Springer Proceedings in Mathematics & Statistics. Volume 226

Transcription

1 Springer Proceedings in Mathematics & Statistics Volume 226

2 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at

3 Carsten Schneider Eugene Zima Editors Advances in Computer Algebra In Honour of Sergei Abramov s 70th Birthday, WWCA 2016, Waterloo, Ontario, Canada 123

4 Editors Carsten Schneider Research Institute for Symbolic Computation Johannes Kepler University Linz Austria Eugene Zima Department of Physics and Computer Science Wilfrid Laurier University Waterloo, ON Canada ISSN ISSN (electronic) Springer Proceedings in Mathematics & Statistics ISBN ISBN (ebook) Library of Congress Control Number: Mathematics Subject Classification (2010): 33F1, 68W30, 33-XX Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 This book is devoted to the 70th birthday of Sergei Abramov, whose classical algorithms for symbolic summation and solving linear differential, difference, and q-difference equations inspired many.

6 Preface The Waterloo Workshop on Computer Algebra (WWCA-2016) was held on July 23 24, 2016, at Wilfrid Laurier University (Waterloo, Ontario, Canada). The workshop provided a forum for researchers and practitioners to discuss recent advances in the area of Computer Algebra. WWCA-2016 was dedicated to the 70th birthday of Sergei Abramov (Computer Center of the Russian Academy of Sciences, Moscow, Russia) whose influential contributions to symbolic methods are highly acknowledged by the research community and adopted by the leading Computer Algebra systems. The workshop attracted world-renowned experts in Computer Algebra and symbolic computation. Presentations on original research topics or surveys of the state of the art within the research area of Computer Algebra were made by Moulay A. Barkatou, University of Limoges, France Shaoshi Chen, Chinese Academy of Sciences, China Mark van Hoeij, Florida State University, USA Manuel Kauers, Johannes Kepler University, Austria Christoph Koutschan, RICAM, Austria Ziming Li, Chinese Academy of Sciences, China Johannes Middeke, RISC, Johannes Kepler University Linz, Austria Mark Round, RISC, Johannes Kepler University Linz, Austria Evans Doe Ocansey, RISC, Johannes Kepler University Linz, Austria Carsten Schneider, RISC, Johannes Kepler University Linz, Austria Eric Schost, University of Waterloo, Canada Vo Ngoc Thieu, RISC, Johannes Kepler University Linz, Austria Eugene Zima, WLU, Canada Success of the workshop was due to the generous support of the Office of the President, Research Office, and Department of Physics and Computer Science of the Wilfrid Laurier University. This book presents a collection of formally refereed selected papers submitted after the workshop. Topics discussed in this book are the latest achievements in algorithms of symbolic summation, factorization, symbolic-numeric linear algebra, vii

7 viii Preface and linear functional equations, i.e., topics of symbolic computations that were extensively advanced due to Sergei s influential works. In Chapter On Strongly Non-singular Polynomial Matrices (Sergei A. Abramov and Moulay A. Barkatou), an algorithm is worked out that decides whether or not a matrix with polynomial entries is a truncation of an invertible matrix with power series entries. Using this new insight, the computation of solutions of higher order linear differential systems in terms of truncated power series is pushed forward. In particular, a criterion is provided when a truncation of a power series solution can be computed. In Chapter On the Computation of Simple Forms and Regular Solutions of Linear Difference Systems (Moulay A. Barkatou, Thomas Cluzeau and Carole El Bacha), first-order linear difference systems with factorial series coefficients are treated. Factorial series, which play an important role in the analysis of linear difference systems, are similar to power series, but instead of z n the power is 1 over the nth rising factorial of z. New reduction algorithms are presented to provide solutions for such systems in terms of factorial series. In Chapter Refined Holonomic Summation Algorithms in Particle Physics (Johannes Blümlein, Mark Round and Carsten Schneider), the summation approach in the setting of difference rings is enhanced by tools from the holonomic system approach. It is now possible to deal efficiently with summation objects that are described by linear inhomogeneous recurrences whose coefficients depend on indefinite nested sums and products. The derived methods are tailored to challenging sums that arise in particle physics problems. In Chapter Bivariate Extensions of Abramov s Algorithm for Rational Summation (Shaoshi Chen), a general framework is developed to decide algorithmically if a bivariate sequence/function can be summed/integrated by solving the bivariate anti-difference/anti-differential equation. The summation/integration problem for double sums/integrals is elaborated completely: Besides the rational case also its q-generalization is treated for both, the summation and integration setting. In Chapter A q-analogue of the Modified Abramov-Petkovšek Reduction (Hao Du, Hui Huang and Ziming Li), an algorithm is provided that simplifies a truncated q-hypergeometric sum to a summable part that can be expressed in terms of q-hypergeometric products and a non-summable sum whose summand satisfies certain minimality criteria. In case that the input sum is completely summable, this representation is computed (i.e., the non-summable part is zero). Experimental results demonstrate that this refined reduction of the Abramov-Petkovšek reduction gains substantial speedups. In Chapter Factorization of C-finite Sequences (Manuel Kauers and Doron Zeilberger), a new algorithm is elaborated that factorizes a linear recurrence with constant coefficients into two non-trivial factors whenever this is possible. Instead of the usage of expensive Gröbner basis computation, the factorization task is reduced to a combinatorial assignment problem. Concrete examples demonstrate the practical relevance of these results.

8 Preface ix In Chapter Denominator Bounds for Systems of Recurrence Equations Using PR-Extensions (Johannes Middeke and Carsten Schneider), a general framework in the setting of difference fields is presented to tackle coupled systems of linear difference equations. Besides the rational and q-rational cases, the coefficients of the system and the solutions thereof might be given in terms of indefinite nested sums and products. In Chapter Representing (q )Hypergeometric Products and Mixed Versions in Difference Rings (Evans Doe Ocansey and Carsten Schneider), algorithms are presented that enable one to represent a finite set of hypergeometric products and more generally q-hypergeometric products and their mixed versions within the difference ring theory of RPR-extensions. As a consequence, one can solve the zero-recognition problem for expressions in terms of such products and obtains expressions in terms of products whose sequences are algebraically independent among each other. In Chapter Linearly Satellite Unknowns in Linear Differential Systems (Anton A. Panferov), an algorithm is worked out that determines for a given system of linear differential equations whether or not a component of a solution can be expressed in terms of the solutions of a fixed set of other components. The solutions of these components can be composed by taking their linear combination and by applying the differential operator to them. The derived knowledge turns out to be useful if one is only interested in parts of the solution. In Chapter Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra (Peter Paule and Silviu Radu), the connection of q-series in partition theory and modular functions, with the Rogers-Ramanujan functions as key players, is worked out. Special emphasis is put on Computer Algebra aspects dealing, e.g., with zero recognition of modular forms, q-holonomic approximations of modular forms, or projections of q-holonomic series. This algorithmic machinery is illuminated by the derivation of Felix Klein s classical icosahedral equation. This book would not have been possible without the contributions and hard work of the anonymous referees, who supplied detailed referee reports and helped authors to improve their papers significantly. Linz, Austria Waterloo, Canada October 2017 Carsten Schneider Eugene Zima

9 Contents On Strongly Non-singular Polynomial Matrices... 1 Sergei A. Abramov and Moulay A. Barkatou On the Computation of Simple Forms and Regular Solutions of Linear Difference Systems Moulay A. Barkatou, Thomas Cluzeau and Carole El Bacha Refined Holonomic Summation Algorithms in Particle Physics Johannes Blümlein, Mark Round and Carsten Schneider Bivariate Extensions of Abramov s Algorithm for Rational Summation Shaoshi Chen A q-analogue of the Modified Abramov-Petkovšek Reduction Hao Du, Hui Huang and Ziming Li Factorization of C-Finite Sequences Manuel Kauers and Doron Zeilberger Denominator Bounds for Systems of Recurrence Equations Using PR-Extensions Johannes Middeke and Carsten Schneider Representing (q )Hypergeometric Products and Mixed Versions in Difference Rings Evans Doe Ocansey and Carsten Schneider Linearly Satellite Unknowns in Linear Differential Systems Anton A. Panferov Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra Peter Paule and Silviu Radu xi