structural properties of polylogarithms · clifford and g. b. preston ... degree in nonlinear...
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Structural Properties of Polylogarithms
MATHEMATICAL Surveys and Monographs
Volume 37
Structural Properties of Polylogarithms
Leonard Lewin Editor
£(fl^^Tvk American Mathematical Society IB Providence, Rhode Island
1980 Mathematics Subject Classification (1985 Revision). Primary 39B50, 33A70, 30D05, 19F27; Secondary 11F67, 39B70, 51M20, 57R20.
Library of Congress Cataloging-in-Publication Data
Structural properties of polylogarithms/Leonard Lewin, editor. p. cm.—(Mathematical surveys and monographs, ISSN 0076-5376; v. 37)
Includes bibliographical references and index. ISBN 0-8218-1634-9 1. Logarithmic functions. I. Lewin, Leonard, 1919- . II. Series.
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10 9 87 6 5 4 3 2 1 95 94 93 92 91
Contents
Preface xiii
Acknowledgments xv
List of Contributors xvii
Chapter 1. The Evolution of the Ladder Concept 1 L. Lewin
1.1 Early History 1 1.2 Functional Equations 2 1.3 More Recent Numerical Results 4 1.4 Current Developments 6 1.5 Base on the Unit Circle and Clausen Function Ladders 8 References 9
Chapter 2. Dilogarithmic Ladders 11 L. Lewin
2.1 Derivation from Kummer's Functional Equation 11 2.2 Relation to Clausen's Function 15 2.3 A Three-Variable Dilogarithmic Functional Equation 17 2.4 Functional Equations in the Complex Plane 18 2.5 Cyclotomic Equations and Rogers' Function 20 2.6 Accessible and Analytic Ladders 21 2.7 Inaccessible Ladders 23 References 25
Chapter 3. Polylogarithmic Ladders 27 M. Abouzahra and L. Lewin
3.1 Kummer's Function and its Relation to the Polylogarithm 27 3.2 Functional Equations for the Polylogarithm 28 3.3 A Generalization of Rogers' Function to the nth Order 31 3.4 Ladder Order-Independence on Reduction of Order 33 3.5 Generic Ladders for the Base Equation if + uq = 1 34 3.6 Examples of Ladders for n < 3 40 3.7 Examples of Ladders for n < 4 44
Vll
viii CONTENTS
3.8 Examples of Ladders for n < 5 45 3.9 Polynomial Relations for Ladders 46 References 47
Chapter 4. Ladders in the Trans-Kummer Region 49 M. Abouzahra and L. Lewin
4.1 Ladder Results to n = 9 for the Base p 49 4.2 Ladder Results to n = 9 for the Base co 53 4.3 Ladder Results to n = 6 for the Base 6 62 4.4 The Nonexistence of Functional Equations at n = 6 with
Arguments Limited to ±zm ( l - z)r{\ + z)5 65 References 67
Chapter 5. Supemumary Ladders 69 M. Abouzahra and L. Lewin
5.1 The Concept of Supemumary Results 69 5.2 Supemumary Results for p = 4 71 5.3 Supemumary Results for p = 5 76 5.4 Supemumary Results for p = 6 78 5.5 Supemumary Results for the Equation-family
„ 6m+l , „ 6 r - l t o n
W + U = 1 OO 5.6 Supemumary Results for an Irreducible Quintic 82 5.7 Supemumary Ladders from a 15-Term Functional Equation 84 5.8 Supemumary Ladders on the Unit Circle 90 References
Chapter 6. Functional Equations and Ladders 97 L. Lewin
6.1 New Categories of Functional Equations 97 6.2 The /^-family of Equations 100 6.3 The a;-family of Equations 109 6.4 The 0-family of Equations 115 Acknowledgements 121 References 121
Chapter 7. Multivariable Polylogarithm Identities 123 G. A. Ray
7.0 Introduction 123 7.1 A General Identity for the Dilogarithm 123 7.2 A General Identity for the Bloch-Wigner Function 135 7.3 A General Identity for the Trilogarithm and D3(z) 141 7.4 Linear Power Relations among Dilogarithms 147 7.5 Cyclotomic Equations and Bases for Polylogarithm Relations 154 7.6 Mahler's Measure and Salem/Pisot Numbers 160 7.7 Recent Results for Supemumary Ladders 165 References 168
CONTENTS ix
Chapter 8. Functional Equations of Hyperlogarithms 171 G. Wechsung
8.1 Hyperlogarithms 171 8.2 Logarithmic Singularities 172 8.3 The Linear Spaces LIn and PLIn 176 8.4 Functional Equations of Hyperlogarithms 177 8.5 A Reduction Problem 181 References 184
Chapter 9. Kummer-Type Functional Equations of Polylogarithms 185 G. Wechsung
9.1 Automorphic Functions 185 9.2 Kummer-Type Functional Equations 186 9.3 A Method to Construct Functional Equations 191 9.4 The Nonexistence of a Kummer-Type Functional Equation
for Li6 197 References 203
Chapter 10. The Basic Structure of Polylogarithmic Equations 205 Z. Wojtkowiak
