http://dx.doi.org/10.

1090/surv/122

Polynomial Identities
and Asymptotic Methods
Mathematical
Surveys
and
Monographs

Volume 122

Polynomial Identities
and Asymptotic Methods
Antonio Giambruno
Mikhail Zaicev

A m e r i c a n M a t h e m a t i c a l Society

*WDED
 EDITORIAL COMMITTEE
Jerry L. Bona Peter S. Landweber
Michael G. Eastwood Michael P. Loss
J. T. Stafford, Chair

2000 Mathematics Subject Classification. Primary 16R10, 16R20, 16R30, 16R40, 16R50,
16P90, 16W22, 16W55, 17B01.

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Giambruno, A.
Polynomial identities and asymptotic methods / Antonio Giambruno, Mikhail Zaicev.
p. cm. — (Mathematical surveys and monographs ; v. 122)
Includes biblographical references and index.
ISBN 0-8218-3829-6 (alk. paper)
1. Pi-algebras. 2. Rings (Algebra). I. Zaicev, Mikhail. II. Title. III. Mathematical surveys
and monographs ; no. 122.
QA251.G43 2005
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 Contents

Preface ix

Chapter 1. Polynomial Identities and PI-Algebras 1

1.1. Basic definitions and examples 1
1.2. T-ideals and varieties of algebras 3
1.3. Homogeneous and multilinear polynomials 5
1.4. Stable identities and generic elements 10
1.5. Special types of identities 12
1.6. Symmetric functions 15
1.7. Identities of matrix algebras 16
1.8. A theorem of Lewin 20
1.9. Identities of block-triangular matrices 24
1.10. Central polynomials in matrix algebras 26
1.11. Structure theorems 29
1.12. Some applications of the structure theorems 35
1.13. The Gelfand-Kirillov dimension of a Pi-algebra 36

Chapter 2. Sn- Represent at ions 43

2.1. Finite dimensional representations 43
2.2. ^-representations 46
2.3. Inducing S^-representations 50
2.4. 5 n -actions on multilinear polynomials 52
2.5. Hooks and symmetric and alternating sets of variables 57

Chapter 3. Group Gradings and Group Actions 61

3.1. Group-graded algebras 61
3.2. Abelian gradings and group actions 63
3.3. G- act ions, G-gradings and free algebras 65
3.4. Wedderburn decompositions 69
3.5. Finite dimensional simple superalgebras 74
3.6. Involutions on matrix algebras 77
3.7. Superalgebras and Grassmann envelopes 80
3.8. Supercommutative envelopes 83

Chapter 4. Codimension and Colength Growth 87

4.1. Codimensions and colengths 87
4.2. An exponential upper bound for the codimensions 94
4.3. Identities of graded algebras 97
4.4. Robinson-Schensted correspondence 101
4.5. Cocharacters of Pi-algebras 104
 CONTENTS

4.6. Capelli polynomials and the strip theorem 107

4.7. Amitsur polynomials and hooks 108
4.8. Finitely generated superalgebras 110
4.9. Colength growth: a polynomial upper bound 115

Chapter 5. Matrix Invariants and Central Polynomials 119

5.1. 5 n -action on tensor space 119
5.2. Trace identities 122
5.3. A primer of matrix invariants 124
5.4. The discriminant 125
5.5. Invariants and central polynomials 128
5.6. Constructing S^-maps 131
5.7. Computing central polynomials 132
5.8. Cocharacters and trace cocharacters 135
5.9. Multialternating polynomials 137
5.10. Asymptotics for the codimensions of k x k matrices 139

Chapter 6. The PI-Exponent of an Algebra 143

6.1. The exponential growth of the codimensions 143
6.2. A candidate for the Pi-exponent 145
6.3. Graded identities and Grassmann envelopes 151
6.4. Gluing Young tableaux 155
6.5. Existence of the exponent 160
6.6. Computing the exponent of some algebras 161

Chapter 7. Polynomial Growth and Low Pi-exponent 165

7.1. The Grassmann algebra and standard polynomials 165
7.2. Varieties of polynomial growth 169
7.3. Locally noetherian varieties 175
7.4. Polynomial growth and bounded multiplicities 179
7.5. Types of polynomial growth 185
7.6. Varieties of exponent two 189

Chapter 8. Classifying Minimal Varieties 193

8.1. Minimal superalgebras 193
8.2. Some examples 196
8.3. The superenvelope of a minimal superalgebra 199
8.4. Products of verbally prime T-ideals 205
8.5. Classifying minimal varieties of exponential growth 207
8.6. Some consequences 211

Chapter 9. Computing the Exponent of a Polynomial 215

9.1. The exponent of standard and Capelli polynomials 215
9.2. An upper bound for the exponent of a polynomial 219
9.3. Powers of standard polynomials 225
9.4. Essential hooks and reduced algebras 238
9.5. The exponent of Amitsur polynomials 242
9.6. The exponent of a Lie monomial 245
9.7. Evaluating polynomials 247
9.8. Asymptotics for the standard and the Capelli identities 251
 CONTENTS vii

Chapter 10. G-Identities and G I Sn-Action 255

10.1. G-identities, G-codimensions and G I S^-action 255
10.2. Decomposable monomials 259
10.3. Essential G-identities. Amitsur's theorem on *-identities 261
10.4. Representations of wreath products 264
10.5. Graded identities and polynomial growth 267
10.6. The Z 2 I 5 n -action 272
10.7. Finite dimensional algebras with (^-action 274
10.8. The Z2-exponent of a finite dimensional algebra 276
10.9. Simple and semisimple ip-algebras 280
Chapter 11. Super algebras, *-Algebras and Codimension Growth 283
11.1. Notation and more 283
11.2. *-varieties of almost polynomial growth 285
11.3. Supervarieties of almost polynomial growth 289
11.4. Capelli identities on superalgebras 292
11.5. Superalgebras and polynomial growth 294
11.6. *-algebras and the Nagata-Higman theorem 296
11.7. Polynomial growth of the *-codimensions 298
11.8. Supervarieties of exponent 2 301
11.9. Further properties 304
Chapter 12. Lie Algebras and Non-associative Algebras 307
12.1. Introduction to Lie algebras 307
12.2. Identities of Lie algebras 309
12.3. Codimension growth of Lie algebras 314
12.4. Exponents of Lie algebras 323
12.5. Overexponential codimension growth 327
12.6. Lie superalgebras, alternative and Jordan algebras 328
12.7. The general non-associative case 330
Appendix A. The Generalized-Six-Square Theorem 333
A.l. The Theorem 333
A.2. Basics 334
A.3. Representations of integers 336
A.4. A crucial lemma 338
A.5. The proof of Theorem A.1.2 339
Bibliography 341

