Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

Existence of invariant bases

Abstract: under the natural map. We shall see that this is so if and only if a certain cocycle of G in the full linear group is trivial. In some applications, a rational structure is added to K and G, namely K is the function field of a principal homogeneous space over a group variety G. We shall show that the cocycle involved is then determined rationally. This leads us into a discussion of rational cocycles in ?3, and of their comparison with the ordinary cocycles of Galois theory, i.e. where G is a finite Galois group. All cocycles involved with coefficients in the full linear group split, and in fact the Galois cohomology (in dimension 1) of the group variety of units in an algebra is trivial (Propositions 2 and 5). 1. The invariant subspace. Let K be a field, and G a group of automorphisms of K. By a (G, K)-space M we shall mean a vector space over K which is also a unitary G-module, such that a(aD) = caaaD for aSK, DEzM and ECG. An element D of Mis said to be invariant under G if oD =D for all ECG. A basis (D) = (Di) of M over K will be called invariant if oDi =Di for every CzG and every i.

De-orbiting and re-entry analysis with generalised intrusive polynomial expansions

Abstract: Generalised Intrusive Polynomial Expansion (GIPE) is a novel method for the propagation of multidimensional compact sets through dynamical systems. It generalises the more widely-known Taylor Differential Algebra in that it allows the use of generic polynomial representations of a multi-dimensional set. In particular the paper proposes the use of truncated Tchebycheff series. Unlike Taylor expansions, that are not generally convergent, Tchebycheff expansions provide fast uniform convergence with relaxed continuity and smoothness requirements, guaranteeing near-minimax approximation. This methodology has proven to be competitive for uncertainty propagation in orbital dynamics, especially when dealing with a large number of uncertain variables. Moreover, it provides the user with a complete polynomial representation of the uncertain region at any point of the propagation, allowing remarkable gain of insight into the underlying properties of the uncertain dynamics. The paper presents the application of the GIPE approach to the end-of-life analysis of Low Earth Orbit satellites, with special emphasis on the case of the de-orbiting and re-entry of GOCE and the de-orbiting of objects with high area to mass ratio. The effect of various sources of uncertainty on the end-of-life dynamics is thus analysed, such as the drag model or the accuracy of the initial orbit determination.

On the field intersection problem of solvable quintic generic polynomials

Abstract: We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multi-resolvent polynomials.