10.1 Introduction 205 10.2 Canonical Unipotent Connection on Pl(C)\{ax, . . . , an+l} 211 10.3 Horizontal Sections 213 10.4 Easy Lemmas about Monodromy 215 10.5 Functional Equations 216 10.6 Functional Equations of Polylogarithms 218 10.7 Functional Equations of Lower Degree Polylogarithms 223 10.8 Generalized Bloch Groups 228 Acknowledgements 231 References 231
Chapter 11. ^-Theory, Cyclotomic Equations and Clausen's Function 233 J. Browkin
11.1 Algebraic Background 233 11.2 Analytic Background 238 11.3 A^-theoretic Background 248 11.4 Examples 251 11.5 Problems and Conjectures 270 References 272
Chapter 12. Function Theory of Polylogarithms 275 S. Bloch
Chapter 13. Partition Identities and the Dilogarithm 287 J. H. Loxton
13.1 Introduction 287
x CONTENTS
13.2 Cyclotomic Equations 290 13.3 Accessible Relations 291 13.4 Partition Identities 292 13.5 Generalisations and Extensions 297 References 299
Chapter 14. The Dilogarithm and Volumes of Hyperbolic Polytopes 301 R. Kellerhals
14.0 Introduction 301 14.1 A Particular Class of Hyperbolic Polytopes 303 14.2 The Volume of a rf-Truncated Orthoscheme 309 14.3 Applications 321 14.4 Further Aspects 328 References 335
Chapter 15. Introduction to Higher Logarithms 337 R. M. Hain and R. MacPherson
15.1 The Problem of Generalizing the Logarithm and the Dilogarithm 337
15.2 The Quest for Higher Logarithms 340 15.3 Higher Logarithms 341 15.4 The Higher Logarithm Bicomplex 343 15.5 Multivalued Deligne Cohomology 346 15.6 Higher Logarithms as Deligne Cohomology Classes 350 Acknowledgements 3 51 References 352
Chapter 16. Some Miscellaneous Results 355 L. Lewin
16.1 Clausen's Function and the Di-Gamma Function for Rational Arguments 355
16.2 An Infinite Integral of a Product of Two Polylogarithms 359 16.3 Cyclotomic and Polylogarithmic Equations for a Salem Number 364 16.4 New Functional Equations 373 References 374
Appendix A. Special Values and Functional Equations of Polylogarithms 377
D. Zagier 0. Introduction 377 1. The Basic Algebraic Relation and the Definition of s/m(F) 378 2. Examples of Dilogarithm Relations 383 3. Examples for Higher Order Polylogarithms 385 4. Examples: Ladders 387 5. Existence of Relations among Polylogarithm Values of
Arbitrarily High Order 390
CONTENTS xi
6. A Conjecture on Linear Independence 391 7. Functional Equations 392 References 399
Appendix B. Summary of the Informal Polylogarithm Workshop, November 17-18, 1990, MIT, Cambridge, Massachusetts 401
R. MacPherson and H. Sah List of Participants 401 Abbreviated Summary 402
Bibliography 405
Index 409
Preface
As editor of this monograph on polylogarithms I would like to take the liberty of commencing with a few personal reminiscences. I first encountered the dilogarithm function many years ago in high school; it was a fascinating discovery for me, and it initiated a romance that has lasted almost sixty years. For the dilogarithm, the transition from its standing as a curious mathematical oddity to its current status as an important element in the fabric of modern mathematical structure began about fifteen years ago with Bloch's studies on its applications in algebraic AT-theory and algebraic geometry. Since then, the pace of discovery has quickened dramatically. In 1980, when I was in the throes of completing my "Polylogarithms and Associated Functions," I became dimly aware that the handful of peculiar numerical identities that had been known since the time of Euler and Landen were, in fact, just the tip of an iceberg of unlimited extent. Thus emerged the new discoveries on cyclotomic equations and their related polylogarithmic "ladders"—a nomenclature that came to me in a dream, after much chewing over of other, more artificial, verbal constructs. Ten years of development in this arena, conducted mostly by the methods of classical analysis with the help of number-crunching computers, ran parallel with other, and more important, discoveries in diverse branches of abstract algebra and algebraic geometry. The confluence of these two streams of thought in the last few years, due to the work of several mathematicians, but particularly to studies of Browkin in Poland and Zagier in Germany, has lead to the present synthesis which I have tried to present in this timely, I hope, monograph.