Index 349
 Preface

One of the main objectives of this book is to show how one can combine methods
of ring theory, combinatorics, and representation theory of groups with an analytical
approach in order to study the polynomial identities satisfied by a given algebra.
The idea of applying analytical methods to the theory of polynomial identities
appeared in the early 1970s and nowadays this approach is one of the most powerful
tools of the theory.
A polynomial identity of an algebra A is a polynomial in non-commuting in-
determinates vanishing under all evaluations in A and the algebras having at least
one such nontrivial relation are called PI-algebras. For instance, xy — yx = 0 is a
polynomial identity for any commutative algebra. Hence, in particular, the polyno-
mial ring in one or several variables is an example of a Pi-algebra. Another natural
example is given by the exterior algebra of a vector space, or Grassmann algebra G
which appears in algebra, analysis, geometry and other branches of modern mathe-
matics. It is easy to see that G satisfies the polynomial identity [[x, y],z] = 0 where
[x, y] — xy — yx is the Lie commutator of x and y. Any nilpotent algebra A such
that An = 0 is a Pi-algebra since it satisfies the polynomial identity X1X2 • • • xn = 0.
All subalgebras, homomorphic images and direct products of algebras satisfying
a given identity / = 0 still satisfy / = 0. Hence the PI-algebras form a quite
wide class including commutative algebras, finite dimensional algebras, algebraic
algebras of bounded exponent and many more.
Non commutative polynomials vanishing on an algebra can be found in the
early papers of Dehn ([De]) and Wagner ([Wa]). The general interest in Pi-theory
started after a paper of Kaplansky ([Kl]) in 1948. In that paper it was proved
that any primitive Pi-algebra is a finite dimensional simple algebra suggesting that
satisfying a polynomial identity is some finiteness condition on a given algebra.
Most of the structure theory of PI-algebras was developed in the 1960s and the
1970s and an account of it can be found in the early books of Jacobson ([JaJ) and
Procesi ([Pr3]). A comprehensive collection of the results on the structure theory
of PI-algebras can be found in the book of Rowen ([Ro2]).
Two years after Kaplansky's theorem, Amitsur and Levitsky proved by purely
combinatorial methods that a certain polynomial, called the standard polynomial of
degree 2/c, is an identity of minimal degree for the algebra of k x k matrices ([AL]).
This theorem was the beginning of a new approach to Pi-theory, the main objective
being the description of the polynomial identities satisfied by a given algebra. A few
years later, Kostant ([Ks]) related the Amitsur-Levitsky theorem to cohomology
theory and to the invariant theory of k x k matrices. In the 1970s the theory of
Pi-algebras was related to the more general theory of trace identities as developed
by Procesi ([Pr2]) via invariant theory and independently by Razmyslov ([R2]).

ix
x PREFACE

Let F(X) denote the algebra of noncommutative polynomials in a given set

of variables X over a field F , i.e., the free algebra on X over F. The polynomial
identities satisfied by an algebra A form an ideal of F(X) invariant under all endo-
morphisms of the free algebra, called a T-ideal. Moreover, every T-ideal of F(X)
is of this type. Hence describing the identities of an algebra means describing the
T-ideals of the free algebra.
Since distinct algebras can have the same ideal of identities, the theory of
T-ideals is linked to the theory of varieties of algebras. Recall that a variety of
algebras is a class of algebras satisfying a given set of identities. The varieties were
introduced by Birkhoff ([Br]) and Malcev ([Mai]) in order to study the identities
of algebraic structures and this seems to be the most natural language in the theory
of identities.
The description of a T-ideal is in general a hard problem. Specht in 1950 ([Sp])
conjectured that over a field of characteristic zero every proper T-ideal of F(X) is
finitely generated as a T-ideal. Many instances of this conjecture were proved in
the following years but a complete proof was only given after a series of papers
by Kemer in 1987 ([Ke5]). His proof is based on some basic structure theory of
the T-ideals which has given a new impetus to the subject. It involves the study
of superidentities of superalgebras and certain graded tensor products with the
Grassmann algebra, called Grassmann envelopes. The theorems and techniques
developed by Kemer are contained mostly in his monograph ([Ke7]) and have
become in recent years some of the basic tools for studying the identities of a given
algebra.
Even if every proper T-ideal is finitely generated, the polynomial identities of
a given algebra, like the algebra o f n x n matrices, are far from being understood.
An important observation is that even if we know the generators of a given T-
ideal, it is quite impossible in general to deduce from them, say, information on the
polynomials of the T-ideal of a given degree. To overcome some of these difficulties
it is natural to introduce a function measuring the growth of the identities of a T-
ideal in some sense. This approach was introduced by Regev in 1972 ([Rel]) and
nowadays it has become not only a main tool but also a major object of research
in the theory of Pi-algebras in characteristic zero.
Since the base field is of characteristic zero, by the well-known polarization
process, every identity is equivalent to a finite set of multilinear ones. Then one
can slice any T-ideal into subspaces Pn of polynomials in a given fixed set of n
variables, n — 1, 2 , . . . , and the function defined by the codimensions of these spaces
is the growth function associated to the given T-ideal. Since T-ideals are invariant
under endomorphisms, the permutation action of the symmetric group Sn turns Pn
into an ^ - m o d u l e and the representation theory of the symmetric group, which is
well-understood in characteristic zero, can be successfully applied.
One associates to each T-ideal a sequence of characters of the symmetric groups
Sn, n = 1, 2 , . . . , called the sequence of cocharacters of the given PI-algebra and a
numerical sequence, called the sequence of codimensions, given by the correspond-
ing degrees measuring the growth of the T-ideal. The most significant results of
the representation theory of the symmetric group and the corresponding combina-
torial theory of Young diagrams such as the branching theorem, the Littlewood-
Richardson rule, the hook formula etc., became an essential tool in the development
of the theory.
 PREFACE xi

The starting point in the investigation of the growth of T-ideals is a theorem

of Regev ([Rel]) stating that the codimension sequence of a PI-algebra is expo-
nentially bounded. By results of Kemer it turns out that this growth is either
polynomial or exponential. Also in recent years ([GZ1], [GZ2]) it has been proved
that the exponent of the growth rate for a proper T-ideal is an integer called the
exponent of the T-ideal or of a corresponding Pi-algebra. Having at hand an in-
teger scale provided by the exponent, the theory has developed in the last years
towards the classification of T-ideals according to the asymptotic behaviour of their
sequence of codimensions and in this book we shall give an account of these results.
Among them, the classification of maximal T-ideals of a given exponent (or minimal
varieties of a given exponent) ([GZ9]), the prominent role played by the standard
and the Capelli polynomials in combinatorial Pi-theory and the precise relation
between the growth of the corresponding T-ideals and the growth of the algebra
of n x n matrices ([GZ8]), etc. This approach to the combinatorial theory of
PI-algebras is also related to other theories of independent interest.
Since every T-ideal is multigraded by the degree, using standard methods the
sequence of cocharacters is strictly related to the corresponding Hilbert or Poincare
series. In particular, the problem of decomposing the cocharacter sequence into
irreducibles, translates into the problem of writing the Hilbert series as a sum of
Schur functions. This relation establishes a precise link between the combinator-
ial theory of Pi-algebras and the theory of symmetric functions ([M]) and many
problems translate in a natural way into that setting.
Another strictly related theory is that of trace identities and the corresponding
invariant theory of n x n matrices as developed by Procesi in [Pr2]. The methods
of invariant theory and the development of the theory of trace identities obtained
independently by Razmyslov in [R2] are one of the basic tools needed in order to
develop the theory of PI-algebras.
It is well known that any field of characteristic zero is a splitting field for the
symmetric group, hence the base field is usually not relevant when studying T-
ideals. This seems to infer that no significant result of number theory should play
a role. Nevertheless, as we shall see, a delicate extension of the well-known four
squares theorem asserting that any integer is the sum of at most four squares, wTill
be crucial in the computation of the growth of some polynomials.

