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ISSN 0361-7688, Programming and Computer Software, 2006, Vol. 32, No. 1, pp. 56–58. © Pleiades Publishing, Inc., 2006.

Original Russian Text © S.A. Abramov, 2006, published in Programmirovanie, 2006, Vol. 32, No. 1.

56

On June 6, 2005, Manuel Bronstein—a prominent

scientist whose contribution to computer algebra and

many other areas of mathematics and computer science

can hardly be overestimated—died of a heart attack. He

was only forty-one.

A truly talented man, who was endlessly devoted to

science, has passed away. Manuel worked with all his

strength, enthusiastically, and was always researching

several difﬁcult problems simultaneously. Everyone

who knew him remembers that he was witty, extremely

keen in intellect, and cheerful. When not working, he

could take part in discussions on diverse topics, and his

partners admired him for his sudden impromptus,

jokes, felicitous remarks, and unexpected viewpoints

on the many little nothings of life.

Manuel was born on August 28, 1963, near Paris.

His father was a physician, and his mother was a sculp-

tor. Having graduated from school in France, he entered

Berkeley University (USA, California), where, in 1987,

he defended his PhD thesis under the supervision of

Professor M. Rosenlicht. For three years, he worked at

the IBM Research Center; then, from 1990 to 1997, in

the Swiss Federal Institute of Technology (ETH), and

since 1997, in France, in the French National Institute

for Research in Informatics and Automation (INRIA)

in Sophia Antipolis.

The dissertation defended at Berkeley was devoted

to a very difﬁcult problem related to symbolic integra-

tion (or integration in ﬁnite terms). Although the theory

of integration was developed by R. Risch (another

Ph.D. student of Rosenlicht) who presented in 1968 an

algorithm for integration of elementary functions, it

turned out that this algorithm was far from effective.

Manuel signiﬁcantly improved it (in particular, by

generalizing B. Trager’s algorithm for algebraic func-

tions to an algorithm for the mixed case of elementary

functions). While working for IBM, he implemented

the integration algorithm in the Axiom system. At that

time, this was the most powerful program for integra-

tion of functions. Manuel presented results of his stud-

ies in a large article published in 1990 in the

Journal of

Symbolic Computation.

Later, he intended to write a

monograph in two volumes devoted to all aspects of

symbolic integration. The ﬁrst volume was written and

went through two editions at the Springer publishing

house in 1997 and 2004. The second volume remained

uncompleted.

It was typical of Manuel to concentrate on urgent

difﬁcult problems. After the problem of integration, he

studied the problem of searching for closed-form solu-

tions to ordinary linear differential equations. In partic-

ular, in 1992, he designed a rather general algorithm for

ﬁnding solutions in the ﬁeld generated by the coefﬁ-

cients of the equation. In construction of these solu-

tions, one usually proceeds from a tower of extensions

of the basic ﬁeld. However, the key point is the possi-

bility of ﬁnding solutions in the basic ﬁeld, which con-

tains the coefﬁcients. This demonstrates the excep-

In Memory of Manuel Bronstein

S. A. Abramov

Computing Center, Russian Academy of Sciences, ul. Vavilova 40, GSP-1, Moscow, 119991 Russia

e-mail: sabramov@ccas.ru

Received September 9, 2005

DOI:

10.1134/S0361768806010063

PROGRAMMING AND COMPUTER SOFTWARE

Vol. 32

No. 1

2006

IN MEMORY OF MANUEL BRONSTEIN 57

tional value of this result by Manuel. Many problems of

differential algebra have analogues in the difference

case. It is also well known that, as a rule, these differ-

ence analogues are much more difﬁcult to solve. Nev-

ertheless, in 2000, Manuel developed an algorithm for

searching for solutions in the ﬁeld of coefﬁcients for the

case of difference equations. Moreover, he constructed

a universal general algorithm that covers differential,

difference, and

q

-difference equations as special cases.

This universality was attained by considering the prob-

lem on the level of noncommutative Ore polynomials.

