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Dive into the research topics where Guillermo Matera is active.

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Featured researches published by Guillermo Matera.


Foundations of Computational Mathematics | 2003

The Hardness of Polynomial Equation Solving

David Tio Castro; Marc Giusti; Joos Heintz; Guillermo Matera; Luis Miguel Pardo

AbstractElimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P


Applicable Algebra in Engineering, Communication and Computing | 2001

On the Time–Space Complexity of Geometric Elimination Procedures

Joos Heintz; Guillermo Matera; Ariel Waissbein

cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.


Foundations of Computational Mathematics | 2009

Deformation Techniques for Sparse Systems

Gabriela Jeronimo; Guillermo Matera; Pablo Solernó; Ariel Waissbein

Abstract. In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input. In this paper we apply this algorithmic concept in order to exhibit an elimination procedure whose space complexity is only quadratic and its time complexity is only cubic in the degree of the input system.


Mathematics of Computation | 2006

Fast computation of a rational point of a variety over a finite field

Antonio Cafure; Guillermo Matera

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration.


Journal of Complexity | 2011

Lower complexity bounds for interpolation algorithms

Nardo Giménez; Joos Heintz; Guillermo Matera; Pablo Solernó

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bezout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.


Computing | 2009

On the solution of the polynomial systems arising in the discretization of certain ODEs

Ezequiel Dratman; Guillermo Matera

We introduce and discuss a new computational model for the Hermite-Lagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known Hermite-Lagrange interpolation problems and algorithms. Like in traditional Hermite-Lagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariskis Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in Hermite-Lagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).


Foundations of Computational Mathematics | 2015

Degeneracy Loci and Polynomial Equation Solving

Bernd Bank; Marc Giusti; Joos Heintz; Grégoire Lecerf; Guillermo Matera; Pablo Solernó

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the diffusion is large enough, then there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the boundary-value problem under consideration. Furthermore, in this case we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is polynomial in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.


Advances in Mathematics of Communications | 2012

Singularities of symmetric hypersurfaces and Reed-Solomon codes

Antonio Cafure; Guillermo Matera; Melina Privitelli

Let


Theoretical Computer Science | 2004

Polynomial equation solving by lifting procedures for ramified fibers

Agustín Bompadre; Guillermo Matera; Rosita Wachenchauzer; Ariel Waissbein


mathematical foundations of computer science | 1998

Combinatorial Hardness Proofs for Polynomial Evaluation

Mikel Aldaz; Joos Heintz; Guillermo Matera; José Luis Montaña; Luis Miguel Pardo

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Antonio Cafure

University of Buenos Aires

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Melina Privitelli

National Scientific and Technical Research Council

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Joos Heintz

University of Buenos Aires

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Joos Heintz

University of Buenos Aires

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Ariel Waissbein

Instituto Tecnológico de Buenos Aires

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Mikel Aldaz

Universidad Pública de Navarra

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Ezequiel Dratman

National Scientific and Technical Research Council

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Eda Cesaratto

National Scientific and Technical Research Council

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