For the reader interested in the general theory of polynomial identities, the
first monographs devoted to the subject were published in the 1970s ([Pr3], [Ja],
([Ro2]). In several books concerning ring theory or other areas of algebra one can
find some parts dedicated to the theory of polynomial identities. Among them we
cite the books of Herstein [H], Cohn [C], Rowen [Ro3], Passman [P], Zhevlakov,
Slinko, Shestakov and Shirshov [ZSSS], Formanek [F4] and Beidar, Martindale and
Mikhalev [BMM]. Polynomial identities of Lie algebras are treated extensively in
the books by Bahturin [Bl] and Razmyslov [R4]. The solution of the Specht
problem is contained in the important monograph by Kemer [Ke7]. The book by
Drensky [D10] is a very good source for a first year graduate course in Pi-theory.
The recent achievements in Pi-theory have also stimulated the appearance of new
monographs and surveys devoted to polynomial identities ([GRZ2], [DF], [BR]).
The book by Belov and Rowen [BR] appeared but there seems to be no significant
overlap between the two books.
Xll PREFACE

The general scheme of the book is as follows. The core of the book is Chapter
6 where we prove the integrality of the exponential growth of any proper variety or
T-ideal. All the previous chapters contain the material needed for this purpose.
In the first chapter, we introduce the basic definitions and we give an ac-
count of the main results of the structure theory of PI-algebras. One of the main
tools for computing the asymptotic behavior of the codimensions is the represen-
tation theory of the symmetric group and we give an account of this theory in
Chapter 2. We present most of the classical results including the branching rules,
the hook formula and the Littlewood-Richardson rule. We then study the permu-
tation action of the symmetric group on the space of multilinear polynomials in a
fixed number of variables and we derive most of the properties of this action that
we shall use throughout.
In Chapter 3 we deal with group gradings and group actions. Group graded
algebras and, in particular, superalgebras play an important role in different areas
of mathematics and theoretical physics. The reason for studying superalgebras and
their identities is twofold. It is an interesting fast growing subject. More impor-
tant, there is a well understood connection between superidentities and ordinary
identities that allows one to reduce some problems to the finite dimensional case,
and this is one of the basic reductions in this book. In this chapter we generalize
Wedderburn theorems to the case of superalgebras and algebras with involution.
We also introduce the Grassmann envelope and the superenvelope of an algebra
and prove their basic properties.
In Chapter 4 we define the basic notions of the theory, namely the sequences
of codimensions and colengths and we prove the most important properties of their
asymptotic behaviour. We also prove a basic structure theorem concerning the
Grassmann envelope of a superalgebra and the well-known hook theorem and strip
theorem.
Chapter 5 is devoted to the introduction of the invariant theory o f n x n matrices
and the consequent theory of trace polynomial identities. This subject is interesting
on its own and is an important area of modern mathematics. In this chapter we
apply results of invariant theory in order to prove the existence of suitable central
polynomials f o r n x n matrices. Such polynomials are used in the subsequent chapter
for finding the precise lower bound of the codimension growth. We also give the
asymptotics of the codimensions of the algebra of n x n matrices.
Chapter 6 is the central chapter of the book and we prove that the sequence
of codimensions of any Pi-algebra (or proper variety) has an integral exponential
growth, called the Pi-exponent of the algebra. We also give a constructive way for
determining it.
In the following chapters we apply the results obtained in order to further
develop the theory. Chapter 7 is mainly devoted to the characterization of varieties
having polynomial growth (or Pi-exponent < 1). The Grassmann algebra and its
properties play a basic role in this description.
In Chapter 8 we classify all varieties minimal of given exponent. This leads to
the notion of minimal superalgebra. We prove that such varieties have an ideal of
identities which is the product of verbally prime T-ideals and are strictly related
to the algebras of block triangular matrices. The classification of minimal varieties
gives an effective way for computing the exponent of a variety. In fact in Chapter 9
we define the exponent of a polynomial, or set of polynomials, as the exponent of the
 PREFACE xm

corresponding variety and we compute it for some significant classes of polynomials

such as standard polynomials, Capelli polynomials, and Amitsur polynomials. This
leads in some significant cases to the determination of a generating algebra for the
corresponding variety.
In Chapter 10 and Chapter 11 we extend our approach to graded algebras and
to algebras with involution. We consider G-identities for an algebra A where G
is a finite group of automorphisms and antiautomorphisms of A. We study such
identities via the representation theory of the wreath product G I Sn and we focus
our attention to the case when G is a group of automorphisms or antiautomorphisms
of A of order two. In this last case A has a structure of superalgebra or of algebra
with involution and we prove that the corresponding G-codimensions have integral
exponential growth in case A is finite dimensional. Chapter 11 is entirely devoted
to superalgebras and algebras with involution and their identities. We characterize
the corresponding varieties of polynomial growth and we prove that no intermediate
growth is allowed for such varieties. We also relate the newly found invariants to
the ordinary ones.
In the last chapter of the book we study our numerical invariants and their
asymptotics in other classes of non-associative algebras. Even in algebras which
are close to being associative, the sequences of codimensions and colengths show a
wild behavior. We deal mostly with Lie algebras and the growth of their identities.
In this setting the sequence of codimensions is no longer exponentially bounded
and we give an account of the various phenomena that can occur. As an outcome,
the combinatorial theory of Pi-algebra seems to be much more developed in the
associative case.

We are very grateful to A. Berele, V. Drensky, P. Koshlukov, S. Mishchenko,

A. Regev and A. Valenti for reading and commenting on parts of the manuscript.
Special thanks are due to A. Regev for preparing Appendix A. We would like to
mention O. M. Di Vincenzo and I. Shestakov for useful discussions and remarks.
We also thank F. Benanti and D. La Mattina for helping during the preparation of
the manuscript.
This project was supported in part by the research grant PRIN 2003 "Algebras
with polynomial identities and combinatorial methods", by the Istituto Nazionale
di Alta Matematica of Italy and by the research grants RFBR No. 02-01-00219 and
SSC-1910.2003.1 of Russia.
 APPENDIX A

The Generalized-Six-Square Theorem

In this appendix we present the proof of a theorem generalizing the classical

Lagrange Theorem stating that any positive integer is the sum of at most four
squares. The proof given here is due ro Regev and is based on the paper [CR].

