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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: This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. [Copyright &y& Elsevier]

Abstract: Faugère’s is one of the fastest known algorithm to compute Gröbner bases (see ). The efficiency of this algorithm comes from two criteria namely criteria, for which it assigns to each polynomial a signature. In this paper, we study the importance of choosing an ordering on the signatures, and we propose a novel ordering on the signatures. Using this ordering, we extend the criteria, and we describe a new algorithm like based on these extended criteria which (despite of ) does not depend on the order of input polynomials. We have implemented our algorithm in Magma for computing the Gröbner basis of a general ideal, and we evaluate its performance via some examples. We show that the new algorithm is more stable and more efficient than , and experimentally it stops at a lower degree than . [ABSTRACT FROM AUTHOR]

Abstract: Let and , where is an effective field and and are given a suitable asymptotic ordering . Consider the mapping , where . For , it is natural to ask how to solve the system . In this paper, we will effectively characterize and show how to compute a so called distinguished right inverse of . We will also characterize the solution space of the homogeneous equation . [Copyright &y& Elsevier]

Abstract: In previous work, we have introduced the technique of relaxed power series computations. With this technique, it is possible to solve implicit equations almost as quickly as doing the operations which occur in the implicit equation. Here “almost as quickly” means that we need to pay a logarithmic overhead. In this paper, we will show how to reduce this logarithmic factor in the case when the constant ring has sufficiently many th roots of unity. [Copyright &y& Elsevier]

Abstract: Let be a linear differential operator, where is an effective algebraically closed subfield of . It can be shown that the differential Galois group of  is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices. In this paper, we present a numeric–symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision. Even though there is no straightforward way to find a “sufficient precision” for guaranteeing the correctness of the end result, it is often possible to check a posteriori whether the end result is correct. In particular, we present a non-heuristic algorithm for the factorization of linear differential operators. [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]