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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.
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
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
Received September 9, 2005
Vol. 32
No. 1
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
-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
The aforementioned paper devoted to this universal
approach is not the only publication by Manuel in
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
system and wrote a family of computer-alge-
braic libraries for it, namely, the
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
library (which contains efficient programs implement-
ing complex modern algorithms for transforming and
solving linear ordinary differential and difference equa-
tions). For the Sum
library, he also developed two
interactive interfaces
, 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
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 (
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.
Vol. 32
No. 1
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
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.
Full-text available
We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of ΠΣ⁎-fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of ΠΣ⁎-field extensions over a difference field that enjoys certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions of a homogeneous linear difference equation that are defined over the arising sums and products.
Full-text available
In this paper by means of computer experiment we study advantages and disadvantages of the heuristical method of “parallel integrator”. For this purpose we describe and use implementation of the method in Mathematica. In some cases we compare this implementation with the original one in Maple.
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