A . l . The Theorem
We deal with the so-called hyperbolic (or super) integers B, that is the elements
of Z x Z with coordinatewise addition and the following multiplication:
(a, 6)(c, d) = (ac + bd, ad + be).
We start with the following definition.
DEFINITION A. 1.1. The set of the generalized squares is
V = {(r 2 , r 2 ), (r 2 + s 2 , 2rs) | r, s G N}.
Here N = {0, 1, 2, . . . } . Note that in particular, (r 2 ,0), (2r 2 ,2r 2 ) as well as
(y2 + {y + k)2, 2y(y + k)) are in V.
Let r, s > 0. If (r, s) is a sum of generalized squares (namely, of elements of
P), then r > s.
It can be shown that for instance (10, 3) is not a sum of 5 generalized squares.
The following is the generalized-six-square theorem.
THEOREM A.1.2. Given r > s > 0 in N, the pair (r, 5) is always a sum of at
most six elements in V.
The basic tool for proving Theorem A. 1.2 is the following classical theorem due
to Legendre and Lagrange (see [D, vol. 2, chapters VII, VIII]).
THEOREM A.1.3.
(1) Every positive integer m G N is a sum of at most four squares.
(2) Every positive integer m G N ; which is not of the form 4u(8/c + 7) with
u, k G N, is a sum of three squares. Moreover, if m G N is not divisible
by 4i then m is a sum of three squares with no common factor: m —
x2 + y2 + z2 and gcd(x, y,z) = 1.
We shall also make use of the following theorem.
THEOREM A. 1.4. Every positive odd integer m G N can be written in the form
m = a2 + b2 + 2c2.
Here we may assume that a is odd (hence a > 1) and that b is even. In addition, if
m > 1, then it has such a presentation with either b > 0 or c > 0.
333
334 A. T H E G E N E R A L I Z E D - S I X - S Q U A R E T H E O R E M

PROOF. Since 4 does not divide 2m, by Theorem A. 1.3 2m = x2 + y2 + z2

and gcd{x,y, z) = 1. Since 2m = 2 (mod 4), we may assume t h a t x,y are odd,
x > y > 1, and 2 even. Define

then it easily follows t h a t

m = a2 + b2 + 2c2.
Assume m > 1. If 6 = c = 0, it implies t h a t z = 0 and t h a t x = y > 1, a
contradiction since gcd{x, y, z) = 1. D
T H E O R E M A. 1.5 ([D]). Every odd integer s G N can 6e written as a sum of
four squares of integers, of which two are consecutive:
s=p2 + q2 + z2 + {z + l)2.

A.2. Basics

D E F I N I T I O N A.2.1. We call the following presentations of s G N quadratic-

ternary presentations.
(1) s = ex2 + rjy2 + 7 * 2 , £, m 7 G 1°. 1, 2}>
(2) s = £x 2 + W 2 + 2z(z + I), o, m G {0,1, 2},
(3) 5 - ex 2 + 22/(2/ + *) + 2*(* + ^), e, G { 0 , 1 , 2}.
T h e length of such a presentation is the number of its non-zero summands; its shift
is 0 in case (1), £2 in case (2) and k2 +£2 in case (3). For example, the length of the
presentation 23 = 3 2 + 2 • 1 • (1 + 2) + 2 • 1 • (1 + 3) is 3, and its shift is 2 2 + 3 2 = 13.

L E M M A A.2.2. Let r > s > 0 in N.

(1) Assume s has a quadratic-ternary presentation of length < 2 with shift
< r — s. Then (r, s) is a sum of six generalized squares, namely, (r, s) is
the sum of six elements ofV.
(2) Assume s has a quadratic-ternary presentation with shift d such that d <
r — s and r — s — d is a sum of three squares in N. Then (r, s) is a sum
of six generalized squares.

P R O O F . 1. T h e proof here follows by expressing r — s—shift as a sum of four

squares in N. Here are the cases:
1.1. s = ex2-\-ny2, s,7] e { 0 , 1 , 2 } (hence shift = 0 ) . In N let r-s = <?2H \-q\\
then
( r , s ) = ( s , s ) + (r - s,0) = {ex2, ex2) + (my 2 ,my 2 ) + (</2,0) + • • • + (g 2 ,0).

1.2. s = ex2 + 2y(y + fc), e, G { 0 , 1 , 2 } and let r - s - s h i f t = r - s - k2 =

qj-\ h g | . Since r = (s + A;2) + (r —s —fc2) and since y2 + {y + k)2 — 2y{y + k) + k2,
we have
(r, s) = {ex2, ex2) + (2/2 + (2/ + A;)2, 22/(2/ + k)) + (</2,0) + • • - + (g 2 ,0).

1.3. s = 22/(2/ + fc) + 2z(z + f). Now r - s - s h i f t = r - s - {k2 + £2) = (<?2,0) +

h (<?2, 0), and from r = (s + /c2 + £ 2 ) -f (r - 5 - {k2 + ^ 2 )) we deduce:

(r, s) = (y 2 + (y + /c) 2 , 2y(y + ^)) + (^2 + (^ + ^ ) 2 , 2^(^ + ^)) + (^?, 0) + • • • + (gl, 0).

2. The proof of part 2 is similar. Here are the details.

 A.2. BASICS 335

2.1. s = ex2 + r]y2 + 7^ 2 , £, rj, 7 G {0,1,2}. Here shifted = 0 while r — s =

2
(g , 0) H h (g2, 0) and we get
(r, 5 ) = (5,s) + ( r - 5 , 0 ) = ( £ - ^
2.2. 5 = ex2 + ?7?/2 + 2z(z + I), £, rj, G {0,1, 2}, so d = £2. By assumption
r - s- £2 = (g 2 ,0) H h (#3,0), and from (r, 5) = (s + £2, s) + (r - 5 - ^2) we get
(r, s) = (ex 2 , ex2) + (TO 2 , TO2) + (^2 + (2 + ^) 2 , 2z(z + £)) + (g2, 0) + • • • + (qh 0).
2.3. s = ex2 + 2y(2/ + *) + 22(2 + ^)? ^7 € {°> 1, 2}. By similar calculations
(r, s) = (s + A: + ^ , s) + (r - 5 - k2 - £2)
2 2

= ( ^ W ) + (^2 + ( y + £ ) 2 ^ ^
•
Note that if M < 4, then Af is a sum of three squares in N.
LEMMA A.2.3. Let 5 < M G N and assume that none of Af, Af - 1 and Af - 4
is a sum of three squares in N. Then M = 4U(8£ + 7) m£/i 0 < £ and 3 < u, hence
M > 4 3 7 = 448. (Note: at a crucial point in the proof of Theorem A. 1.2 we shall
use the fact that that number is > 149; see the case a > 7 in the last section here).
PROOF.By Theorem A.1.3 Af = 4U(8^ + 7) with 0 < £, u. Similarly, Af - 1 =
4 (8k + 7), hence 4W(8^ + 7) = 4v(8fe + 7) + 1. If 1 < v, we must have u = 0
v