One of the biggest problems has been the pace of new research; it is obviously extremely difficult to produce a book that is current when new discoveries are taking place all the time and making already-written material partially outdated—though it is also a sign of a very flourishing field when things go this way. During the approximately twelve months that the book has been in active preparation many new discoveries were made. I have endeavored to keep the material up-to-date by the last minute inclusion of two appendices: one on a special workshop on polylogarithms held in November 1990; and one on very recent discoveries on the relation of functional equations to polylogarithmic ladders, Dedekind's zeta function; and including the remarkable discovery by D. Zagier and H. Gangl at the Max-Planck-Institut fur Mathe-
Xlll
XIV PREFACE
matik of a two-variable functional equation for the hexalogarithm—the first significant advance in this area since Kummer's work of 150 years ago. In my earlier (1958) book on dilogarithms, talking about the difficulty of making much further progress in this area, I had written "But the complexity of the present results makes a completely new approach imperative if much progress is to be made." It is now clear what this new approach is entailing: on the one hand the structural analysis arising from algebraic AT-theory and related fields; and on the other the extensive use of computers, both for high precision numerical work and also for machine computation using symbolic logic. It is doubtful that many of the new and interesting formulas could have been found by hand alone; powerful computer programs are becoming almost as important as mathematical skills and the ability to generate new constructive conjectures.
This book could not have been written without the splendid help and cooperation of the several contributors who gave generously of their time and effort. Many helpful suggestions and contacts were made. I would particularly like to thank Richard Hain for his assistance in the compilation of the bibliography, Don Zagier for his extensive up-to-date appendix, and Han Sah and Robert MacPherson for their report on the recent polylogarithm workshop.
Authors have very individual styles of writing and it is not practical, for the purpose of uniform presentation, to constrain them into one common pattern of text organization. Even so, I think the overall volume has not suffered from any ensuing tendency to be "patchy," and I hope that, the disparate contributions notwithstanding, the material as a whole is sufficiently coherent to give the entire work the integrity that I, as editor, have sought.
Most authors have written their chapters in the absence of knowing in detail what others were writing. This has given rise to a small amount of redundancy which I have not thought fit to try to remove; I do not think the work has suffered in any way from this. Rather, it has been interesting to see how similar ideas have arisen independently and received corresponding treatment. The whole subject is now in a state of rapid transition; even as I write, new discoveries vie for admission. With reluctance I have had to call a halt to the inclusion of a flood of new material. It will be fascinating to see what further developments the coming decade will bring. Don Zagier once wrote that "the dilogarithm is the only mathematical function with a sense of humor." As this subject matures and gets more important, and more serious, I hope it manages to retain its once light-hearted beginnings. Its ability over the years to attract and hold the interest of so many mathematicians, many of them of the finest caliber, has been outstanding. I hope that its capacity for fruitful exploration will continue unabated for a long time to come.
Leonard Lewin January 1991
Acknowledgments
Much credit for the preparation of this volume is due the various contributors, who, together with their affiliations, are listed on the following pages.
