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The aim of this paper is to exploit the algorithms of paper Experimental Math. 8 (1999) in order to produce a new algebraic method for computing efficiently absolute Lagrange resolvent, a fundamental tool in constructive algebraic Galois theory. This article is composed of two parts.The main idea of the first part is to break up the calculation of absolute resolvent into smaller computations. Since a multi-resolvent is a factor of a resolvent, the whole resolvent may be computed by means of several multi-resolvents.The idea of the second part is that an irreducible polynomial over Z might be reducible over Z /pZ for certain integer p. So the first part can be applied and then, the Chinese remainder theorem allows to lift up the resolvent over Z. [Copyright &y& Elsevier]

Abstract: Algebraic surfaces–which are frequently used in geometric modelling–are represented either in implicit or parametric form. Several techniques for parametrizing a rational algebraic surface as a whole exist. However, in many applications, it suffices to parametrize a small portion of the surface. This motivates the analysis of local parametrizations, i.e., parametrizations of a small neighborhood of a given point of the surface . In this paper we introduce several techniques for generating such parametrizations for nonsingular cubic surfaces. For this class of surfaces, it is shown that the local parametrization problem can be solved for all points, and any such surface can be covered completely. [Copyright &y& Elsevier]

Abstract: Let be a finite group. It is easy to compute the character of corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation. Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups. We shall describe an algorithm for computing an irreducible matrix representation which affords a given character of a given group . The algorithm uses properties of the structure of which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. All results in this paper have been implemented in the GAP package REPSN. [Copyright &y& Elsevier]

In this paper, we propose a semi-numerical algorithm for computing absolute factorization of multivariate polynomials. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, this method provides an efficient probabilistic algorithm for medium degrees. Two implementations are presented and compared to other algorithms. [Copyright &y& Elsevier]