Software Engineering and Complexity in Effective Algebraic Geometry

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434

Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page

Elimination for generic sparse polynomial systems

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate affine space with l < n. The complexity of the algorithm depends polynomially on combinatorial invariants associated to the supports.Comment: 22 page

The computational complexity of the Chow form

We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.Comment: 60 pages, Latex2

The initiation to architectural analysis viewed by a group of architect teachers

Ponencia presentada a Session 4: Investigar los procesos de diseño: etnografías y análisis de dialogías sociales / Research through the design processes: etnographic and social dialogical perspectivesThis article is about a pedagogical experience in architecture workshop teaching first-year student? at the National School of architecture of Tunis (ENAU). It focuses, in particular, on the initiation of the student to the architectural analysis process which is a major step in his course. The present work is based on a comparative study between the statements of the exercises related to the topics studied in the workshop. This comparison covers a period of eight years of teaching for the same group of teachers, and deals with their conception of architectural analysis and their way to approaching this initiation to their students. For this purpose, the Group of teachers has implemented an analysis grid that serves, to guide students in their work, and provides a good understanding of the architectural analysis as a process and brain action summoning both the senses and the mind. For this, the Group of teachers made the choice that the parameters to be analyzed concern only the geometry and topology of architectural form levels. They built their grid of architectural analysis on the basis of a postulate stating that “an architectural project is a complex act”. Thus, they consider the architectural project as a whole composed of a multitude of elements; a unit that draws its essence from the plurality. They formulate this complexity by the following equation: [An architectural project = A = 1 unit = 1+1+1+1+1...] Where the (1) represents the components of the project and the (+), the relationships that binds them to each other