Some of the formulas on ladders had already appeared earlier in the literature, and acknowledgment is gratefully made to the publishers for permission to use material from some of their publications of the past decade. In particular, credit is due to the Academic Press, publisher of the Journal of Number Theory, for permission to use material from: The inner structure of the dilog-arithm in algebraic fields, J. Number Theory, 19 (1984), 345-373, and The polylogarithm in algebraic number fields, J. Number Theory 21 (1985), 214-244. Credit is also due to Birkhauser Verlag AG, publisher of Aequationes Mathematicae, for permission to use material from: The order-independence of the polylogarithmic ladder structure, Aequationes Math. 30 (1986), 1-20; Polylogarithmic functional equations, Aequationes Math. 31 (1986), 223-242; The polylogarithm in the field of two irreducible quintics, Aequationes Math. 31 (1986), 315-321; Polylogarithms in the field of omega, Aequationes Math. 33 (1987), 23-45; and Supernumary polylogarithmic ladders and related functional equations, Aequationes Math. 39 (1990), 210-253.
Much gratitude is also due to the reviewers of the original proposal for this book for making helpful suggestions, many of which have been incorporated into the current version.
XV
List of Contributors
Mohamed D. Abouzahra, Ph.D. MIT Lincoln Laboratory, Lexington, MA 02173, USA
Spencer Bloch, Ph.D. Department of Mathematics, University of Chicago, Chicago, IL 60637,
USA Jerzy Browkin, Ph.D.
Institute of Mathematics, Warsaw University, ul. Banacha 2, PL-00-913, Warsaw 59, Poland
Richard M. Hain, Ph.D. Department of Mathematics, University of Washington, Seattle, WA
98195, USA Ruth Kellerhals, Ph.D.
Max-Planck-Institut ftir Mathematik, Gottfried-Claren-StraBe 26, 5300 Bonn 3, Germany
Leonard Lewin, D.Sc. Professor Emeritus, Campus Box 425, University of Colorado, Boulder,
CO 80309, USA John H. Loxton, Ph.D.
Head of School of Mathematics, Macquarie University, NSW 2109, Australia
Robert MacPherson, Ph.D. Department of Mathematics, MIT, Cambridge, MA 02139, USA
Gary A. Ray, Ph.D. University of Washington, Seattle, WA 98195, USA (currently at Boeing
High Technology Center, Seattle, WA 98124, USA) C. Han Sah, Ph.D.
Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
Gerd Wechsung, Ph.D. Prorektor fur Naturwissenschaft und Technik, Friedrich-Schiller-
Universitat, 6900 Jena, Germany
XVll
XV111 LIST OF CONTRIBUTORS
Zdzislaw Wojtkowiak, Ph.D. Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440
Bures-Sur-Yvette, France. (Formerly at Max-Planck-Institut fiir Mathematik, Gottfried-Claren-StraBe 26, 5300 Bonn 3, Germany)
Don Zagier, Ph.D. Max-Planck-Institut fiir Mathematik, Gottfried-Claren-StraBe 26, 5300
Bonn 3, Germany
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[BGSV] Beilinson, A. A., Goncharov, A. B., Schechtman, V. V., and Varchenko, A. N., Aomoto dilogarithms, mixed Hodge structures, and motivic cohomology of pairs of triangles on the plane, The Grothendieck Festschrift, vol. 1, Birkhauser, Progress in Mathematics Series, vol. 86, Boston, 1990.
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[Bl] Bloch, S., Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, International Symposium on Algebraic Geometry, Kyoto, 1977, Kinokuniya Book-Store Co. Ltd., Tokyo (1978), pp. 103-114.
[B2] , Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, unpublished manuscript, 1978.
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[Rs] Richmond, B., and Szekeres, G., Some formulas related to dilogarithms, the zeta function and the Rogers-Ramanujan identities, J. Austral. Math. Soc. (A) 31 (1981), 362-373.
[V] Varchenko, A., Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry, ICM, Kyoto, 1990.
[Wl] Wojtkowiak, Z., A construction of analogs of the Bloch-Wigner function, Math. Scand. 64(1989), 140-142.
[W2] , A note on functional equations of polylogarithms, Max-Planck-Institut fiir Mathematik, Bonn, 1990 preprint.
[W3] , A note on the monodromy representation of the canonical unipotent connection on Pl(C) \ {a{, . . . , an} . Max-Planck-Institut fur Mathematik, Bonn, 1990 preprint.
[W4] , A note on functional equations of the p-adic polylogarithms, Institut des Hautes Etudes Scientifiques, France, 1991 preprint.
