Table Of ContentDifferential and Difference Dimension Polynomials
Mathematics and Its Applications
Managing Editor:
M.HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 461
Differential and
Difference Dimension
Polynomials
by
M. V. Kondratieva
Department of Mechanics and Mathematics,
Moscow State University,
Moscow, Russia
A. B. Levin
Department of Mathematics,
The Catholic University ofA merica,
Washington, D.C., U.S.A.
A. V. Mikhalev
Department of Mechanics and Mathematics,
Moscow State University,
Moscow, Russia
and
E. V. Pankratiev
Department of Mechanics and Mathematics,
Moscow State University,
Moscow, Russia
II
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. CataIogue record for tiris book is available from the Library of Congress.
ISBN 978-90-481-5141-7 ISBN 978-94-017-1257-6 (eBook)
DOI 10.1007/978-94-017-1257-6
Printed on acid-free paper
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©1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
No part of the material protected by tiris copyright notice may be reproduced or
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CONTENTS
Preface vii
Chapter I. Preliminaries 1
1.1. Notation and Conventions 1
1.2. Some Basic Notions and Results of the Theory of Com-
mutative Rings 5
1.3. Graded and Filtered Rings and Modules 19
1.4. Conservative Systems 32
1.5. Derivations and Differentials 37
Chapter II. Numerical Polynomials 45
2.1. Definition and some Properties of Numerical Polynomi-
~ %
2.2. Subsets on~m and their Dimension Polynomials. Dimen-
sion Polynomials of a Matrix 53
2.3. Algorithms for Computation of Dimension Polynomials 64
2.4. Ordering of Kolchin Dimension Polynomials 92
2.5. Dimension Polynomials of Subsets of zm 107
Chapter III. Basic Notion of Differential and Difference
Algebra 123
3.1. Rings with Operators 123
3.2. Basis Notions of Differential Algebra 125
3.3. Basic Notions of Difference Algebra 135
3.4. Inversive Difference Rings and Modules 158
3.5. Differential-Difference Structures 174
Chapter IV. Grabner Bases 191
4.1. Grobner Bases for Polynomial, Differential and Differ-
ence Modules 183
4.2. Basic Algorithms of Computation of Grobner Bases 203
4.3. Application of Grobner Bases to the Computation of
Characteristic Polynomials 209
Chapter V. Differential Dimension Polynomials 223
5.1. Characteristic Polynomials of Excellently Filtered Dif-
ferential Modules 223
5.2. Differential Dimension 228
5.3. Autoreduced Sets of Differential Polynomials. Charac-
teristic Sets 231
v
vi CONTENTS
5.4. Differential Dimension Polynomial of a Finitely Gener-
ated Differential Field Extension 237
5.5. Coherent Autoreduced Sets. Ritt-Kolchin's Algorithm 244
5.6. Invariants of Differential Dimension Polynomials 257
5.7. Minimal Differential Dimension Polynomial 267
5.8. Jacobi's Bound for a System of Algebraic Differential
Equations 273
Chapter VI. Dimension Polynomials in Difference and
Difference-Differential Algebra 281
6.1. Characteristic Polynomials of Graded Difference Module 281
6.2. Dimension Polynomials of Filtered Difference Modules.
Difference Dimension 284
6.3. Characteristic Polynomials of Inversive Difference Mod-
ules and their Invariants 290
6.4. Dimension Polynomials of Extensions of Difference and
Inversive Difference Fields 304
6.5. Linear (1'* -Ideals and their Dimension Polynomials 320
6.6. Computation of Dimension Polynomials in the Case when
the Basic Set Consists of Two Translations 332
6.7. Characteristic Polynomials of Finitely Generated Diffe-
rence-Differential Modules and their Invariants 344
Chapter VII. Some Application of Dimension Polyno-
mials in Difference-Differential Algebra 355
7.1. Type and Dimension of Difference-Differential Vector
Spaces 355
7.2. Type and Dimension of Finitely Generated Difference-
Differential Algebras 361
7.3. Difference-Differential Local Algebras 370
Chapter VIII. Dimension Polynomials of Filtered G-
modules and Finitely Generated G-fields Exten-
sions 377
8.1. Rings with a Group of Operators. G-modules 377
8.2. Dimension Polynomials of Excellently Filtered
G-modules 380
8.3. Some Generalizations for Differential G-structures 387
Chapter IX. Computation of Dimension Polynomials 397
9.1. Description of the Program Complex 397
9.2. Computation of Dimension Polynomials for some Sys-
tems of Differential Equations 398
References 405
Index 417
PREFACE