At the same time, he signiﬁcantly advanced in the

development of the theory of unimonomial ﬁeld exten-

sions, whose foundations were laid by M. Karr in the

early 1980s. These results allowed a number of well-

known algorithms for searching for various solutions of

linear ordinary equations with polynomial coefﬁcients

to be generalized to much more complicated situations.

As for the Ore polynomials, it should be emphasized

that the very idea of using them in computer algebra

was ﬁrst proposed by Manuel (together with M. Pet-

kov ek) in a paper published in

Programming and

Computer Software

in 1994. This idea was important

not only from the theoretical standpoint; it also demon-

strated the possibility of designing universal computer

programs adjustable to the differential, difference, and

some other cases. This approach is widely used nowa-

days by developers of computer algebra algorithms and

systems.

The aforementioned paper devoted to this universal

approach is not the only publication by Manuel in

Pro-

gramming and Computer Software.

In 1992, he pub-

lished a survey of methods for solving ordinary differ-

ential equations and integration in this journal. With the

help of this survey, many specialists actively working in

related scientiﬁc areas managed to penetrate into this

involved subject. In 1993, Manuel was a co-editor of a

special issue of

Programming and Computer Software

devoted to computer algebra.

Far from intending to give here a complete survey of

Manuel’s results, we mention only that he obtained many

profound and valuable results not only on integration and

ordinary differential and difference equations, but also

on special functions, partial differential equations, oper-

ator factorization, and reducibility of systems of equa-

tions to special forms. He also published nice works on

linear algebra, algebraic geometry, etc.

Manuel was a brilliant programmer. He artistically

implemented all his algorithms in a number of com-

puter algebra systems. Recently, he actively worked on

the

Aldor

system and wrote a family of computer-alge-

braic libraries for it, namely, the

libaldor

and

Algebra

libraries (which provide the user with basic data struc-

tures and their operation procedures that are necessary

for applications of computer algebra) and the Sum

it

library (which contains efﬁcient programs implement-

ing complex modern algorithms for transforming and

solving linear ordinary differential and difference equa-

s

^

tions). For the Sum

it

library, he also developed two

interactive interfaces

bernina

and

shasta

, which made

the functions of this library available from other com-

puter algebra systems. These libraries and interactive

interfaces are high-quality tools that are widely used in

many research centers.

As noted earlier, since 1997, Manuel worked at

INRIA. At this institute (his last place of work), he

headed a research group consisting of ﬁrst-rate special-

ists. Each of them worked on his or her particular sci-

entiﬁc problem, and witnesses of his discussions with

collaborators were amazed by a deep insight of Manuel

into all these problems and by his ability to easily pass

in these discussions from one problem to another. The

intellectual virtuosity that he demonstrated in these dis-

cussions was magniﬁcent.

Manuel was a member of Editorial Boards of some

leading journals and scientiﬁc series, for instance, the

Journal of Symbolic Computation

and the series

Algo-

rithms and Computation in Mathematics.

He was a

member of the program and organizing committees of

several respected conferences and often chaired these

committees. This particularly relates to the annual

international ISSAC conference. He was also a vice-

president of SIGSAM, the international group on sym-

bolic and algebraic manipulation. In this role, he pro-

posed and realized many fruitful ideas. For instance, for

the ISSAC’05 conference, which was held in July of

2005, he had prepared a CD that contained not only

texts of all the talks given at the conference but also

some new software and other information valuable for

everyone interested in computer algebra and its appli-

cations. Unfortunately, he was not to take part in that

conference. That CD was distributed to all the partici-

pants of the conference and will remind them of Manuel.

