Computing Local Artin Maps, and Solvability of Norm Equations - Acciaro, Vincenzo; Klüners, Jürgen (Journal of Symbolic Computation)
Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.
Keywords: Computational Mathematics; Algebra and Number Theory

Explicit Galois Realization of Transitive Groups of Degree up to 15 - Klüners, Jürgen; Malle, Gunter (Journal of Symbolic Computation)
We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of ℚ (t), and in each case compute a polynomial f ∈ ℚ [ x ] with Gal(f) = G.
Keywords: Computational Mathematics; Algebra and Number Theory

A Polynomial with Galois GroupSL2(11) - Klüners, Jürgen (Journal of Symbolic Computation)
We compute a polynomial with Galois group SL₂(11) over ℚ. Furthermore we prove that SL₂(11) is the Galois group of a regular extension of ℚ (t).
Keywords: Computational Mathematics; Algebra and Number Theory

The computation of lyapunov exponents via spatial integration with application to blowout bifurcations - Aston, Philip J.; Dellnitz, Michael (Computer Methods in Applied Mechanics and Engineering)

On the Approximation of Complicated Dynamical Behavior - Dellnitz, Michael; Junge, Oliver (SIAM Journal on Numerical Analysis)

Computation of Essential Molecular Dynamics by Subdivision Techniques - Deuflhard, Peter; Dellnitz, Michael; Junge, Oliver; Schütte, Christof (Computational Molecular Dynamics: Challenges, Methods, Ideas)

Visualizing Complicated Dynamics - Bürkle, David; Dellnitz, Michael; Junge, Oliver; Rumpf, Martin; Spielberg, Michael (Proceedings of Visualization 99)

On Polynomial Decompositions - Klüners, Jürgen (Journal of Symbolic Computation)
We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.
Keywords: Computational Mathematics; Algebra and Number Theory

Computing Automorphisms of Abelian Number Fields - Klüners, Jürgen; Acciaro, Vincenzo (Mathematics of Computation)
Let L = ℚ(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on p-adic techniques. Given mα(x) and a rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α.

An adaptive subdivision technique for the approximation of attractors and invariant measures - Dellnitz, Michael; Junge, Oliver (Computing and Visualization in Science)

Almost Invariant Sets in Chua's Circuit - Dellnitz, Michael; Junge, Oliver (International Journal of Bifurcation and Chaos)
Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.

Exploring invariant sets and invariant measures - Dellnitz, Michael; Hohmann, Andreas; Junge, Oliver; Rumpf, Martin (Chaos: An Interdisciplinary Journal of Nonlinear Science)

An adaptive method for the approximation of the generalized cell mapping - Guder, Rabbijah; Dellnitz, Michael; Kreuzer, Edwin (Chaos, Solitons & Fractals)

A subdivision algorithm for the computation of unstable manifolds and global attractors - Dellnitz, Michael; Hohmann, Andreas (Numerische Mathematik)

KANT V4 - DABERKOW, M.; FIEKER, C.; Klüners, Jürgen; POHST, M.; ROEGNER, K.; SCHÖRNIG, M.; WILDANGER, K. (Journal of Symbolic Computation)
The software packageKANT V4for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.
Keywords: Computational Mathematics; Algebra and Number Theory

On Computing Subfields - Klüners, Jürgen; Pohst, Michael (Journal of Symbolic Computation)
The purpose of this article is to determine all subfields ℚ(β) of fixed degree of a given algebraic number field ℚ(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in ℚ[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of ℚ(α) obtained from the van der Waerden criterion.
Keywords: Computational Mathematics; Algebra and Number Theory

Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper (Dissertation) - Klüners, Jürgen

The Computation of Unstable Manifolds Using Subdivision and Continuation - Dellnitz, Michael; Hohmann, Andreas (Nonlinear Dynamical Systems and Chaos)

Symmetry Breaking Bifurcations of Chaotic Attractors - Aston, Philip J.; Dellnitz, Michael (International Journal of Bifurcation and Chaos)
In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.

Admissible symmetry increasing bifurcations - Dellnitz, M; Heinrich, C (Nonlinearity)