The role of Hilbert polynomials in commutative and homological algebra as
well as in algebraic geometry and combinatorics is well known. A similar role in
differential algebra is played by the differential dimension polynomials. The notion
of differential dimension polynomial was introduced by E. Kolchin in 1964 [KoI64]'
but the problems and ideas that had led to this notion (and that are reflected in this
book) have essentially more long history. Actually, one can say that the differential
dimension polynomial describes in exact terms the freedom degree of a dynamic
system as well as the number of arbitrary constants in the general solution of a
system of algebraic differential equations. The first attempts of such description
were made at the end of 19th century by Jacobi [Ja890] who estimated the number
of algebraically independent constants in the general solution of a system of linear
ordinary differential equations. Later on, Jacobi's results were extended to some
cases of nonlinear systems, but in general case the problem of such estimation
(that is known as the problem of Jacobi's bound) remains open. There are some
generalization of the problem of Jacobi's bound to the partial differential equations,
but the results in this area are just appearing.
At the beginning of the 20th century algebraic methods in the theory of differen-
tial equations were actively developed by F. Riquier [RiqlO] and M. Janet, [Jan21]
and [J an29] , whose works can be considered as the first stones in the basement of
differential algebra. It should be noted that many of the results obtained in this
period were connected with the concept of differential dimension. In particular,
we mention the Janet conjecture whose formulation in modern differential algebra
describes some properties of the differential dimension polynomial.
Differential algebra as a separate area of mathematics is largely due to its founda-
tor J.F. Ritt (1893-1951) and E.R. Kolchin (1916-1991). In the preface to his book
An Introduction to Differential Algebra (1957) I. Kaplansky wrote: "Differential al-
gebra is easily described: it is (99 percent or more) the work of Ritt and Kolchin".
In 30th J. Ritt published a series of articles (see [RR39], [Rit31], [Rit32], [Rit35aJ-
[Rit39], and [Rit45]) where he introduced the basic notions of differential algebra
(such as the algebra of differential polynomials, autoreduced sets (chains), perfect
differential ideals, and differential manifolds), proved some fundamental theorems
(including an important case of the basis theorem that was later generalized by
H. Raudenbuch) and outlined the main directions of research in this area. In 1950
J. Ritt published the first monograph [Rit50] on differential algebra where he sum-
marized all essential results obtained in this field by that time and formulated a
series of problems for the further study many of which remain unsolved up to now.
J. Ritt's research was continued and extended by E. Kolchin who not only created
and developed new fruitful theories in differential algebra but recasted the whole
vii
viii PREFACE
subject in the style of modern algebraic geometry with the additional presence of
derivation operators. In particular, Ellis Kolchin developed the theory of differen-
tial fields and created the differential Galois theory (see [KoI39-KoI55] and [KoISO])
where finite dimensional algebraic groups played the same role as finite groups play
in the theory of algebraic equations. In this connection he developed general the-
ory of algebraic groups that became later a field of intensive research with various
and deep applications. We should mention here Kolchin's monograph [KoIS5] and
works by A. Buium [Bui92], Ph. Cassidy [Cass72-CassS9], J. Kovacic [Kov69-
KovS6], E. Pankrat'ev [Pa71]' J.-F. Pommaret [PoS3], C. Mitschi and M. Singer
[MS96a], [MS96b], H. Umimura [Um96a], [Um96b], and some other mathematicians
that have formed the contemporary differential Galois theory and the modern the-
ory of differential algebraic groups. Among the other fields of Kolchin's activity in
differential algebra one should mention the study of singular solutions of algebraic
differential equations [KoI65], extensions of differential specializations, rational ap-
proximation [KoI59]' [KoI92]' and, of course, the theory of differential dimension
where the central role is played by Kolchin's differential dimension polynomials.