(otherwise 4 divides 1), hence l.h.s. is congruent to 3 modulo 4 while the r.h.s.
is congruent to 1, a contradiction. Thus v = 0, so Au(8£ + 7) = 8(k + 1), which
implies that 2 < u. Similarly, we have Af = Af - 4 + 4 = Aw(8m + 7) + 4, therefore
4U(8£ + 7) = 4w(8m + 7) + 4; since 0 < n, deduce that 1 < w. By cancellation,
4"" 1 (8^ + 7) = 4 W - 1 (8m + 7) + 1 and since 2 < n, this implies that w-l = 0. Thus
4U~1(8£ + 7) = 8(ra + 1) which implies that 2 < u - 1. D
LEMMA A.2.4. Let 5 < Af G N and assume none of Af, Af - 1 and Af - 4 is a
simz 0/ £/iree squares. Let t > 0 be an integer that modulo 8 is congruent to either
2 or b. Then M — t is a sum of three squares.
PROOF. If M-t is not a sum of three squares, then Af = 4 n (8^+7) = A f - t + t =
9
4 (8r + 7) +t for some 0 < £, g, r and with 3 < u. Deduce a contradiction as follows.
If q = 0, it implies that t = l(mod8), a contradiction. If q = 1, then modulo 8,
t is congruent to 4, a contradiction. Finally, if 2 < g, then t = 0(mod8), again a
contradiction. •
Note that the proof implies more: if modulo 8 t is not congruent to 0, 1 or 4
(hence, if modulo 8 t is congruent to 2, 3, 5, 6 or 7), then Af — t is a sum of three
squares.
LEMMA A.2.5. Let 9 < Af G N and assume none of Af, Af - 1 and M - 4 is a
sum of three squares. Then Af — 8 is a sum of three squares.
PROOF. Again, assume this is not the case and deduce a contradiction. Thus
Af - 4U{8£ + 7) = A f - 8 + 8 = 4^(8s + 7) + 8
where 3 < u and 0 < £,k, s. Since both sides are even, 1 < y. Dividing by 4 we
have
4u-\8£ + 7) = 4 ^ (8s + 7) + 2.
336 A. T H E G E N E R A L I Z E D - S I X - S Q U A R E THEOREM

Since 3 < u, by parity we must have 2 < y. Now reducing this equation modulo 4
yields 0 = 2(mod 4), a contradiction. •

A.3. Representations of integers

In this section we prove some lemmas on the representations of integers. These
lemmas are then applied in the proof of Theorem A. 1.2.

LEMMA A.3.1. Every integer s £ N can be represented as s = x2 + y2 -f ez2,

x,y,z GN and e £ {0,1, 2}.
P R O O F . The case s is odd is given by Theorem A. 1.4 (with e — 2), so let s be
even. If s = 2(mod 4) then s cannot be of the form s = 4U(8£ + 7), hence is a sum
of three squares and we are done. So assume s = 0(mod 4). If s is a sum of three
squares, we are done. Otherwise, s = 4U(8^ + 7) with 1 < u, and we can write
s = 2 • Au~l{U + 7) + 2 • 4 U ~ 1 ( ^ + 7).
Since 2 • Au~l(U + 7) is not of the form 4v(8/c + 7), by Theorem A.1.2 it is a sum
of three squares: 2 • 4 u-1 (8-£ + 7) = x2 + y2 + z2, which implies that
s = 2x2 + 2y2 + 2z2 = (x + y)2 -f (x - y)2 + 2z2
as desired. •
LEMMA A.3.2. Every s G N can be represented as s = ex2 + y2 + 2z(z + 1),
where x , i / , z G N and £ £ {0,1, 2}.
PROOF. Case 1: s is even. By Theorem A.1.5 s + 1 = p2 + g2 + z2 + (2 + l ) 2 ,
therefore 8 = p 2 + q2 4- 2:2 + 2z(z + 1) and we are done.
Case 2: 8 is odd. Hence either s = 4fc -f 1 or 5 = 4/c + 3. Assume first that
s = 4fc + l. By Theorem A.1.2 8/c + 3 is a sum of three squares: 8/c + 3 = a2 + b2 + c2.
Reducing modulo 4 implies that a, 6, c are all odd. Therefore we can write 8/c + 4 =
1 + a2 + 62 + c2 as

8A; + 4 = i ((a - l ) 2 + (a + l ) 2 + (6 - c) 2 + (6 + c) 2 ) .
Thus

which implies that

a ,
..« + l- a (^)(i±l) + (tif) +(^) -2.,. + I) + M » + »»,
where e = 1.
Finally, consider the case s = 4/c+3. The argument here is similar: By Theorem
A.1.4, Sk + 7 = a2 + 62 + 2c2, where a > 1 is odd and 6 > 0 even. Thus

8/c + 8 = 1 + a2 + 62 + 2c2 = ^ ((a - l ) 2 + (a + l) 2 ) + b2 + 2c2.

This implies that

« + 4 _(i-)' + (-i)> + < ,

 A.3. REPRESENTATIONS OF INTEGERS 337

hence

as desired. •
LEMMA A.3.3. Every s G N can be represented as s = ex2 + y2 + 2z(z -f- 2),
w/iere x,y,z eN and e G {0,1, 2}.
PROOF. The proof is divided into several cases and subcases, and in each we
show that s has the desired form.
Case 1. s is odd. By Theorem A. 1.4 s + 2 = a2 + b2 + 2c2 with a odd, 6 even,
and either b > 0 or c > 0.
Subcase 1.1: c ^ 0. It follows that s — a2 + £>2 + 2(c — l)(c + 1) and we are done.
Subcase 1.2: c = 0, so 5 + 2 = a2 + 62 and 6 > 0. Thus b = 2k, k > 1, so
5 + 2 = a2 + (2/c)2 = a 2 + 2k2 + 2/c2,
which implies that
5 = a 2 + 2/c2 + 2(/c- l)(fc + l),
and we are done.
Case 2: s is even, hence 5 + 2 = 2*v where £ > 1 and v odd. By Theorem A.1.4,
v — p2 + g2 + 2r 2 where, say, p is odd, so p ^ 0. Write
2 2e i; = (2 e p) 2 + (2eq)2 + 2(2 e r) 2 = A2 + B2 + 2C 2
where A — (2ep)2 etc., and A ^ 0 since p ^ 0 .
Subcase 2.1: £ = 2e + 1, i.e., odd. Then
5 + 2 = 2- 22ev = 2A2 + 2 £ 2 + (2C) 2
with A > 0, so
5 = 2(A - 1)(A + 1) + 2B2 + (2C) 2
as desired.
Subcase 2.2: £ = 2e, is even, with e > 1. Now
s + 2 = 2 % = (2 e p) 2 + (2 e g) 2 + 2(2 e r) 2 .
Subcase 2.2.1: r / 0 . In that case
s = (2ep)2 + (2 e ^) 2 + 2(2 e r - l)(2 e r + 1)
is the desired presentation.
Subcase 2.2.2: r = 0. In that case v = p2+q2 with p > 0. Here 5+2 = (2 e p) 2 + (2 e 4) 2
and e > 1. With w = 2e~1p ^ 0 and w = 2eq we have
8 + 2 = 2u2 + 2 w 2 + w 2 ,
hence
5 = 2(u - l)(u + 1) + 2ix2 + w 2 ,
as required. •
COROLLARY A.3.4. Let r > s > 0 be integers and let M = r — s. If at least
one of M, M — 1, M — A is a sum of three squares in N, £/ien (r, 5) 25 a 5tira of
six generalized squares. In particular, if r — s < 448, then (r, 5) is a 5tm?, of six
generalized squares.
338 A. T H E G E N E R A L I Z E D - S I X - S Q U A R E T H E O R E M