He participated fruitfully in international research

projects. For instance, in the 1990s, he was one of the

leaders of the European projects Cathode 1 and Cath-

ode 2 devoted to computer-algebraic methods for solv-

ing ordinary differential equations. During the last ten

years, Manuel co-headed some projects involving Rus-

sian scientists, namely, “Computer algebra and linear

functional equations” (RFBR–INTAS), “Direct com-

puter-algebraic methods for explicit solution of sys-

tems of linear functional equations” (French—Russian

Lyapunov Center), and “Computer algebra and (

q

-)

hypergeometric terms” (Eco-Net program of the

French Ministry of Foreign Affairs). His last voyage

abroad was to Russia on May 15–19, 2005, within the

framework of the Eco-Net program.

It should be noted that Manuel was particularly

interested in Russia and events there. It is appropriate to

mention that his father’s family had Russian roots and

Manuel himself had chosen Russian as the foreign lan-

guage to study at high school (he told that, on the ﬁnal

exam, he had to read a passage from “Second Lieuten-

ant Kizhe” by Yu.N. Tynyanov). Later, he read scientiﬁc

journals in Russian and even translated some papers.

58

PROGRAMMING AND COMPUTER SOFTWARE

Vol. 32

No. 1

2006

ABRAMOV

And when he met his future wife Karola in Leipzig in

1990, the Russian language helped them to communi-

cate, although it was not a native tongue to either.

Remembering the joint work with Manuel, we

would like to mention his remarkable ability to grasp

instantly mathematical ideas and the extraordinary

mental agility, which followed from his acute analytical

sense. If a problem that arose in a discussion at the

blackboard or was proposed by somebody was of inter-

est to Manuel, he, as a rule, immediately proposed sev-

eral approaches to solving it, including quite unusual

and promising ones. Having outlined these approaches,

he immediately started to develop them in detail. He

made some calculations on the blackboard so fast that,

sometimes, it was hard to follow them. As a result of

such an improvisation, either the question was com-

pletely answered or real obstacles for further investiga-

tion were found. And Manuel often performed such

analyses without any intention to be a coauthor of the

work. He was a benevolent man and readily gave

detailed answers to questions of people whom he

scarcely knew, who asked him for a consultation or

advice during a break of a conference.

Of course, Manuel’s scientiﬁc interests were not

restricted to only difﬁcult classical problems. Computer

algebra is known to have at its disposal complete algo-

rithms for solving a number of such problems. How-

ever, the computational complexity of these algorithms

is very high, and they are hard to implement. Manuel

was interested in consideration of special cases of these

problems and in simplifying and reﬁning algorithms by

using heuristics and other methods. The results of his

work in this area included a new version of the algo-

rithm of parallel integration (the ﬁrst versions of the

algorithm of parallel integration were proposed in the

late 1970s and early 1980s by A. Norman, P. Moore,

and J. Davenport; here, the term “parallel” does not

relate to multiprocessor execution, and Manuel sug-

gested replacing this term with “ﬂat integration”). In

general, this algorithm is not as powerful as the com-

plete version of the Risch—Bronstein algorithm for

symbolic integration; however, it may be implemented

in just a hundred lines of code. A note on the algorithm

of parallel integration is published in this issue of

Pro-

gramming and Computer Software.

This note is an

extended abstract of Manuel’s talk at the joint seminar

on computer algebra of the MSU and JINR (Joint Insti-

tute for Nuclear Research) in Dubna on May 18, 2005.

It was submitted for publication in

Programming and

Computer Software

on June 3, three days before his

death overtook him outside his hometown, in Montpel-

lier. He went there for a few days to discuss with biolo-

gists the possibility of describing some biological mod-

els by recurrence relations of a special form. Manuel

was going to try to solve these relations using an origi-

nal approach he was working on in his last days. The

stock of his ideas and intentions seemed to be endless…

Providing for his large family (he was the father of

six children), he was always ready to support his

friends, colleagues, and associates, and helped them

any time when he felt that they needed his assistance or

sympathy. He never stopped being friendly to people

around him.

Manuel was just as benevolent and kind as he was

outstandingly talented. His name and his accomplish-

ments in computer algebra have already found their

high place in sciences. His death is a grievous, irre-

placeable loss for everyone who was lucky to work with

him or just be acquainted with him.