At the International Congress of Mathematicians in Moscow (1966) Kolchin formu-
lated the main problems and outlined the most perspective directions of research
connected with the differential dimension. Later on the results obtained in this
area were included into his famous monograph Differential Algebra and Algebraic
Groups that hitherto remains the most fundamental work on differential algebra.
The role of Kolchin in the development of differential algebra cannot be understood
completely if we do not mention his Differential Algebra Seminar at Columbia Uni-
versity that has been working for over thirty years. Various and deep results in the
classical differential algebra and in the related fields were obtained by Ph. Cassidy,
J. Johnson, W. Keigher, J. Kovacic, S. Morrison, W. Sit and many other Kolchin's
disciples.
Discussing the history of creation of the differential dimension theory, it should
be noted that in 1953 A. Einstein [Ei53] introduced a notion of the 'strength' of
a system of differential equations that is a certain function of integer argument
associated with the system. Another work that should be mentioned in connection
with the history of differential dimension polynomial is the paper by I. Zuckerman
[Zu65] who (independently of Kolchin and at almost the same time) found two
invariants of a partial differential field extension that turned out to be what we now
call the differential type and typical differential dimension (they define the degree
and the leading coefficient of the differential dimension polynomial, respectively).
In 1980 A. Mikhalev and E. Pankrat'ev [MP80] showed that this function actually
coincides with the appropriate differential dimension polynomial and found the
strength of some well-known systems of partial differential equations using methods
of differential algebra.
The intensive study of Kolchin's differential dimension polynomials began at the
end of 60th with the series of works by J. Johnson [Jo69a-Jo78] who showed that
the differential dimension polynomial of a differential field extension coincides with
the Hilbert characteristic polynomial of the filtered module of Kahler differentials
associated with the extension. In this connection J. Johnson developed the tech-
nique of dimension polynomials for differential modules and applied it to the study
of some classical problems of differential algebra. In particular, he characterized
PREFACE ix
the Krull dimension of finitely generated differential algebras, developed the the-
ory of local differential algebras, and proved a special case of Janet conjecture.
The study of differential dimension polynomials was continued by M. Kondrat'eva
[Kon88], [Kon89], A. Levin, A. Mikhalev, and Pankrat'ev [LM87], [LM92a]' [MP73],
[MP80], W. Sit [Si78], [Si92] and some other mathematicians. One of the most im-
portant directions of this study was the search for new invariants of the differential
dimension polynomial. Here we should mention the results of W. Sit [Si75] who
showed that the set of all differential dimension polynomials is well-ordered with
respect to some natural ordering and introduced the notion of the minimal dif-
ferential dimension polynomial associated with a differential field extension. This
important invariant was studied by M. Kondrat'eva [Kon88], [Kon89]; in particular,
she proved that every differential dimension polynomial is minimal for some differ-
ential field extension. There were some other results on invariants of a differential
dimension polynomial, but many important questions in this area are still open.
Another direction of activity that is related to the differential dimension polyno-
mials and that caught a second wind in 70th is the study of the problem of Jacobi's
bound. Some new results in this area were obtained by B. Lando [Land70], S.
Tomasovic [T076], R. Cohn [CoR80], [CoR83], and M. Kondrat 'eva, A. Mikhalev
and E. Pankrat'ev [KMP82]. At present the interest in the problem of Jacobi's
bound increases because the information on this bound leads to the estimates of
the complexity of the algorithms that are used for computation of differential di-
mension polynomials.
In much the same way as differential algebra arises from the study of algebraic
differential equations, difference algebra arises from the study of algebraic differ-
ence equations and therefore bears a considerable resemblance to its "differential
counterpart". Difference algebra was born in the same period as differential alge-
bra and it had the same foundator, J. Ritt, whose works [Rit34], [Rit41], [RD33]
and [RR39], as well as works of H. Raudenbush [RR39] and F. Herzog [Herz35],
laid the groundwork for the theory of algebraic difference equations. (It should be
noted that the first attempts of the study of equations in finite differences from the
algebraic standpoint were made at the end of nineteenth century by F. Casorati
[Cas882].) In 1934-1939 J. Ritt and H. Raudenbush introduced the basic concepts
of difference algebra (such as the ring of difference polynomials, perfect difference
ideals and difference manifolds) and described some properties of difference ideals
and manifolds in the spirit of differential algebra they were developing at that time.