PROOF. If M is a sum of three squares, apply Lemma A.3.1. Similarly, if

M — 1 is a sum of three squares, apply Lemma A.3.2, and if M — 4 is a sum of three
squares, apply Lemma A.3.3. For example, assume M — 1 = q2 -f q2 + q2 in N. By
Lemma A.3.2 s = EX2 + y2 + 2z(z + 1), £ G { 0 , 1 , 2}, hence
(r, 5 ) = ( C T 2 , ex2) + (y 2 , i/2) + ( s 2 + (z + l ) 2 , 2 s ( s + 1)) + (ql 0) + (ql 0) + («£, 0).

Similarly, assume A f - 4 = ^ i + ^ + ^ i - By Lemma A.3.3, 5 = ££ 2 +?/ 2 +22:(z+2),

e G { 0 , 1 , 2 } , hence
(r, s) = (5x 2 , £x 2 ) + (y\y2) + (^ 2 + (z + 2) 2 , 2z(z + 2)) + (^ 2 ,0) + (g 2 ,0) + (ql 0).
The case M is a sum of three squares is left to the reader. •

C O R O L L A R Y A.3.5. Let r > s > 0 be integers with

S = EX 2
+ 2y(y + 1) + 2z(z + 2), e G {0,1, 2},
£/ien (r, 5) is a sum of six generalized squares.

P R O O F . Let M = r - s. If at least one of M , M - 1, M - 4 is a sum of three

squares in N, then, by Corollary A.3.4, (r, s) is a sum of six generalized squares.
Thus, assume none of M, M — 1, Af — 4 is a sum of three squares in N. In particular,
it implies t h a t M > 5. By Lemma A.2.4 with £ = 5, M — 5 = r — s — 5 = qf + q'i + q2,.
Now proceed as in Corollary A.3.4: r — (s + 1 -f 4) + (r — 5 — 5), and with s as in
the lemma we get
(r, s) = (EX2, EX2) + (y 2 + (y + l ) 2 , 23/(2/ + 1))
+ ( ^ 2 + (2 + 2) 2 , 2z(* + 2)) + (qlO) + (g 2 , 0) + (ql 0).
•
A.4. A crucial l e m m a

In Lemma A.4.1 below we show t h a t most s G N admit the presentation s =

EX2 + 2y(y + 1) + 2z(z + 2 ) , £ G { 0 , 1 , 2}. Note t h a t neither 3 nor 23 admit such a
presentation.

L E M M A A.4.1. Let s G N and assume 2 s + 5 cannot be written as 2 s + 5 = a 2 + 2 c 2

with both a and c odd. Then s can be written in the form
S = EX2 + 2y(y + 1) + 2z(z + 2), e G {0,1, 2}.

P R O O F . By Theorem A. 1.4, 2s + 5 = a 2 + fr2 + 2c 2 , and we may assume t h a t

a > 1 is odd, 6 > 0 even, and either b > 0 or c > 0.
Case 1: 6 7^ 0. Since 6 is even, b>2. It follows t h a t

2 S + 6 = 1 + a2 + 62 + 2c 2 = i ((a - l ) 2 + (a + l ) 2 + 2b2 + (2c) 2 ) ,

therefore

which implies t h a t
+
as required.
">(^)(^MH(H *
 A.5. T H E P R O O F O F T H E O R E M A.1.2 339

Case 2: Assume 6 = 0, then c > 0, a n d is even by t h e assumptions of the

lemma. Thus
2s + 5 = a 2 + 2c 2 ,
a > 1 odd and c > 2 even. From

2s + 6 = ^ ((a - l ) 2 + (a + l ) 2 + 4c 2 )

it follows t h a t

and we conclude t h a t

s .2(^)(^)+2(£_l)(£ + l)+2(£)=,

which completes the proof. •

C O R O L L A R Y A.4.2. Let r > s > 0 and assume s satisfies the assumptions of

Lemma A.J^.l (namely, 2s + 5 cannot be written as 2s -f 5 = a2 -f 2c 2 wz£ft both a
and c odd). Then (r, s) is a sum of six generalized squares.

PROOF. Apply L e m m a A.4.1 and Corollary A.3.5. •

A.5. T h e proof of T h e o r e m A.1.2

Recall t h a t we want t o prove t h a t given r > s > 0 in N, the pair (r, s) is always
a sum of a t most six elements in V.

P R O O F O F T H E O R E M A . 1 . 2 . Denote M = r — s. By Corollary A.3.4, if one of

M, M — 1, M — 4 is a sum of three squares in N, we are done. T h e same if M < 448.
By Corollary A.4.2 the theorem holds if 2 s + 5 cannot be written as 2 s + 5 = a 2 + 2c 2
with b o t h a and c odd. It therefore remains t o prove the theorem in the following
case:
None of M,M — l , A f — 4 is a sum of three squares in N, M > 448, and
2s + 5 = a2 + 2c 2 with both a, c > 1 odd.
Case 1: a > 7. Then
2s - 44 = a2 - 49 + 2c 2 = (a - 7)(a + 7) + 2c 2 ,
so

which implies the presentation

The shift of this presentation is 10 2 + 7 2 = 149; it is congruent t o 5 modulo 8, and

149 < 448 < M. By Lemma A.2.4 it implies t h a t M - 149 is a sum of three squares
in N, hence, by part 2 of Lemma A.2.2, (r, s) is a sum of six generalized squares.
Case 2: a = 5. Then 2s -f 5 = 25 + 2c 2 so s = 10 -f c 2 which implies t h e
presentation
s = 2 - l - 2 + 2 - l - 3 + c2.
340 A. T H E G E N E R A L I Z E D - S I X - S Q U A R E THEOREM

