Direct symbolic transformation from 3D cartesian into hyperboloidal coordinates

A direct transformation from cartesian coordinates into hyperboloidal coordinates (considered for biaxial hyperboloids) is presented in this paper. The transformation problem is reduced to the problem of finding the smallest positive root of a fourth degree polynomial. The analysis of the polynomial’s roots is performed by an algebraically complete stratification, based on symbolic techniques (mainly Sturm–Habicht sequences and its properties related to real root counting), of a planar region situated in the positive quadrant. Two approaches for computing the polynomial’s roots are presented, one based on the Merriman method and the other one obtained using the Computer Algebra System Maple. Our approach improves the solution presented in Feltens (2011) [1], being reduced to a few evaluations of symbolic expressions

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]

Recursive Polynomial Remainder Sequence and its Subresultants

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.Comment: 30 pages. Preliminary versions of this paper have been presented at CASC 2003 (arXiv:0806.0478 [math.AC]) and CASC 2005 (arXiv:0806.0488 [math.AC]

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