Philippe Aubry

On the Theories of Triangular Sets

Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.

Triangular Sets for Solving Polynomial Systems

Four methods for solving polynomial systems by means of triangular sets are presented and implemented in a unified way. These methods are those of Wu (1987), Lazard (1991), Kalkbrener (1991) and Wang (1993b). They are compared on various examples with the emphasis on efficiency, conciseness and legibility of the output.

Using Galois Ideals for Computing Relative Resolvents

In this paper we show that some ideals which occur in Galois theory are generated by triangular sets of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may and should be exploited in order to obtain more efficient algorithms, and it enables us to present a new algebraic method for computing relative resolvents which works with any polynomial invariant.

PARALLEL COMPUTATION OF RESOLVENTS BY MULTIMODULAR TECHNIQUES AND DECOMPOSITION FORMULA

In this paper we significantly extend the limits of existing algebraic algorithms for computing Lagrange resolvents. Such algorithms are fundamental in effective Galois theory. However, they usually are of high complexity. For the case of absolute resolvents, N. Rennert has shown the value of a multimodular approach to reduce the complexity in space and time. We improve and generalize his work to the computation of any resolvent. In addition, we introduce a decomposition formula which allows us to split modular resolvents into resolvents of smaller degrees, thus both speeding computations and making it possible to efficiently parallelize them.