Difference algebra as a separate mathematical area with its own ideas and methods
is mainly due to R. Cohn whose works [CoR4S-CoRS6aJ not only raised the differ-
ence algebra to the level comparable with the level of development of differential
algebra, but also clarified why many ideas that are fruitful in differential algebra
cannot be successfully applied in difference case, as well as many methods of dif-
ference algebra cannot have differential analogs. R. Cohn's book [CoR65] hitherto
remains the only fundamental monograph on difference algebra. Since 60th vari-
ous problems of difference algebra were developed by A. Babbitt [Bab62], 1. Balaba
[BaS4], [Ba87], 1. Bentsen [Be71], A. Bialynicki-Birula [Bia62]' P. Evanovich [Ev73),
[Ev84]' C. Franke [Fr63], B. Greenspan [Gre59], R. Infante [InSO], [InSl]' B. Lando
[Land72]' and the authors [KPSS5], [Le7S-LeS5b], [MPS7], [MP89], [Pa72a-PaS9].
The first steps in constructing the difference dimension theory were made by
x PREFACE
R. Cohn who introduced the notion of limit degree of an ordinary difference field
extension and found some interesting properties of this characteristic. Some gener-
alizations of the concept of limit degree for partial difference extensions were made
by P. Evanovich [EvS4], but the intensive study of the problems of dimension in
difference algebra began in SOth when the technique of dimension polynomials were
applied to the difference algebraic structures. The concept of difference dimension
polynomial was introduced by A. Levin first for difference field extension [Le7S]
and then for inversive difference field extensions and for difference and inversive
difference modules [LeSOa-LeS5a]. As it should be expected, difference dimension
polynomials play the same role in difference algebra as Hilbert polynomials in com-
mutative algebra or differential dimension polynomials in differential algebra. In
particular, a difference dimension polynomial of a difference field extension f{ C; L
carries some important invariants of the extensions (such as the difference transcen-
dence degree, type, and typical transcendence degree of Lover K). The properties
of difference dimension polynomials and their invariants were studied in the works
of I. Balaba [BaS4], [BaS7], M. Kondrat 'eva, E. Pankrat 'ev, and R. Serov [KPSS5],
A. Levin [LeSOa-LeS5b], E. Pankrat'ev [PaS9] and some other authors.
In the last ten years there were obtained some generalizations of the results
on difference dimension polynomials to the case of difference-differential algebraic
structures (see, for example, works [LMSSa], [LM91b]) and to the case of algebraic
and differential algebraic structures on which a finitely generated commutative
group acts (see [LMSSb], [LMS9a], [LM91a-LM92b]). These generalizations allow
to develop an algebraic approach to study the systems of algebraic differential
equations with delay (in particular, they allow to determine the strength of such a
system in the sense of A. Einstein [Ei53]).
The increasing role of dimension polynomials in the study of differential and
difference algebraic structures stimulated the development of methods of compu-
tation of such polynomials and their invariants. It turned out that the processes
of computation of differential and difference dimension polynomials are similar;
such a process can be divided into two steps: constructing a characteristic set of a
prime differential (difference) ideal and solving a certain combinatorial problem for
a subset of Nm . The development of the constructive methods in differential and
difference algebra was also stimulated by the recent development of computers and
their applications to the problems of algebraic computations. One can say that a
new science, computer algebra, has arisen on the junction of the mathematics and
computer science.
One of the main problems of computer algebra is the problem of construction
of "good" bases of polynomial ideals. Such bases (i.e., finite systems of generators
of an ideal that satisfy certain conditions) arouse in the works of F.S. Macaulay
[Mac16] and G. Herman [He26]. The interest to this kind of problems greatly
increased after the work of B. Buchberger [Bu65], [Bu70], [Bu79] who introduced
the notion of Grabner basis of a polynomial ideal and proposed an algorithm for its
computation. The explicit presentations of Grabner bases allows to answer many
questions of the theory of polynomial ideals. In particular, one can algorithmically
verify whether a polynomial belongs to a given polynomial ideal, solve the problem
of compatibility of a system of algebraic equations, explicitly construct the free
resolutions and so on. It should be noted that there are many monographs on
