After my paper [2] was electronically published by Mathematics of Computation, I came across the PhD thesis of professor I. Y. Potemine [6]. In Section 4.3 of his thesis, an algorithm for factoring polynomials is proposed which is equivalent to the algorithm discussed in my paper. Potemine’s algorithm is acknowledged in my PhD thesis [1]. Our algorithms were found independently, both as analogu...

Several classifications of solvable Lie algebras of small dimension are known. Up to dimension 6 over a real field they were classified by G. M. Mubarakzjanov [Mubarakzjanov 63a, Mubarakzjanov 63b], and up to dimension 4 over any perfect field by J. Patera and H. Zassenhaus [Patera and Zassenhaus 90]. In this paper we explore the possibility of using the computer to obtain a classification of s...

Abstract We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit graduated orders introduced by Plesken and Zassenhaus. These are classified polytrope region. advance ideal theory introducing their class polytropes. This article emphasizes examples computations. It offers first steps in geometric com...

The purpose of these notes is to give a substantially self-contained introduction to the factorization of polynomials over number fields. In particular, we present Zassenhaus’ algorithm and a factoring algorithm using lattice reduction, which were, respectively, the best in practice and in theory, before 2002. We give references for the van Hoeij-Novocin algorithm, currently the best both in pr...

The general structure of a q-analogue of the Zassenhaus formula, the dual of the Baker-Campbell-Hausdorff formula for combining exponential operators, is derived and the first few terms of the disentanglement of a q-exponential operator are explicitly given.

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...

A long-standing problem, posed first by Graham Higman in 1940 [Hi] and later by Brauer [Br], is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG N ZH implies G N H? Many authors have worked on this question but progress has been difficult [Sal. Perhaps the best positive result was that of Whitcomb in 1968 [Wh], who showed that the implication G...