Parallel computation of Lagrange resolvents by multi-resolvents
Résumé
The goal of this paper is the parallel computation of Lagrange resolvents of a univariate polynomial. The computation of Lagrange resolvents of a univariate polynomial has significance for Galois theory. Since Lagrange's algorithms, many other algorithms for computing some particular resolvents, called absolute, were developed from the fundamental theorem of symmetric functions. The algebraic algorithms for non absolute resolvents are few and recent because they use galoisian ideals that were introduced recently. However these algorithms become time and space consuming when the degree of the polynomial increases. This motivates their parallelisation. In 2004 N. Rennert presented a multimodular method for computing absolute resolvents of polynomials with integer coefficients. We show that the same techniques can be extended to any resolvent. This method is naturally parallelisable. Moreover, we give a decomposition formula of resolvents which makes possible another level of parallelisation. This leads to an algorithm with a doubly parallel character.