Constructive Arithmetics in Ore Localizations of Domains

For a non-commutative domain $R$ and a multiplicatively closed set $S$ the (left) Ore localization of $R$ at $S$ exists if and only if $S$ satisfies the (left) Ore property. Since the concept has been introduced by Ore back in the 1930's, Ore localizations have been widely used in theory and in applications. We investigate the arithmetics of the localized ring $S^{-1}R$ from both theoretical and practical points of view. We show that the key component of the arithmetics is the computation of the intersection of a left ideal with a submonoid $S$ of $R$. It is not known yet, whether there exists an algorithmic solution of this problem in general. Still, we provide such solutions for cases where $S$ is equipped with additional structure by distilling three most frequently occurring types of Ore sets. We introduce the notion of the (left) saturation closure and prove that it is a canonical form for (left) Ore sets in $R$. We provide an implementation of arithmetics over the ubiquitous $G$-algebras in \textsc{Singular:Plural} and discuss questions arising in this context. Numerous examples illustrate the effectiveness of the proposed approach.Comment: 24 page

Note on Integer Factoring Methods IV

This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.Comment: 20 Pages, New Versio

PBW bases, non-degeneracy conditions and applications

Abstract. We establish an explicit criteria (the vanishing of non–degeneracy conditions) for certain noncommutative algebras to have Poincaré–Birkhoff– Witt basis. We study theoretical properties of such G–algebras, con-cluding they are in some sense ”close to commutative”. We use the non–degeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras. The famous Poincaré–Birkhoff–Witt (or, shortly, PBW) theorem, which ap-peared at first for universal enveloping algebras of finite dimensional Lie algebras ([7]), plays an important role in the representation theory as well as in the the-ory of rings and algebras. Analogous theorem for quantum groups was proved by G. Lusztig and constructively by C. M. Ringel ([6]). Many authors have proved the PBW theorem for special classes of noncom-mutative algebras they are dealing with ([17], [18]). Usually one uses Bergman’s Diamond Lemma ([4]), although it needs some preparations to be done before ap-plying it. We have defined a class of algebras where the question ”Does this algebra have a PBW basis? ” reduces to a direct computation involving only basic polyno-mial arithmetic. In this article, our approach is constructive and consists of three tasks. Firstly, we want to find the necessary and sufficient conditions for a wide class of algebras to have a PBW basis, secondly, to investigate this class for useful properties, and thirdly, to apply the results to the study of certain special types of algebras. The first part resulted in the non–degeneracy conditions (Theorem 2.3), the second one led us to the G – and GR–algebras (3.4) and their properties (Theorem 4.7, 4.8), and the third one — to the notion of G–quantization and to the descrip-tion and classification of G–algebras among the quadratic and diffusion algebras

Bounds and definability in polynomial rings

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials $f_1,...,f_n\in R[X_1,...,X_N]$ with coefficients in a Pr\"ufer domain $R$ can be generated by elements whose degrees are bounded by a number only depending on $N$, $n$ and the degree of the $f_j$. This implies that if $R$ is a B\'ezout domain, then the generators can be parametrized in terms of the coefficients of $f_1,...,f_n$ using the ring operations and a certain division function, uniformly in $R$.Comment: 36 page

Developments in Random Matrix Theory

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given.Comment: 22 pages, Late

Survey on the DD-module fsf^s

In this survey we discuss various aspects of the singularity invariants with differential origin derived from the $D$-module generated by $f^s$.Comment: 30 page

The Jacobian Conjecture, together with Specht and Burnside-type problems

We explore an (unpublished) approach to the famous Jacobian Conjecture by means of identities of algebras, discovered by the brilliant deceased mathematician, Alexander Vladimirovich Yagzhev (1951{2001). This approach also indicates some very close connections between mathematical physics, universal algebra and automorphisms of polynomial algebrasComment: 48 page

Calculating power integral bases by using relative power integral bases

Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields

A Jordan-Hoelder Theorem for Differential Algebraic Groups

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.Comment: 39 pages; typos corrected, and more detailed explanations added. This is the final version to be published in the Journal of Algebr

Complex symmetric Hamiltonians and exceptional points of order four and five

In the broad context of physics ranging from classical experimental optics to quantum mechanics of unitary as well as non-unitary systems there emerge interesting phenomena related to the presence of the so called Kato's exceptional points in the space of parameters. An elementary linear-algebraic method of their localization is proposed and applied to the class of tridiagonal $N$ by $N$ complex symmetric toy-model generators of evolution $H=H(\gamma)$. The implementation of the method is shown to provide new models with the exceptional points $\gamma=\gamma^{(EP)}$ of higher orders. Two distinct areas of applicability are expected to lie (1) in quantum mechanics of non-Hermitian (open as well as closed) systems, and (2) in the experiments using the coupled classical optical waveguides simulating the EP-related effects in the laboratory.Comment: 25 pp., 2 figure