Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation

A new approach to Buss’s NC¹ algorithm [Proc. 19thACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123-131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for NC¹ over AC¬ reductions. This approach is then used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits.

Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.

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.

This paper is devoted to the complexity analysis of certain uniformity properties owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which is parsimonious with respect to branchings and divisions must necessarily have a non-polynomial sequential time complexity, even if highly ecient data structures (as e.g. the arithmetic circuit encoding of polynomials) are used.

. Viewing a parallel execution as a set of tasks that execute on a set of processors, a main problem is to find a schedule of the tasks that provides an efficient execution. This usually leads to divide algorithms into two classes: static and dynamic algorithms, depending on whether the schedule depends on the indata or not. To improve this rough classification we study, on some key applications of the Stratag` eme project [21, 22], the different ways schedules can be obtained and the associated overheads. This leads us to propose a classification based on regularity criteria i.e. measures of how much an algorithm is regular (or irregular). For a given algorithm, this expresses more the quality of the schedules that can be found (irregular versus regular) as opposed to the way the schedules are obtained (dynamic versus static). These studies reveal some paradigms of parallel programming for irregular algorithms. Thus, in a second part we study a parallel programming model that takes i...

This paper describes the parallelization of Constraint Dependency Grammar (CDG) parsing. Though CDG provides a flexible framework for text-based and spoken language parsing and has an expressivity strictly greater than context-free grammars (CFGs), it also has a relatively slow serial running time (i.e., O(n 4 )). However, a parallelization for this algorithm is derived which uses O(n 4 ) processors to parse in O(k) time for a CRCW P-RAM, where n is the number of words in the sentence and k, the number of constraints, is a grammatical constant. Additionally, the paper describes an implementation of the algorithm on the MasPar MP-1, which uses the special features of the machine (particularly the global router) and O(n 4 ) processors to obtain an O(k + log(n)) running time. Because the average length of an English sentence is on the order of 10 words, the MasPar MP-1 has sufficient processors (i.e., 16,000) for parsing a typical sentence. Previous work in parallel parsing has fo...

We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) space. The boolean circuits themselves are developed using techniques based on the computation of a primitive element of a suitable zero-dimensional algebra and diophantine considerations. Our algorithms improve...

This paper is devoted to the complexity analysis of a particular property, called geometric robustness owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which owns this property must necessarily have an exponential sequential time complexity even if highly performant data structures (as e.g. the straight--line program encoding of polynomials) are used. The paper finishes with the motivated introduction of a new non-uniform complexity measure for zero-dimensional polynomial equation systems, called elimination complexity.

this paper we introduce a computational model and call it multiprocessor algebraic computation which on the one hand is a special case of the very general concept of arithmetic network [G86]. On the other hand multiprocessor algebraic computations extend the usual models of algebraic PRAM ([M88], [M94]) or accepting network ([MP93]). The crucial new feature for multiprocessor algebraic computation is that we allow the use of polynomials from a shared pool, these polynomials are computed along with the whole computation (one can view these polynomials as reserved in the hard disc), and the number of processors is restricted (one could view the processors as the random access memory). The processors can be used for control needs, say for branching, in order to reach some node in the computation, as is usually done in computation trees (e.g. while recognizing membership to some set).