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Index
A*(X), 211 a is dependent on A, 251 Abel, 2, 4, 6, 185, 193, 194 Abel equation, 2 Abouzahra, 123, 370 absolute Hodge cohomology, 347 accessibility, 4, 17, 18, 64, 65, 71, 97 accessible ladder, 40, 44 accessible relations, 291 accessible results, 62 Acreman, 298 Adamchik, 359 adjoint, 114 adjoint equation, 108 adjoint set, 115, 119 Alg(G(X))9 216 Andrews, 19, 297, 298 Apery, 46 automorphic functions, 11, 29, 185 automorphism groups, 185
Barrucand, 24 base, 6 base T, 152 base Q, 151 base p, 125, 150 base equation, 11, 34, 84, 87, 90 base on the unit circle, 8 Bass theorem, 158 Beilinson, 275, 383, 391 Birkhoff, 155 Bloch, xiii, 15, 134, 136, 141, 163, 290,
333, 334, 382 Bloch dilogarithm function, 8 Bloch-Wigner, 161 Bloch-Wigner dilogarithm, 245, 334, 345,
380 Bloch-Wigner function, 123, 135, 141 Bombieri, 46, 141 Borel, 330, 382 Boyd, 160, 165 Browkin, xiii, 9, 23, 69, 91, 92, 151, 167 Bohm, 302
C[[HX (X,C)]], 212 C[[X]], 212 C[[X]]*, 212 Catalan constant, 9, 243, 358, 362 Cheeger-Simons Chern class, 345, 346 Chen, 275 Chern class, 338, 340 Chudnovsky, 1, 46, 141 circle method, 293 Clausen, 8, 90, 91 Clausen component ladders, 91, 94 Clausen function, 8, 15, 16, 90, 92, 135,
141, 239, 315, 355, 358 Clausen function ladders, 8, 9 Clausen functional equation, 84 Clausen ladder, 94, 95 cocycle condition, 340 Cohen, 387, 389 complete system of equivalent points, 186,
195 component-ladder, 8, 35 congruence, 268 conjecture of Birch and Tate, 250 conjecture of Lichtenbaum, 250 conjecture of Milnor, 272 conjecture on linear independence, 391 Coxeter, 4, 5, 11, 49, 50, 52, 53, 97, 99,
151, 302 Coxeter polytopes, 306, 321 Coxeter simplexes, 322 cross-ratio, 393, 394, 396 cyclic symmetry, 393 cyclotomic equation, 6, 7, 20, 21, 22, 24,
34, 36, 39, 40, 41, 42, 43, 44, 54, 69, 72, 74, 80, 82, 90, 91, 148, 154, 159, 236, 288, 290, 296, 365, 367
cyclotomic equations with a factor, 264 cyclotomic polynomial, 255 cyclotomic relation, 387
Damiano, 342 de Doelder, 358 de Rham complex, 343, 351
409
410 INDEX
Dedekind zeta function, xiv, 330, 377, 384 deg / , 209 Dehn invariant, 331 Deligne cohomology, 350 Deligne, 275, 383 dependence, 254 di-gamma function, 355 dilogarithms, 238 Dirichlet L functions, 140, 164 Dirichlet L-series, 161 Dirichlet theorem, 377 dodecahedron, 328 ^/-truncated orthoschemes, 301 duplication formula, 1, 29, 32, 236, 240 Dupont, 4, 5, 50, 53, 333, 334, 335
ew, 213 en, 218 Eastham, 46 Erdos-Stewart-Tijdeman, 390 essential, 307 Euler, xiii, 1, 4, 5, 12, 20, 36, 38, 40, 287 Euler dilogarithm, 238, 301, 314 exceptional, 256 exp(u>), 213 exp, 213 exponent form, 11, 117
f~l(a\ 206 factorization formula, 154 factorization theorem, 202 Fettis, 358 fifteen-term functional equation, 16 5-cycle, 236 5-cycles equivalent, 236 5 term equation, 335 flow chart, 39, 44, 45, 52, 61, 65 Fox //-function, 360 function of Schlafli, 309 functional equations, 28, 247, 281, 340,
342, 392 functor of Milnor, 233 fundamental relations, 313
G(X), 214, 216 Gangl, xiv, 4, 6, 23, 29, 39, 49, 98, 151,
290, 373, 374, 378, 396, 398, 399 Gauss sum, 248 G-functions, 287 Gelfand-MacPherson higher logarithm, 342 general identity for the trilogarithm, 141 generalization of Rogers function, 31 generalized /?-logarithm, 351 generalized Rogers function, 34 generalized, 236 generic ladders, 34 generic part of the grassmannian, 342
golden ratio, 2 Goncharov, 375, 382 good 5-units, 234 Gordon, 297 Gram matrix, 304
Haagerup, 324 Hain, xiv harmonic group, 11, 28, 31, 100, 105, 116 heptalogarithm, 374, 384, 385 hexalogarithm, xiv, 378, 398, 399 higher logarithms, 337, 341, 350 Hilbert's Third Problem, 331 Hill, 3, 5, 11, 18, 28, 116 Hodge filtration, 275, 276, 277, 347 Humbert's formula, 330 hypergeometric series, 292 hyperlogarithms, 171
T> 205 Jx fZ , 205
Jx,y ideal points, 304 imaginary part of the dilogarithm, 15, 16 inaccessibility, 5, 6, 9, 11, 110 inaccessible ladders, 23, 42, 44 index, 4 interated integrals, 205 inversion formula, 29, 32 inversion relations, 1 inversion theorem, 202 iterated integral, 340 Jensen formula, 160 Jorgensen, 328
Klein, 185 Kneser, 301, 310 ^-theory, xiii, xiv, 8, 373, 381 Kubert identities, 157 Kummer, xiv, 3, 4, 5, 6, 8, 9, 11, 15, 16,
27, 28, 29, 39, 49, 50, 71, 100, 108, 151, 153, 171, 185, 187, 188, 190, 193, 373, 375, 395
Kummer equations, 7, 13, 15, 17, 18, 19, 20, 21, 33, 34, 44, 54, 56, 57, 62, 64, 76, 80, 84, 97, 102, 103, 105, 107, 108, 110, 114, 116, 118, 334
Kummer formulas, 113 Kummer function and its relation to the
polylogarithm, 27 Kummer functional equation, 11, 16, 37,
39, 165 Kummer two-variable functional equation,
32 Kummer type, 12, 196 Kummer-type functional equation, 186,
187, 196, 203
INDEX 411
Kolbig, 359
&v{z;x), 217 &v(z,x,y), 217 &en{z;x), 220 L(ni(X9x))9 212 /f (z;x), 215 /J(z;*,y), 215 /*(*;*)), 213 Wz;jc,y)),215 ladder, xiii, 4, 6, 387 ladder clusters, 154 ladder order-independence on reduction of
order, 33 ladders from functional equations, 21, 23 ladders from quintic equations, 24 ladders on the unit circle, 90 kx(z\x))9 213 ^(z;jc,y) , 213 A x (£^ , . . . , e^ ) (z ) , 213 Lambert cube, 308, 311 Landen, xiii, 1, 2, 4, 5, 12, 31, 36, 38, 40,
52, 287 /.</.*.(«), 209 Lehmer, 160 Lerch function, 361 Lewin, 5, 123, 124, 126, 151, 152, 154,
158, 161, 275, 288, 291, 292, 298, 378, 387
LI„(Z;JC), 219 Lin(z\x,y), 218 Lichtenbaum, 9, 92 Lie{ni{Y,y)), 217 linear power relation, 123, 148, 152, 165 Ljungren, 53 Lobachevsky, 301 Lobachevsky function, 302, 314, 315 logarithm-removal property of Rogers
function, 20 logarithmic integral, 171 logarithmic singularity, 174 Lorentz space, 303 Loxton, 5, 6, 11, 22 L-series, 161
MacPherson, xiv MACSYMA, 28, 67, 100, 101, 103, 112,
121 Mahler measure, 123, 160, 162, 164, 389 Mantel, 17 Max-Planck-Institut fiir Mathematik, xiv maximal subscheme, 311 Meijer's (/-function, 360 Mellin transform, 359 Meyerhoff, 322, 330 Milnor conjecture, 391 minimal set, 237, 251
mixed Hodge structure, 275, 276, 277 mixed motives, 275 modified external product, 233 modified function, 380 modified ladder, 7 module Pnk, 194 module P^k{H0), 195 monodromy group, 340 monodromy of the polylogarithms, 279 monodromy operator, 338, 339 multi-variable equation, 3 multiple-angle formulas, 18 multiplication theorem, 20 multivalued Deligne cohomology, 346, 350 multivalued Deligne complex, 347 multivalued differential form, 343 multivalued function, 340 Munkholm, 324
