivariable hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. I will discuss, in particular, joint work with A. Dickenstein and B. Sturmfels on rational functions and the connections with discriminants and multivariable residues.

We propose two algorithms which compute an exact absolute factorization of a bivariate polynomial from an approximate one. These algorithms are based on some properties of the algebraic integers over Z. The first relies on an efficient heuristic method for computing the denominator of an algebraic integer over Z which uses only gcd calculations. The second is a deterministic algorithm and relies on a representation of the algebraic integers which uses f'(a) as denominator (f is the minimal polynomial of a).

We give a criterion to decide if a homogeneous polynomial in several variables with real coefficients is positive, generalizing the Sylvester criterion for quadratic forms.

The implicitization problem of rational surfaces is a central challenge in CAGD. We propose an algorithm to  solve it, based on the residue calculus in the general multivariate setting. The proposed approach allows us to treat  surfaces with base points (without geometric hypothesis on the  zero-locus of base points).

A description of the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic (monomial) transformation in any dimension, and geometric presentations in terms of fans.

The classical assumption of differential algebra and differential elimination is that the derivations do commute. That is the standard case arising from systems of partial differential equations. We inspect here the case where the derivations satisfy nontrivial commutation rules. That situation arises We inspect here the case where the derivations satisfy nontrivial commutation rules. That situation arises when we consider a system of equations on the differential invariants of a Lie group action. This work is part of a project initiated by E. Mansfield and in collaboration with I. Kogan to treat differential systems that are too symmetric to be handled by classical differential elimination algorithms.

We will consider the problem of computing a system of generators for the ideal of a smooth equidimensional affine variety. First, a result showing the existence of a system of generators with 'few' polynomials of 'low' degree will be presented. Using this result, we will derive an algorithm for the computation of these generators in single exponential time from a given set of polynomials whose set of common zeroes is V. (Joint work with C. Blanco and P. Solernó).

Polynomial systems of equations arising naturally in Mathematics exhibit some sort of symmetry that can be accounted for rigorously using group-theoretic ideas such as representations and invariant theory of finite groups. I will discuss some aspects of a joint work with R. Corless and K. Gatermann on solving efficiently polynomial systems with symmetries. The method will be illustrated with an application from Chaos Theory, namely the determination of bifurcation points of the logistic map.

I will present different applications of Hensel's lemma in Computer Algebra.

Many constructions of hypersurfaces with many singularities start by taking the equation of a family of hypersurfaces depending on a small number of parameters. E.g., Endrass found parameters s.t. the corresponding octic in a 5 parameter family has exactly 168 nodes. Our method for searching for good parameters within a given family is based on computer algebra computations over small finite fields. In the above family, we find Endrass' 168- and van~Straten's 165-nodal octic.

This talk will discuss applications of computational algebra to the analysis of dynamic networks over finite fields. In particular, applications to several biological networks will be discussed.

The cost of homotopy type algorithms for solving systems of polynomial equations is related to the following issues: the number of solutions, the number of steps of the homotopy path, and the cost of obtaining a "starting" system. The generic number of solutions of a system of equations with a given suppport is precisely the "volume" of a certain "toric variety" associated to the support. The number of steps can be bounded in terms of condition numbers, which depend on the metric of this toric manifold. The obtention of a suitable starting system is a major current bottleneck, but some geometric insight is available.

We shall exhibit estimates on the number of rational points on a variety over a finite field, which improve the existing estimates, and efficient algorithms for the computation of one rational point.

In this work, we study properties of a class of fractal sets generated by polynomials. In paticualr we prove that the fractals are recognized by finite state automata.

The use of polynomial matrices to solve control problems in industrial processes is something that is being studied. It is possible to demonstrate the advantages of polynomial methods compared with traditional ones.

We present an overview of a convex optimization framework for semialgebraic problems. Along the way, we'll learn how to compute sum of squares decompositions for polynomials using semidefinite programming, and how to obtain infeasibility certificates for real solutions of polynomial equations. The developed techniques, based on results from real algebraic geometry, unify and generalize many well-known existing methods. The ideas and algorithms will be illustrated with examples from a broad range of domains, and the use of the SOSTOOLS software (developed in collaboration with Stephen Prajna, Antonis Papachristodoulou, and Pete Seiler).

We describe the first algorithm to compute the gcd of two n-bits integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step that replaces U by (U-bV)/p, where p is one of the moduli and b is the unique integer in the interval (-p/2,p/2) such that b = U/V (mod p). A naive model for the average case analysis of the algorithm fits experimental data and matches the fastest algorithms presently known.