Here the shift is l 2 -f 2 2 = 5. By Lemma A.2.4 r — s—shift = M — 5 is a sum of

three squares in N, hence we are done by part (2) of Lemma A.2.2.
Case 3: a = 3. From 2s + 5 = 9 + 2c2 deduce that
s = 2 - l + c2.
The length of this presentation is 2, and by part (1) of Lemma A.2.2 we are done.
Case 3: a = 1. Then 2s + 5 = 1 + 2c2, so 2s + 4 = 2c2 and
s = c2 - 2.
Since s > 1 and c is odd, it implies that c > 3, so we can write

This is a presentation of the form

s = l + 2t/(y + 2) + 2 ^ + 2),
whose shift is 8. By Lemma A.2.5 r — s—shift = M — 8 is a sum of three squares in
N, and by part (2) of Lemma A.2.2 (r, s) is a sum of six generalized squares. This
completes the proof. D
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 Index

*-exp(A), 284 T 2 -ideal, 80

*-var(A), 284 TA,47
C(fc,n), 125 UT(du...,dm), 24
C(k,n)mult, 125 UT2(F), 88
CTA,49 t / T | r , 289
Capm, 13 UTn(F), 2
£>A,47 [a, 6], 2
FSn, 46 A « i , . . . , f f c 2 ) , 125
F(Y,Z), 69 a (= n, 50
F{X | G), 65 X I # , 46
F ( X , * ) , 69 X T G, 46
F ( X , T r ) , 122 r
x ; P ' ( M f c ( F ) ) , 135
F(X>, 1 XA,46
F ( X ) * , 56 *£(>*), 265
F ( X ) » r , 66 Xn! n fc (A), 266
F{£h 12 deg/, 1
G?Sn,257 deg x . / , 1
G-identity, 66 exp(A), 144
G^ r , 289 e x p ( / ) , 215
i / ( d , 0 , 58 G, 63
/ A , 47 A h n, 15
L fe (x ;2 /), 129 <S>T, 4
M T n , 122 (A), 264
M§)N, 50 A x !,...,CC r , 1 8
M f c (F), 11 V, 4
M fc (G), 83 V*, 82
Mn(F®cF), 75 Aut(A), 63
M f c ) i (^), 75 Aut*(A), 65
M M ( G ) , 83 E n d A ( M ) , 29
Mkxl(F), 251 GKdim(A), 39
P T n , 122 Id(A), 3
Fn, 53 I d G ( A ) , 66
Pn(A), 54 Id*(A), 284
F n (V), 54 Id^ r (A), 66
P G , 256 supexp(A), 284
Fn*, 284 adcc, 309
P ^ , 268 det(a), 2
F ^ 2 , 283 tr d e g ( K / F ) , 39
F ^ 2 ( A ) , 273 tr(a), 2
9?-ideals, 73
FTA , 49 c n (A), 88
5(A), 84 <£(A), 256
Stm(xi ), 13 c ^ 1 " " ' ' 0 ^ ) , 98

349
350 INDEX

< ( A ) , 284 dense set, 29

4 r ( A ) , 268 Dilworth, 94
c3nup(A), 283 discriminant, 125
cl\A), 273 Drensky, 20, 28, 137, 185, 187, 207
^A,47
eA,47 element
e T A , 49 skew-symmetric, 69
/ = 0, 2 symmetric, 69
h(d,l,t), 57 elementary symmetric function, 15
/n(A), 90 essential G-identity, 262
Z 2 -exp(A), 276 essential hook, 241
exponent, 144
algebra *-exponent, 284
algebraic of bounded degree, 14 Z2-exponent, 276
center, 31 of a polynomial, 215
of generic elements, 11 superexponent, 284
of generic matrices, 12
of invariants, 124 First Fundamental Theorem, 125
supercommutative, 83 Formanek, 27, 28, 40, 128, 134, 137
verbally prime, 83 free G-graded algebra, 66
Amitsur, 35, 36, 104, 106, 108, 215, free algebra
242-245 with G-action, 65
Amitsur identity, 109 with involution, 69
Amitsur trick, 100 with trace, 122
Amitsur-Levitzki Theorem, 18 free associative algebra, 1
antiautomorphism, 65 free Lie algebra, 310
artinian ring, 29 free superalgebra, 69
AT-algebra, 317 free supercommutative algebra, 83
Frobenius reciprocity, 46
Berele, 38, 117, 212, 274
Birkhoff Theorem, 4 Gelfand-Kirillov dimension, 37
block-triangular matrix algebra, 24 generalized square, 244
Branching Theorem, 50 generic division ring, 40
generic element, 11
Capelli identity, 13 generic matrix, 12
Capelli polynomial, 13 graded algebra, 5, 61
central localization, 34 G-graded algebra, 61
character, 45 Z2-graded algebra, 65
induced, 46 graded identity, 66
inner product, 45 graded subalgebra, 61
irreducible, 45 graded subspace, 61
cocharacter, 54 Grassmann algebra, 2, 90
*-cocharacter, 284 Grassmann envelope, 81
G-cocharacter, 265 GSPI-algebra, 316
graded cocharacter, 238 Gurevich, 125
mixed trace cocharacter, 135
pure trace cocharacter, 123 Halpin, 28
codimension, 88 Herstein, 29
*-codimension, 264, 284 homogeneous
G-codimension, 256 component, 61
Z 2 -codimension, 273, 284 element, 61
graded codimension, 268, 283 hook, 58
supercodimension, 283 Hook Formula, 48
trace codimension, 123 hook number, 48
colength, 90 Hook Theorem, 105
column-stabilizer, 49
commuting ring, 29 idempotent, 45
central, 45
decomposable monomial, 259 essential, 49
 INDEX 351

minimal, 45 consequence, 7
minimal graded, 194 equivalent, 7
indecomposable monomial, 259 homogeneous, 5
induced module, 46 linear, 7
involution, 69 multialternating, 138
exchange, 77 multihomogeneous, 5
symplectic, 77 multihomogeneous component, 6
transpose, 77 multilinear, 7
polynomial growth, 171
Jacobson, 29 polynomial identity, 2
Posner's Theorem, 34
Kaplansky, 27
power sums symmetric function, 15
Kaplansky's Theorem, 31
prime ring, 34
Kasparian, 28
primitive ring, 29
Kemer, 20, 83, 110, 112, 113, 169
Procesi, 27, 40, 123, 125
Koshlukov, 20
product of varieties, 327
Kostant, 120
pure trace polynomial, 122
Krull dimension, 39
Razmyslov, 19, 20, 27, 28, 123
lattice permutation, 51
Razmyslow-Kemer-Braun Theorem, 35
Latyshev, 94
Regev, 94-96, 108, 117, 128, 141, 184, 212
Lewin Theorem, 21
relatively free algebra, 4
Lie algebra, 307
representation, 43
abelian, 308
completely reducible, 44
adjoint representation, 309
equivalent, 44
center, 309
irreducible, 44
nilpotent, 308
left regular, 44
representation, 309
Robinson-Schensted Correspondence, 102
simple, 308
Rosset, 18
solvable, 308
row insertion algorithm, 101
universal enveloping algebra, 309
row-stabilizer, 49
Lie commutator, 2
Rowen, 33
Lie ideal, 9
Lie identity, 310 Schur, 120
Littlewood-Richardson rule, 51 Schur's Lemma, 29
lower exponent, 144 Second Fundamental Theorem, 125
semiprime ring, 32
Maschke's Theorem, 44
semistandard tableau, 51
Mishchenko, 184, 268, 289
Sibirskii, 125
mixed trace polynomial, 122
skew-tableau, 51
monomial, 1
Skolem-Noether Theorem, 78
multilinearization process, 7
SPI-algebra, 314
multipartition, 264
splitting field, 30, 45
Newton's formulas, 15 stable identity, 10
Noether Normalization Theorem, 39 standard Lie polynomial, 311
standard polynomial, 13
outer tensor product, 50 standard tableau, 47
Strip Theorem, 107
partition, 46 subdirect product, 33
conjugate, 47 superalgebra, 65
permutation minimal, 194
d-bad, 94 reduced, 240
d-good, 94 superenvelope, 84
Pi-algebra, 2 supervariety, 80
Pi-exponent, 144 symmetric algebra, 124
polynomial symmetric function, 15
G-polynomial, 66 symmetric polynomial, 15
alternating, 12
central, 26 T-ideal, 3
352 INDEX

