We investigate the integration of C implementation of fast arithmetic operations into MAPLE, focusing on triangular decomposition algorithms. We show substantial improvements over existing MAPLE implementations; our code also outperforms MAGMA on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements. 1

We discuss the parallelization of algorithms for solving polynomial systems symbolically by way of triangular decompositions. We introduce a component-level parallelism for which the number of processors in use depends on the geometry of the solution set of the input system. Our long term goal is to achieve an efficient multi-level parallelism: coarse grained (component) level for tasks computing geometric objects in the solution sets, and medium/fine grained level for polynomial arithmetic such as GCD/resultant computation within each task.

Abstract. Triangular decompositions for systems of polynomial equations with n variables, with exact coefficients are well-developed theoretically and in terms of implemented algorithms in computer algebra systems. However there is much less research about triangular decompositions for systems with approximate coefficients. In this paper we discuss the zero-dimensional case, of systems having finitely many roots. Our methods depend on having approximations for all the roots, and these are provided by the homotopy continuation methods of Sommese, Verschelde and Wampler. We introduce approximate equiprojectable decompositions for such systems, which represent a generalization of the recently developed analogous concept for exact systems. We demonstrate experimentally the favourable computational features of this new approach, and give a statistical analysis of its error. Keywords. Symbolic-numeric computations, Triangular decompositions, Dimension zero, Polynomial system solving.

We introduce the concept of comprehensive triangular decomposition for a parametric polynomial system and propose an algorithm for its computation. Using this decomposition, we solve the following problems: describe the sets of all parameter values, for which the system has, respectively, an empty, finite, or infinite set of solutions; partition the finite part into cells, so that in each cell the system has a constant number of solutions (the complex root counting problem). We generalize the Vincent-Collins-Akritas algorithm to solve the real root isolation problem for zerodimensional squarefree regular chains. Combining this algorithm with the comprehensive triangular decomposition, we obtain a solution to the real root counting problem for parametric polynomial systems with real coefficients. 1.

Symbolic methods are powerful tools in scientific computing. The implementation of symbolic solvers is, however, a highly difficult task due to the extremely high time and space complexity of the problem. In this thesis, we study and apply fast algorithms, modular methods, parallel approaches and software engineering techniques to improve the efficiency of symbolic solvers for computing triangular decomposition, one of the most promising methods for solving non-linear systems of equations symbolically. We first adapt nearly optimal algorithms for polynomial arithmetic over fields to direct products of fields for polynomial multiplication, inversion and GCD compu-tations. Then, by introducing the notion of equiprojectable decomposition, a sharp modular method for triangular decompositions based on Hensel lifting techniques is obtained. Its implementation also brings to the Maple computer algebra system a unique capacity for automatic case discussion and recombination. A high-level categorical parallel framework is developed, written in the Al-dor language, to support high-performance computer algebra on symmetric multi-

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Triangular decompositions are one of the major tools for solving polynomial systems. For systems of algebraic equations, they provide a convenient way to describe complex solutions and a step toward isolation of real roots or decomposition into irreducible components. Combined with other techniques, they are used for these purposes by several computer algebra systems. For systems of partial differential equations, they provide the main practicable way for determining a symbolic description of the solution set. Moreover, thanks to Rosenfeld’s Lemma, techniques from the algebraic case apply to the differential one [3]. Research in this area is following the natural cycle: theory, algorithms, implementation, which will be the main theme of this tutorial. We shall also concentrate on the algebraic case and mention the differential one among the applications.

Since the discovery of Gröbner bases, the algorithmic advances in Commutative Algebra have made possible to tackle many classical problems in Algebraic Geometry that were previously out of reach. However, algorithmic progress is still desirable, for instance when solving symbolically a large system of algebraic non-linear equations. For such a system, in particular if its solution set consists of geometric components of different dimension (points, curves, surfaces, etc) it is necessary to combine Gröbner bases with decomposition techniques, such as triangular decompositions. Ideally, one would like each of the different components to be produced by an independent processor, or set of processors. In practice, the input polynomial system, which is hiding those components, requires some transformations in order to split

We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of algebraic numbers in small characteristic.

arithmetic for triangular sets: