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Gröbner Bases

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Page 1: Gröbner Bases - Springer978-4-431-54574... · 2017-08-27 · Sturmfels demonstrated an exciting application of Gröbner bases to algebraic statistics. With these backgrounds, in

Gröbner Bases

Page 2: Gröbner Bases - Springer978-4-431-54574... · 2017-08-27 · Sturmfels demonstrated an exciting application of Gröbner bases to algebraic statistics. With these backgrounds, in
Page 3: Gröbner Bases - Springer978-4-431-54574... · 2017-08-27 · Sturmfels demonstrated an exciting application of Gröbner bases to algebraic statistics. With these backgrounds, in

Takayuki HibiEditor

Gröbner Bases

Statistics and Software Systems

123

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EditorTakayuki HibiDepartment of Pure and Applied

MathematicsOsaka UniversityToyonaka, Osaka, Japan

ISBN 978-4-431-54573-6 ISBN 978-4-431-54574-3 (eBook)DOI 10.1007/978-4-431-54574-3Springer Tokyo Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013954414

© Springer Japan 2013This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe 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 informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed. Exempted from this legal reservation are brief excerpts in connectionwith reviews or scholarly analysis or material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work. Duplication ofthis publication or parts thereof is permitted only under the provisions of the Copyright Law of thePublisher’s location, in its current version, and permission for use must always be obtained from Springer.Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violationsare liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

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Preface

The present volume is an English translation of the Japanese mathematics book“Gröbner Dojo” (Kyoritsu Shuppan Co., Ltd., September 2011). The dojo is aJapanese traditional term, which represents, in general, the place for the trainingof the judo, an Olympic sport. Our book “Gröbner Dojo” invites the reader to theGröbner world, a fascinating research area of mathematics, where three aspectsof Gröbner bases, viz., theory, application and computation, are linked effectivelyand systematically. A beginner including a first year graduate student can learn theABC’s of Gröbner bases from “Gröbner Dojo.” In addition, “Gröbner Dojo” can bea how-to book for users of Gröbner bases such as scientists engaging in statisticalproblems as well as engineers being active in industrial society. This is the reasonwhy we select the term dojo for the title of our book.

An idea of Gröbner bases was apparently studied by Francis Sowerby Macaulayin 1927; he succeeded in finding a combinatorial characterization of the Hilbertfunctions of homogeneous ideals of the polynomial ring. Later, current definitionof Gröbner bases was independently introduced by Heisuke Hironaka in 1964 andBruno Buchberger in 1965. However, after the discovery of the notion of Gröbnerbases by Hironaka and Buchberger, no activity had been done for about twentyyears. A first breakthrough was done by David Bayer and Michael Stillman inthe middle of 1980s, who created the computer algebra system Macaulay with thehelp of Gröbner bases. In 1995 the second breakthrough was achieved by BerndSturmfels, who discovered the fascinating relation between regular triangulationsof convex polytopes and Stanley–Reisner ideals of initial ideals of toric ideals.Furthermore, the third breakthrough arose in 1998 when Persi Diaconis and BerndSturmfels demonstrated an exciting application of Gröbner bases to algebraicstatistics.

With these backgrounds, in October 2008, the JST1 CREST2 Hibi project startedtoward the progress of theory and application of Gröbner bases together with the

1Japan Science and Technology Agency.2Core Research for Evolutional Science & Technology.

v

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vi Preface

development of their algorithms. The publication of “Gröbner Dojo” was alreadyannounced in the original research plan of the project.

“Gröbner Dojo” is a comprehensive textbook to learn algebraic statistics basedon Gröbner bases. First, in Chap. 1, starting from Dickson’s Lemma, a classicalresult in combinatorics, we explain the division algorithm, Buchberger criterion andBuchberger algorithm. Then the theory of elimination follows and toric ideals areintroduced. In addition, the basic theory of Hilbert functions is discussed. Moreover,the historical background of Gröbner bases is surveyed.

Chapter 2 is a warming-up drill for learning the basic ideas of using mathematicalsoftware. We choose the mathematical software environment named “MathLibre.”It is a collection of mathematical software and free documents. MathLibre is a kindof Live Linux system. Linux is a system compatible with UNIX, a traditional OSfor specialists. Many mathematical research systems are developed on UNIX. Thebasic usages and fundamental ideas of UNIX are introduced.

Chapter 3 discusses how to compute various objects related to Gröbner basesexplained in Chap. 1. After introducing fundamental tools for efficient Gröbner basiscomputation, we illustrate fundamental computations related to Gröbner bases byusing Macaulay2, SINGULAR, CoCoA and Risa/Asir.

In writing Chaps. 1–3, we do not assume that the reader is familiar with theoryand computation of Gröbner bases. If the reader has an experience of handlingGröbner bases, then these chapters may be skipped partly. On the other hand, sincethe latter Chaps. 4–6 are written independently, after reading the former Chaps. 1–3,the reader can read Chaps. 4–6 in any order.

Chapter 4 is devoted to algebraic statistics. This field was initiated by the workof Diaconis and Sturmfels in 1998 and the work of Pistone and Wynn in 1996, bothapplying Gröbner basis theory to statistics. Since then the field has been developingrapidly with providing challenging problems to both statisticians and algebraists.

Chapter 5 plays the introduction to two fascinating rainbow bridges betweenthe world of Gröbner bases and that of convex polytopes. One is the big theoryof Gröbner fans and state polytopes. The other is the reciprocal relation betweeninitial ideals of toric ideals and triangulations of convex polytopes.

Recently, Gröbner bases of rings of differential operators turn out to be usefulto numerical evaluations of a broad class of normalizing constants in statistics. Themethod is called the holonomic gradient method, which is a rapidly growing areain algebraic statistics. Chapter 6 is a self-contained exposition to invite readers tothese topics. Nobuki Takayama, the author of Chap. 6, thanks Professor FranciscoCastro-Jiménez for providing useful comments to a draft of Chap. 6.

Finally, Chap. 7 provides a collection of rich problems and their answers byutilizing various software systems, such as Risa/Asir, 4ti2, polymake, R, and so on.Chapter 7 complements Chaps. 4–6, and is helpful for readers to understand how touse software systems to study or apply Gröbner bases.

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Preface vii

On behalf of the JST CREST Hibi project, I would express our thanks to JSTfor providing financial support, which made it possible to organize internationalconferences and to employ promising young researchers. Finally, I am grateful toMs. Kaoru Yamano for her administrative job for Hibi project.3

Toyonaka, Osaka, Japan Takayuki Hibi

3All computer programs appearing in this volume and a list of corrections are available athttp://www.math.kobe-u.ac.jp/OpenXM/Math/dojo-en

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Contents

1 A Quick Introduction to Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Takayuki Hibi1.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Monomials and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Dickson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 Monomial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.5 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.6 Hilbert Basis Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.1 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Reduced Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Buchberger Criterion and Buchberger Algorithm .. . . . . . . . . . . . . . . . . . . 161.3.1 S -Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.2 Buchberger Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.3 Buchberger Algorithm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4 Elimination Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.1 Elimination Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.2 Solving Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.5 Toric Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.5.1 Configuration Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.5.2 Binomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.3 Toric Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.4 Toric Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.6 Residue Class Rings and Hilbert Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 411.6.1 Residue Classes and Residue Class Rings . . . . . . . . . . . . . . . . . . 411.6.2 Macaulay’s Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.3 Hilbert Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.7 Historical Background.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

ix

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x Contents

2 Warm-Up Drills and Tips for Mathematical Software . . . . . . . . . . . . . . . . . . 55Tatsuyoshi Hamada2.1 Using MathLibre.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.1 How to Get MathLibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.1.2 How to Boot and Shut Down MathLibre . . . . . . . . . . . . . . . . . . . 562.1.3 Various Mathematical Software Packages . . . . . . . . . . . . . . . . . . 56

2.2 File Manager .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2.1 New Folder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.2.2 New Text File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3 Terminal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.3.1 Files and Directories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3.2 Text Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.3 Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.4 Character Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4 How to Write Mathematical Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.4.1 Writing a TEX Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.4.2 Making a PDF File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.4.3 Brief Introduction to TEX Source Code . . . . . . . . . . . . . . . . . . . . . 662.4.4 Math Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.4.5 graphicx Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.5 Various Math Software Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5.1 KSEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5.2 GeoGebra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.5.3 Surf Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.5.4 Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.5.5 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.5.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.6 Emacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.6.1 Starting Emacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.6.2 Cut and Paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.6.3 Editing Multiple Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.6.4 Remove Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.6.5 Point, Mark, and Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.6.6 Undo, Redo, and Etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.6.7 Command and Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.6.8 Math Software Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.7 Other Ways of Booting MathLibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.7.1 Various Virtual Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.7.2 Making a USB-Bootable MathLibre . . . . . . . . . . . . . . . . . . . . . . . . 1042.7.3 How to Install MathLibre to an Internal Hard Disk . . . . . . . . 105

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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3 Computation of Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Masayuki Noro3.1 Improving the Efficiency of the Buchberger Algorithm . . . . . . . . . . . . . 108

3.1.1 Elimination of Unnecessary S-Pairs . . . . . . . . . . . . . . . . . . . . . . . . 1103.1.2 Strategies for Selecting S-Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.1.3 Homogenization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.1.4 Buchberger Algorithm (an Improved Version) . . . . . . . . . . . . . 113

3.2 Using Macaulay2, SINGULAR, and CoCoA . . . . . . . . . . . . . . . . . . . . . . . . 1153.2.1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2.2 Packages, Libraries, and Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.2.3 Rings, Term Orderings, and Polynomials . . . . . . . . . . . . . . . . . . . 1203.2.4 Computation of Gröbner Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2.5 Computation of Initial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.2.6 Computation of Quotient and Remainder .. . . . . . . . . . . . . . . . . . 127

3.3 Operations on Ideals by Using Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . 1303.3.1 Elimination Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.3.2 Sum, Product, and Intersection of Ideals. . . . . . . . . . . . . . . . . . . . 1323.3.3 Radical Membership Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.3.4 Ideal Quotient and Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.3.5 Computation of a Radical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.4 Change of Ordering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.4.1 FGLM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.4.2 Hilbert-Driven Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.5 Computation of Gröbner Bases for Modules . . . . . . . . . . . . . . . . . . . . . . . . . 1423.5.1 Term Orderings for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.5.2 Buchberger Algorithm for Modules . . . . . . . . . . . . . . . . . . . . . . . . . 1433.5.3 Computation of Syzygy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.6 Computation in Risa/Asir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.6.1 Starting Risa/Asir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.6.2 Help Files and Manuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.3 Reading and Writing Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.4 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.5 Term Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.6.6 Computation of Gröbner Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483.6.7 Computation of Initial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.6.8 Computation of the Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.6.9 Elimination .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.6.10 Computation of Minimal Polynomials . . . . . . . . . . . . . . . . . . . . . . 1503.6.11 Change of Orderings for Zero-Dimensional Ideals. . . . . . . . . 1513.6.12 Ideal Operations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.7 An Example of Programming in Macaulay2 . . . . . . . . . . . . . . . . . . . . . . . . . 1533.7.1 Primary Decomposition of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.7.2 SYCI Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543.7.3 Implementation in Macaulay2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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3.8 Additional Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.9 Answers to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4 Markov Bases and Designed Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Satoshi Aoki and Akimichi Takemura4.1 Conditional Tests of Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.1.1 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.1.2 2 � 2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.1.3 Similar Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.1.4 I � J Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2 Markov Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.2.1 Markov Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.2.2 Examples of Markov Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.2.3 Markov Bases and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.3 Design of Experiments and Markov Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.3.1 Two-Level Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054.3.2 Analysis of Full Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3.3 Analysis of Fractional Factorial Designs . . . . . . . . . . . . . . . . . . . 213

4.4 Research Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.4.1 Topics with Markov Bases for Models without

Three-Factor Interactions for Three-WayContingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

4.4.2 Topics Related to the Efficient Algorithmfor a Markov Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.4.3 Topics on Modeling Experimental Data . . . . . . . . . . . . . . . . . . . . 219References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5 Convex Polytopes and Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Hidefumi Ohsugi5.1 Convex Polytopes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.1.1 Convex Polytopes and Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2245.1.2 Faces of Convex Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.1.3 Polyhedral Complices and Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.2 Initial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305.2.1 Initial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.2.2 Weight Vectors and Monomial Orders . . . . . . . . . . . . . . . . . . . . . . 2315.2.3 Universal Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.3 Gröbner Fans and State Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.3.1 Gröbner Fans of Principal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.3.2 Gröbner Fans and State Polytopes of

Homogeneous Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2375.4 State Polytopes of Toric Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5.4.1 Circuits and Graver Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2445.4.2 Upper Bounds on the Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

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5.4.3 Lawrence Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2505.4.4 Computations of State Polytopes .. . . . . . . . . . . . . . . . . . . . . . . . . . . 251

5.5 Triangulations of Convex Polytopes and Gröbner Bases . . . . . . . . . . . . 2525.5.1 Unimodular Triangulations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.5.2 Regular Triangulations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.5.3 Initial Complices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2565.5.4 Secondary Polytopes and State Polytopes . . . . . . . . . . . . . . . . . . 261

5.6 Ring-Theoretic Properties and Triangulations . . . . . . . . . . . . . . . . . . . . . . . 2645.6.1 Lexicographic Triangulations and Unimodular

Configurations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2645.6.2 Reverse Lexicographic Triangulations

and Compressed Configurations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.6.3 Normality of Toric Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.7 Examples of Configuration Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.7.1 Configuration Matrices of Finite Graphs . . . . . . . . . . . . . . . . . . . 2725.7.2 Configuration Matrices of Contingency Tables. . . . . . . . . . . . . 275

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6 Gröbner Basis for Rings of Differential Operatorsand Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Nobuki Takayama6.1 Gröbner Basis for the Ring of Differential Operators

with Rational Function Coefficients R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.2 Zero-Dimensional Ideals in R and Pfaffian Equations .. . . . . . . . . . . . . . 2866.3 Solutions of Pfaffian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2896.4 Holonomic Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2996.5 Gradient Descent for Holonomic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.6 Gröbner Bases in the Ring of Differential Operators

with Polynomial Coefficients D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3046.7 Filtrations and Weight Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.8 Holonomic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3146.9 Relationship Between D and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166.10 Integration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3186.11 Finding a Local Minimum of a Function Defined

by a Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.12 A-Hypergeometric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3306.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

7 Examples and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Hiromasa Nakayama and Kenta Nishiyama7.1 Software .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.2 Markov Bases and Designed Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.2.1 Conditional Tests of Contingency Tables (Sect. 4.1) . . . . . . . 3487.2.2 Markov Basis (Sect. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3507.2.3 Design of Experiments and Markov Basis (Sect. 4.3) . . . . . . 363

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7.3 Convex Polytopes and Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3677.3.1 Convex Polytopes (Sect. 5.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3687.3.2 Initial Ideals (Sect. 5.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.3.3 Gröbner Fans and State Polytopes (Sect. 5.3) . . . . . . . . . . . . . . 3797.3.4 State Polytopes of Toric Ideals (Sect. 5.4) . . . . . . . . . . . . . . . . . . 3847.3.5 Triangulations of Convex Polytopes and

Gröbner Bases (Sect. 5.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.3.6 Ring-Theoretic Properties and Triangulations (Sect. 5.6) . . 4037.3.7 Examples of Configuration Matrices (Sect. 5.7) . . . . . . . . . . . . 407

7.4 Gröbner Basis of Rings of Differential Operatorsand Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4137.4.1 Gröbner Basis for the Ring of Differential

Operators with Rational Function CoefficientsR (Sect. 6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

7.4.2 Zero-Dimensional Ideals in R and PfaffianEquations (Sect. 6.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

7.4.3 Solutions of Pfaffian Equations (Sect. 6.3). . . . . . . . . . . . . . . . . . 4327.4.4 Holonomic Functions (Sect. 6.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4397.4.5 Gradient Descent for Holonomic Functions (Sect. 6.5) . . . . 4407.4.6 Gröbner Bases in the Ring of Differential

Operators with Polynomial Coefficients D (Sect. 6.6) . . . . . 4417.4.7 Holonomic Systems (Sect. 6.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4457.4.8 Relationship of D and R (Sect. 6.9) . . . . . . . . . . . . . . . . . . . . . . . . 4507.4.9 Integration Algorithm (Sect. 6.10) . . . . . . . . . . . . . . . . . . . . . . . . . . 4527.4.10 Finding a Local Minimum of a Function

Defined by a Definite Integral (Sect. 6.11) . . . . . . . . . . . . . . . . . . 462References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

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List of Contributors

Satoshi Aoki Department of Mathematics and Computer Science, KagoshimaUniversity, Korimoto, Kagoshima, Japan

Tatsuyoshi Hamada Department of Applied Mathematics, Fukuoka University,Nanakuma, Fukuoka, Japan

Takayuki Hibi Department of Pure and Applied Mathematics, Graduate School ofInformation Science and Technology, Osaka University, Toyonaka, Osaka, Japan

Hiromasa Nakayama Department of Mathematics, Graduate School of Science,Kobe University, Rokko, Nadaku, Kobe, Japan

Kenta Nishiyama School of Management and Information, University ofShizuoka, Shizuoka, Japan

Masayuki Noro Department of Mathematics, Graduate school of Science, KobeUniversity, Kobe, Japan

Hidefumi Ohsugi Department of Mathematics, College of Science, RikkyoUniversity, Toshima-ku, Tokyo, Japan

Nobuki Takayama Department of Mathematics, Graduate School of Science,Kobe University, Rokko, Nadaku, Kobe, Japan

Akimichi Takemura Department of Mathematical Informatics, Graduate Schoolof Information Science and Technology, University of Tokyo, Bunkyo, Tokyo, Japan

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