^-dimensional hyperbolic space, 303 nonaccessibility, 71 nonexistence of a Kummer-type functional
equation for Li6, 197 nontrivial, 236 nonvalid ladders, 70 notation, 6 rc-variable identities, 124
cox, 212 order-independence property, 7, 8 order-reduction property, 7 ordinary cyclotomic equation, 237 orthoscheme of degree, 307
P(X)9 212 partial Clausen's function, 241 partition identities, 292 permanent, 305 Phillips, 50 Pisot number, 160, 164, 165 n(X), 212 ^-logarithm function, 344 /?-logarithm, 343, 344 points at infinity, 304 polar hyperplane, 304 polylogarithm extension of Q-mixed Hodge
structures, 279 polylogarithm workshop, xiii, xiv polylogarithms, 340 Pontrjagin class, 337 Pontryagin classes, 275 principal parameter, 311 principal vertices, 307 pseudodifferentiation, 7 pseudointegrate transparently, 52 pseudointegration, 7, 35, 39, 43, 46, 56,
83
412 INDEX
pure form, 28 pure Hodge structure, 275, 276 purity property, 28, 33
Ramakrishnan, 141, 142 Ramanujan, 298 Ray, 4, 12, 29, 71, 86, 90, 96, 290, 292 real part of the dilogarithm, 18 redefinition of Rogers function, 20 reduction formula, 310 redundant results, 78 regulator, 346 regulator mappings, 275 relation of dependence, 251 Richmond, 19, 292, 297, 298 Rogers, 3, 4, 8, 9, 16, 28, 123, 124, 287,
381, 383, 385 Rogers dilogarithm, 239 Rogers function, 7, 20, 27, 28, 102, 106,
110, 111, 112, 117 Rogers-Ramanujan partition functions, 5 Rogers-Ramanujan partition identities, 19 Rogers-Ramanujan, 292, 297 Rost, 121
Sah, xiv, 334, 335 Salem, 164 Salem number, 160, 164, 364, 389 Salem/Pisot numbers, 123, 160 Sandham, 29 Sandham «-variable identity, 141 Schaeffer, 3, 11, 28, 29, 52, 116 schematic type, 306, 310 scheme, 305 scheme of a polytope, 304 Schinzel, 158, 164 Schlafli, 301 scissors congruence groups, 331 second-degree ladders, 38 Siegel, 287 six-fold symmetry, 393 Slater, 292, 296 Smyth, 161 special exponents, 388 Spence, 3 Steinberg symbol, 233 Stewart, 290 sums, 268 S-units, 234 supernumary, 12, 69 supernumary component-ladders, 38
supernumary cyclotomic equation, 69, 78 supernumary ladders, 123, 165 Suslin, 382 symmetry group, 393, 395 Szekeres, 19, 60, 292, 293, 297, 298, 370
Tate Hodge structures, 276 the Schlafli differential formula, 309 three-term base equation, 12 three-term equation, 24 three-variable functional equation, 17 Thurston, 326, 328, 333 totally asymptotic regular hexahedron, 326 totally asymptotic regular octahedron, 327 totally asymptotic regular simplex, 323 totally asymptotic simplex, 322, 334 trans-Kummer range, 31, 33, 45, 49, 53,
58, 61, 65, 72, 97, 98, 114 trans-Kummer results, 36 transparency, 8, 35, 70 transparency property, 44 trivial, 12 Tverberg, 53 two-term base equation, 11 two-variable functional equations, 29 type A, 308 type £, 308
unit circle, 16
v*, 217 valid ladder, 7 Vandiver, 155 vertex polytope, 305 volume differential, 301 volume form, 344 volume spectrum, 328 volumes of hyperbolic 3-folds, 328
Watson, 5, 19, 21, 22, 287, 288, 291, 328 Wechsung, 12, 17 Weeks manifold, 328 weight filtration, 276, 277, 343 Wigner, 333, 334
Xiao Hongnian, 121
Zagier, xiii, xiv, 33, 61, 69, 82, 92, 142, 155, 275, 290, 330, 365, 368
zeta function, 247