verbally prime, 82
trace identity, 122
trace polynomial, 122
transcendence degree, 39
trivial grading, 62

unirational, 40
unordered partition, 50
upper exponent, 144

Valenti, 128, 289, 292

Vandermonde matrix, 6
variety
distributive, 177
left noetherian, 176
minimal, 205
non-trivial, 4
of almost polynomial growth, 171
of polynomial growth, 171
prime, 83
proper, 4
variety of algebras, 4
Von Neumann Lemma, 50

Wedderburn, 29
Wedderburn-Artin Theorem, 29
Wedderburn-Malcev Theorem, 71
Weyl, 120
wreath product, 257

Young diagram, 47
Young tableau, 47
Young's Rule, 50
Young-Frobenius Formula, 48

Zorn, 30
Titles in This Series
122 A n t o n i o G i a m b r u n o and Mikhail Zaicev, Editors, Polynomial identities and
asymptotic methods, 2005
121 A n t o n Zettl, Sturm-Liouville theory, 2005
120 Barry S i m o n , Trace ideals and their applications, 2005
119 Tian M a and S h o u h o n g Wang, Geometric theory of incompressible flows with
applications to fluid dynamics, 2005
118 A l e x a n d r u B u i u m , Arithmetic differential equations, 2005
117 V o l o d y m y r N e k r a s h e v y c h , Self-similar groups, 2005
116 A l e x a n d e r K o l d o b s k y , Fourier analysis in convex geometry, 2005
115 Carlos Julio M o r e n o , Advanced analytic number theory: L-functions, 2005
114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005
113 W i l l i a m G. D w y e r , Philip S. Hirschhorn, Daniel M. K a n , and Jeffrey H. S m i t h ,
Homotopy limit functors on model categories and homotopical categories, 2004
112 Michael Aschbacher and S t e p h e n D . S m i t h , The classification of quasithin groups
II. Main theorems: The classification of simple QTKE-groups, 2004
111 Michael Aschbacher and S t e p h e n D . S m i t h , The classification of quasithin groups I.
Structure of strongly quasithin X-groups, 2004
110 B e n n e t t Chow and D a n Knopf, The Ricci flow: An introduction, 2004
109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups,
2004
108 Michael Farber, Topology of closed one-forms, 2004
107 J e n s Carsten J a n t z e n , Representations of algebraic groups, 2003
106 Hiroyuki Yoshida, Absolute CM-periods, 2003
105 Charalambos D . Aliprantis and O w e n Burkinshaw, Locally solid Riesz spaces with
applications to economics, second edition, 2003
104 G r a h a m Everest, Alf van der P o o r t e n , Igor Shparlinski, and T h o m a s Ward,
Recurrence sequences, 2003
103 Octav Cornea, Gregory Lupton, J o h n Oprea, and Daniel Tanre,
Lusternik-Schnirelmann category, 2003
102 Linda R a s s and J o h n Radcliffe, Spatial deterministic epidemics, 2003
101 Eli Glasner, Ergodic theory via joinings, 2003
100 P e t e r D u r e n and A l e x a n d e r Schuster, Bergman spaces, 2004
99 Philip S. Hirschhorn, Model categories and their localizations, 2003
98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps,
cobordisms, and Hamiltonian group actions, 2002
97 V . A. Vassiliev, Applied Picard-Lefschetz theory, 2002
96 Martin Markl, S t e v e Shnider, and J i m Stasheff, Operads in algebra, topology and
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95 Seiichi K a m a d a , Braid and knot theory in dimension four, 2002
94 M a r a D . N e u s e l and Larry S m i t h , Invariant theory of finite groups, 2002
93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:
Model operators and systems, 2002
92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1:
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91 Richard M o n t g o m e r y , A tour of subriemannian geometries, their geodesies and
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90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant
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89 Michel Ledoux, The concentration of measure phenomenon, 2001

88 Edward Frenkel and D a v i d Ben-Zvi, Vertex algebras and algebraic curves, second
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87 B r u n o Poizat, Stable groups, 2001
86 Stanley N . Burris, Number theoretic density and logical limit laws, 2001
85 V . A. Kozlov, V . G. Maz'ya, and J. R o s s m a n n , Spectral problems associated with
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84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001
83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant
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82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000
81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential
operators, and layer potentials, 2000
80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module
theory, 2000
79 J o s e p h A. C i m a and W i l l i a m T. R o s s , The backward shift on the Hardy space, 2000
78 Boris A. K u p e r s h m i d t , KP or mKP: Noncommutative mathematics of Lagrangian,
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77 Fumio Hiai and D e n e s P e t z , The semicircle law, free random variables and entropy,
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76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000
75 Greg Hjorth, Classification and orbit equivalence relations, 2000
74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,
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73 J o h n Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000
72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999
71 Lajos P u k a n s z k y , Characters of connected Lie groups, 1999
70 C a r m e n Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems
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69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition,
1999
68 D a v i d A. C o x and Sheldon K a t z , Mirror symmetry and algebraic geometry, 1999
67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and
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66 Yu. Ilyashenko and W e i g u Li, Nonlocal bifurcations, 1999
65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,
1999
64 R e n e A. C a r m o n a and Boris Rozovskii, Editors, Stochastic partial differential
equations: Six perspectives, 1999
63 Mark H o v e y , Model categories, 1999
62 Vladimir I. B o g a c h e v , Gaussian measures, 1998
61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic
algebra for ordinary differential and quasi-differential operators, 1999
60 Iain R a e b u r n and D a n a P. W i l l i a m s , Morita equivalence and continuous-trace
C*-algebras, 1998

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