Galois theory, splitting fields and computer algebra

AbstractWe provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.Comment: 34 pages, lot of figure

Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

This paper presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Gr\"obner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results clearly show that our algorithms improve the aforementioned iterative methods, in terms of both efficiency and accuracy.Comment: 17 page

A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

Abstract Let A be a commutative ring and M be a projective module of rank k with n generators. Let h = n − k. Standard computations show that M becomes free after localizations in`n k´c omaximal elements (see Theorem 5). When the base ring A contains a field with at least hk + 1 non-zero distinct elements we construct a comaximal family G with at most (hk + 1)(nk + 1) elements such that for each g ∈ G, the module Mg is free over A[1/g]

The Berlekamp-Massey Algorithm revisited

We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.Comment: in English and French version

On the implicit equation of conics and quadrics offsets

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in ?A Treatise on the Analytic Geometry of Three Dimensions? by G. Salmon for offsets to non-degenerate conics and central quadrics.This research was funded by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P