Parallel and distributed Gr\"obner bases computation in JAS

This paper considers parallel Gr\"obner bases algorithms on distributed memory parallel computers with multi-core compute nodes. We summarize three different Gr\"obner bases implementations: shared memory parallel, pure distributed memory parallel and distributed memory combined with shared memory parallelism. The last algorithm, called distributed hybrid, uses only one control communication channel between the master node and the worker nodes and keeps polynomials in shared memory on a node. The polynomials are transported asynchronous to the control-flow of the algorithm in a separate distributed data structure. The implementation is generic and works for all implemented (exact) fields. We present new performance measurements and discuss the performance of the algorithms.Comment: 14 pages, 8 tables, 13 figure

Purely exponential parametrizations and their group-theoretic applications

This paper is mainly motivated by the analysis of the so-called Bounded Generation property (BG) of linear groups (in characteristic $0$), which is known to admit far-reaching group-theoretic implications. We achieve complete answers to certain longstanding open questions about Bounded Generation (sharpening considerably some earlier results). For instance, we prove that linear groups boundedly generated by semi-simple elements are necessarily virtually abelian. This is obtained as a corollary of sparseness of subsets which are likewise generated. In the paper in fact we go further, framing (BG) in the more general context of (Purely) Exponential Parametrizations (PEP) for subsets of affine spaces, a concept which unifies different issues. Using deep tools from Diophantine Geometry (including the Subspace Theorem), we systematically develop a theory showing in particular that for a (PEP) set over a number field, the asymptotic distribution of its points of Height at most $T$ is always $\sim c(\log T)^r$, with certain constants $c>0$ and $r\in \mathbb{Z}_{\geq 0}$. (This shape fits with a well-known viewpoint first put forward by Manin.

Torsion points, Pell's equation, and integration in elementary terms

The main results of this paper involve general algebraic differentials $\omega$ on a general pencil of algebraic curves. We show how to determine if $\omega$ is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of André and Hrushovski and with the Grothendieck-Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin-Mumford type allied to the Zilber-Pink conjectures: we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations $A^2-DB^2=1$ over a polynomial ring. We determine whether the Pell equation (with squarefree $D$) is solvable for infinitely many members of the pencil

Braids: A Survey

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

Triangular modular curves of low genus and geometric quadratic Chabauty

This manuscript consists of two parts. In the first part, we study generalizations of modular curves: triangular modular curves. These curves have played an important role in recent developments in number theory, particularly concerning hypergeometric abelian varieties and approaches to solving generalized Fermat equations. We provide a new result that shows that there are only finitely many Borel-type triangular modular curves of any fixed genus, and we present an algorithm to list all such curves of a given genus. In the second part of the manuscript, we explore the problem of computing the set of rational points on a smooth, projective, geometrically irreducible curve of genus g\u3e1 over Q. We study the geometric quadratic Chabauty method, which is an effective method for producing a finite set of p-adic points containing the rational points of the curve. This method is due Edixhoven and Lido. We overview the method and discuss explicit algorithms for finding rational points. We also present a comparison is with the classical (cohomological) quadratic Chabauty method

Non-commutative Noetherian Unique Factorisation Domains.

The commutative theory of Unique Factorisation Domains (UFDs) is well-developed (see, for example, Zariski¬-Samuel[75], Chapter 1, and Cohn[2l], Chapter 11). This thesis is concerned with classes of non-commutative Noetherian rings which are generalisations of the commutative idea of UFD. We may characterise commutative Unique Factorisation Domains amongst commutative domains as those whose height-l prime ideals P are all principal (and completely prime ie R/P is a domain). In Chatters[l3], A.W.Chatters proposed to extend this definition to non-commutative Noetherian domains by the simple expedient of deleting the word commutative from the above. In Section 2.1 we describe the definition and some of the basic theory of Noetherian UFDs, and in Sections 2.2, 2.3, and 2.4 demonstrate that large classes of naturally occuring Noetherian rings are in fact Noetherian UFDs under this definition. Chapter 3 develops some of the more surprising consequnces of the theory by indicating that if a Noetherian UFD is not commutative then it has much better properties than if it were. All the work, unless otherwise indicated, of this Chapter is original and the main result of Section 3.1 appears Gilchrist-Smith[30]. In the consideration of Unique Factorisation Domains the set C of elements of a UFD R which are regular modulo all the height-l prime ideals of R plays a crucial role, akin to that of the set of units in a commutative ring. The main motivation of Chapter 4 has been to generalise the commutative principal ideal theorem to non-commutative rings and so to enable us to draw conclusions about the set C. We develop this idea mainly in relation to two classes of prime Noetherian rings namely PI rings and bounded maximal orders. Chapter 5 then returns to the theme of unique factorisation to consider firstly a more general notion to that of UFD, namely that of Unique Factorisation Ring (UFR) first proposed by Chatters-Jordan[17]. In Section 5.2 we prove some structural results for these rings and in particular an analogue of the decomposition R -snT for R a UFD. Finally section 5.3 briefly sketches two other variations on the theme of unique factorisation due primarily to Cohn[ 20], and Beauregard [4], and shows that in general these theories are distinct

Proceedings of the 4th International Conference on Principles and Practices of Programming in Java

This book contains the proceedings of the 4th international conference on principles and practices of programming in Java. The conference focuses on the different aspects of the Java programming language and its applications

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout