Eurocal '87 : European Conference on Computer Algebra Leipzig, GDR, June 2–5, 1987 Proceedings 3540515178

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Lecture Notes in Computer Science Edited by G. Goos and J. Hartmanis

378 J.H. Davenport (Ed.)

EUROCAL '87 European Conference on Computer Algebra Leipzig, GDR, June 2-5, 1987 Proceedings

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Editorial Board

D. Barstow W. Brauer P. Brinch Hansen D. Gries D. Luckham C. Moler A. Pnueli G. Seegmüller J. Stoer N. Wirth Editor

James H. Davenport University of Bath, School of Mathematical Sciences Claverton Down, Bath BA2 7 AY, UK

CR Subject Classification (1987): I. 1, J. 2, G. 4

ISBN 3-540-51517-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51517-8 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2145/3140-543210 - Printed on acid-free paper

Preface

This volume contains the papers presented at EUROCAL ’87, the sixth in a series of Interna­ tional Conferences on Computer Algebra, held in Europe with proceedings published by SpringerVerlag in this series. Previous conferences were: EUROSAM ’79 (Marseille; vol. 72), EUROCAM ’82 (Marseille; vol. 144), EUROCAL ’83 (Kingston-on-Thames; vol. 162), EUROSAM ’84 (Cam­ bridge; vol. 174) and EUROCAL ’85 (Linz; vol. 203 and 204). EUROCAL ’87 took place from the second to the fifth of June 1987, at the Karl-Marx Univer­ sity of Leipzig, GDR. There were invited lectures in the mornings, many of which are reproduced in the early part of this volume, and contributed papers and abstracts in the afternoons. There were also demonstrations of various software systems, a poster session and numerous meetings and informal encounters that cannot be repesented in a formal record, but which contributed greatly to the success of the conference. In addition, two papers presented at the Conference actually ap­ peared elsewhere: Operational semantics of behavioural covers based on narrowing by H. Reichel, which appeared in Springer Lecture Notes in Computer Science 332, pp. 235-248, and What is the rank of the Demjanenko matrix? by H. G. Folz and H. G. Zimmer, which appeared in J. Symbolic Computation 4 (1987) pp. 53-67. The General Chairman of the Conference was Prof. Dr. W. Lassner, of the Mathematics Department of the Karl-Marx University. To him and his colleagues is due the success of the local arrangements. Prof. Dr. N.J. Lehmann’s international experience was invaluable in organising the many facets of a large intercontinental conference. Generous sponsorship was provided by the Robotron organisation, and by the Rank Xerox group. Like any scientific conference, the quality is assured by the many referees who read the papers, and whose comments often greatly improve the presentation of the conference. To all of these people the Programme Committee extends its sincere thanks. I personally am grateful to many colleagues, notably Prof. J.P. Fitch, for their assistance in the preparation of these proceedings. Bath, April 1989

J.H. Davenport

Contents

Invited Papers Computer algebra in physical research of Joint Institute for Nuclear Research................................. 1 R. N. Fedorova, V. P. Gerdt, N. N. Govorun, V. P. Shirikov Complexity of quantifier elimination in the theory of ordinary differential equations .... 11 D. Yu. Grigor’ev Groups and polynomials................................................................................................................................. 26 G. C. Smith Symbolic computation in relativity theory............................................................................................... 34 M. A. H. MacCallum A zero structure theorem for polynomial-equations-solving and its applications.......................... 44 W. Wu (Abstract) Some algorithms of rational function algebra...........................................................................................45 S. A. Abramov

Applications and Systems The computer algebra system SIMATH................................................................................................... 48 R. Boffgen, M. A. Reichert Converting SAC-2 code to LISP ................................................................................................................ 50 L. Langemyr Computer algebra system for continued fractions manipulation........................................................ 52 V. Tomov, M. Nisheva, T. Tonev Computing a lattice basis from a system of generating vectors........................................................ 54 J. Buchmann, M. Pohst Expression optimization using high-level knowledge.............................................................................. 64 M. P. W. Mutrie, B. W. Char, R. H. Bartels CATFACT: Computer algebraic tools for applications of catastrophe theory...............................71 R. G. Cowell, F. J. Wright Computer algebra application for investigating integrability of nonlinear evolution systems . 81 V. P. Gerdt, A. B. Shabat, S. I. Svinolupov, A. Yu. Zharkov Computer classification of integrable seventh order MKdV-like equations....................................... 93 V. P. Gerdt, A. Yu. Zharkov Symbolic computation and the finite element method......................................................................... 95 J. P. Fitch, R. G. Hall Application of Lie group and computer algebra to nonlinear mechanics ....................................... 97 D. M. Klimov, V. M. Rudenko, V. F. Zhuravlev Hierarchical symbolic computations in the analysis of large-scale dynamical systems .... 107 J. Paczynski SCHOONSCHIP for computing of gravitino interaction cross sections in n=2 supergravity . 116 N. I. Gurin Creation of efficient symbolic-numeric interface.................................................................................... 118 N. N. Vasiliev Automatic generation of FORTRAN-coded Jacobians and Hessians.............................................. 120 P. van den Heuvel, J. A. van Hulzen, V. V. Goldman Laplace transformations in REDUCE 3................................................................................................. 132 C. Kazasov REDUCE 3.2 on iAPX 86/286-based personal computers ............................................................... 134 T. Yamamoto, Y. Aoki

VI

Some extensions and applications of REDUCE System........................................................................ 136 M. Spiridonova Infinite structures in SCRATCHPAD II.................................................................... 138 W. H. Burge, S. M. Watt Application of a structured LISP system to computer algebra........................................................... 149 J. Smit, S. H. Gerez, R. Mulder Number-theoretic transforms of prescribed length................................................................................ 161 R. Creutzburg, M. Tasche A Hybrid algebraic-numeric system ANS and its preliminary implementation............................. 163 M. Suzuki, T. Sasaki, M. Sato, Y. Fukui The calculation of QCD triangular Feynman graphs in the external gluonic field using REDUCE-2 system................................................................................................................................172 L. S. Dulyan Computer algebra application for determining local symmetries of differential equations . .174 R. N. Fedorova, V. V. Kornyak Trace calculations for gauge theories on a personal computer........................................................... 176 J. Ranft, H. Perlt Evaluation of plasma fluid equations collision integrals using REDUCE...........................................178 R. Liska, D. Drska (Abstract) Computerised system of analytic transformations for analysing of differential equations . . 179 V. L. Katkov, M. D. Popov Integral equation with hidden Eigenparameter solver: REDUCE and FORTRAN in tandem..................................................................................................................................................... 186 E. Shablygin Combinatorial aspects of simplification of algebraic expressions.......................................................192 A. Ya. Rodionov, A. Yu. Taranov Dynamic program improvement.................................................................................................................. 202 P. D. Pearce, J. P. Fitch Computer algebra and numerical convergence......................................................................................... 204 K. -U. Jahn Computer algebra and computation of Puiseux expansions of algebraic functions........................ 206 V. P. Gerdt, N. A. Kostov, Z. T. Kostova Boundary value problems for the Laplacian in Euclidean space solved by symbolic computation............................................................................................................................................ 208 F. Brackx, H. Serras The methods for symbolic evaluation of determinants and their realization in the planner-analytic, system.......................... .■........................................................................................ 216 V. A. Eltekov, V. B. Shikalov Transformation of computation formulae in systems of recurrence relations................................. 223 E. V. Zima DIMREG - The package for calculations in the dimensional regularization with 4-dimensional 7®-matrix in quantum field theory........................................................................225 V. A. Ilyin, A. P. Kryukov CTS - Algebraic debugging system for REDUCE programs............................................................... 233 A. P. Kryukov, A. Ya. Rodionov Applications of computer algebra in solid modelling............................................................................244 A. Bowyer, J. H. Davenport, P. S. Milne, J. A. Padget, A. F. Wallis Implementation of a geometry theorem proving package in SCRATCHPAD II ........................ 246 K. Kusche, B. Kutzler, H. Mayr Collision of convex objects...........................................................................................................................258 B. Roider, S. Stiffer

VII

Polynomial Algorithms Solving algebraic equations via Buchberger’s algorithm ................................................................... 260 S. R. Czapor Primary ideal decomposition.......................................................................................................................270 H. Kredel Solving systems of algebraic equations by using Grôbner bases...................................................... 282 M. Kalkbrener Properties of Grôbner bases under specializations................................................................................ 293 P. Gianni The computation of polynomial greatest common divisors over an algebraic number field . . 298 L. Langemyr, S. McCallum An extension of Buchberger’s algorithm to compute all reduced Grôbner bases of a polynomial ideal....................................................................................................................................300 K.-P. Schemmel Singularities of moduli spaces...................................................................................................................... 311 B. Martin, G. Pfister Radical simplification using algebraic extension fields....................................................................... 313 T. J. Smedley Hermite normal forms for integer matrices............................................................................................. 315 R. J. Bradford Mr Smith goes to Las Vegas: Randomized parallel computation of the Smith Normal Form of polynomial matrices.............................................................................................................. 317 E. Kaltofen, M. S. Krishnamoorthy, B. D. Saunders Fonctions symétriques et changements de bases.................................................................................... 323 A. Valibouze Complexity of standard bases in projective dimension zero............................................................... 333 M. Giusti Grôbner bases for polynomial ideals over commutative regular rings.............................................. 336 V. Weispfenning Some algebraic algorithms based on head term elimination over polynomial rings.................... 348 T. Sasaki Algorithmic determination of the Jacobson radical of monomial algebras..................................... 355 T. Gateva-Ivanova A recursive algorithm for computation of the Hilbert polynomial.................................................. 365 M. V. Kondratéva, E. V. Pankratév An afiine point of view on minima finding in integer lattices of lower dimensions.................... 376 B. Vallée A combinatorial and logical approach to linear-time computability .............................................. 379 P. Scheffler, D. Seese Complexity of computation of embedded resolution of algebraic curves............................. 381 J. P. G. Henry, M. Merle Polynomial factorization: An exploration of Lenstra’s algorithm...................................................... 391 J .A. Abbott, J. H. Davenport

Advanced Algorithms A matrix-approach for proving inequalities............................................................................................. 403 A. Ferscha Using automatic program synthesizer as a problem solver: Some interesting experiments . . 412 P. Nâvrat, L. Molnar, V. Vojtek Strong splitting rules in automated theorem proving............................................................................424 M. Baaz, A. Leitsch

VIII

Towards a refined classification of geometric search and computation problems........................ 426 T. Fischer Matrix-Pade fractions....................................................................................................................................438 G. Labahn, S. Cabay Computation of generalized Pade approximants.................................................................................... 450 G. Nemeth, M. Zimanyi A critical pair criterion for completion modulo a congruence.......................................................... 452 L. Bachmair, N. Dershowitz Shortest paths of a disc inside a polygonal region................................................................................ 454 G. Werner Rabins width of a complete proof and the width of a semialgebraic set......................................... 456 T. Recio, L. M. Pardo Practical aspects of symbolic integration over Q(x)............................................................................ 463 D. M. Gillies, B. W. Char Integration: Solving the Risch differential equation............................................................................ 465 J. A. Abbott Computation and simplification in Lie fields.........................................................................................468 J. Apel, W. Lassner A package for the analytic investigation and exact solution of differential equations .... 479 T. Wolf An algorithm for the integration of elementary functions................................................................... 491 M. Bronstein Index of authors............................................................................................................................................ 498

COMPUTER ALGEBRA IN PHYSICAL RESEARCH OF JINR

R.N. Fedorova, V.P. Gerdt, N.N.Govorun, V,P. Shirikov Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research Head Post Office, P.O, Box 79, Moscow, USSR

I. Originally computers were designed for numeric computation. Computer structure and instruction system were directed just at this application field, though instructions from the very beginning include those ones for logical operations, that allowed one to exceed the bounds of numeric information processing. Against a breakdown of predominant numeric methods the separate papers [1-3] on computer application to algebraic formula manipulati­ ons began to appear about 30 years ago. However, the two reasons delayed the development of computer algebra programs. First, unsufficient training of experts and unpreparedness of their scientific fi­ elds for computer algebra usage. Secondly, difficulties of computer adaptation to formula manipulations, since a user was obliged by himself to train a computer to perfom (i.e. to design a compiler,or an interpreter) algebraic transformations and only after that to use it for solving his problem. Therefore it was necessary to combine in one person two different professions system and applied programmer. It is also necessary to note one more not the least of the factors connected with unsufficient core memory and slow performance of the first generation computers for algebraic manipulations. Years went by, computer possibilities are rapidly extended from generation to generation, with increasing the efficiency of its usage in different applied fields. All that furthers the appearance of needs in algebraic manipulati­ ons by computer which could already be effectively enought implemented in serial computers. It was getting more and more Soviet and foreign papers on computer algebra (see,for example,reviews [4-7]) .Algorithmic and program methods were developed. Languages and compilers for symbolic information processing were created. Among them LISP was the most widely spreaded. Many early and some present-day CAS had been written in a code or assembler language, sometimes including elements of such numerical

2

languages as FORTRAN,ALGOL,PL/1 and others.lt raised an immobility of such a system. An availability of high-level symbolic languages like LISP allowed one create mobile systems. Just LISP underlies of the most developed and universal CAS, for example, widely-distributed systems REDUCE and MACSYMA. Such a system requires large computer resourses itself, i.e. typically 1 megabyte computer memory to say nothing of a problem to be solved, Howewer, the big universal systems could relatively easy be adopted at another computer. Moreover those CAS permit an extension by the use of addition to a program written in LISP or in "Bourse" language for a given system, Inapite of the appearance of those and other powerful universal CAS [7] (SMP, SCRATCHPAD-II, MAPLE, etc.) the process of creation, development and usage of special purpose CAS is in progress. In its own field such a system could in a sufficiently full measure satisfy the user's requirements. At the same time the special purpose systems gave, as a rule, the most effective algorithms and optimum internal representation for data, corresponding to the mathematical expressions and operations from the field of specialization for a given CAS. Therefore the specialized systems are usually much more effective in computer resourseB then the powerful universal CAS, Ab the stricking example of the specialized CAS it should be noted the system SCHOONSCHIP [8] which was developed more than 20 years ago (the first SCHOONSCHIP version has been created by Dutch physicist M.Veltman in 1965) and intended for computations in quantum field theory. Inspite of its "middle-aged" SCHOONSCHIP is till now beyond comparison in high energy physics. Just by means of its usage the record (with respect to amount of computation) results in quantum field theory were obtained. Below some of such computation will be briefly described. In JINR up to now the most part of the problems connected with computer algebra application is solved with the help of SCHOONSCHIP. Among others such general purpose CAS as REDUCE and FORMAC are more

widely used. II. Comparing computer algebra development in the USSR and abroad one can note the same nature of its basic stages: 1. Appearence of separate papers which were pioneers in the field. 2. Creation of tens of specialized and general purpose CAS written in assembler languages or algorithmic languages FORTRAN, ALGOL, PL/1,

3

LISP and others. 3. Intensive development of the algorithmic base of the present-day CAS and in first place of the general purpose systems. Appearance of new, considerably improved versions of such systems (REDUCE, ANALITIK, SCRATCHPAD).Creation of developed computer algeebra software for miniand microcomputers. A shift should be noted in time of coming either stage in the USSR and abroad. If the first papers on computer algebra appeared in our country and abroad approximately at the same time (first stage), then mass creation of CAS in the USSR (second stage) has begun in fact more later. It became possible only with appearance of the BESM-6 and then-serial ES computers. Now, several tens of CAS were created in the USSR (see review [6] and also proceedings of the conferences on computer algebra held in the Soviet Union [9-15]). Among the Soviet special purpose CAS, ones intended to mechanical problems, are predominated [6], While from general purpose systems: AVTO-ANALITIC [16],SIRIUS-SPUTNIK [17],AUM [18],ANALITIK [19] (see also [6]) the later is most widely used. In contrast to other Soviet CAS the language ANALITIK has been implemented by hardware, initially for MIR computers, then for the special processor SM-2410, which is a part of the two-processors complex SM-1410, and for the special processors ES-2680 distined for ES computers. Those special processors interprete by hardware the language ANALITIK-79 (SM-2410) and ANALITIK-82 (ES-2680). It is remarcable that such an approach to support by hardware of a high-level language started to be developed in the Institute of Cybernetics of the Ukrainian Akademy of Sciences as early as the sixties.For some time past one can see a sharp rise of interest to the symbolic processors as abroad where the greatest attention was paid to the different LISP and PROLOG dialects, as also in the Soviet Union. Among home works it should be noted the investigations on creation of the symbolic processor which are carried on at the Institute of Applied Mathematics of the USSR Academy of Sciences [20], REDUCE is the most widely-distributed foreign CAS in the USSR. It is used in different institutions for solving the scientific and applied problems. With the help of JINR REDUCE was adapted to ES computers in more than 50 Soviet institutions. Soviet papers based on REDUCE usage are sufficiently enough represented in references [10-15].

4

III. Let ub go on to abrief description of basic works on computer algebra carried out at the Joint Institute for Nuclear Research. The first investigations on computer realizations of symbolic mathematical operations were carried out at the JINR as long ago as early 1960b. Kim Ze Phen in 1963 has created the program for definite integration of some class of rational functions. At the same time H. Kaizer has developed computer algorithms and program [21] for the algebra of Dirac r-matrices. In 1964 V.I, Sharonov has done extention of ALGOL-60 to make some formula manipulations and, in particular, to calculate the trace of the r-matrices product [22]. The next step in computer algebra investigations at JINR began in the middle of 1970b after the first CAS SCHOONSCHIP had been obtained. It was in 1975 and created the favourable conditions for making investigations in computer algebra more active. In 1976 there were the first successful attempts [23,24] to use SCHOONSCHIP in quantum field theory. Now JINR has 12 different CAS [25] for the ES-1060, ES-1061, CDC-6500 and BESM-6 computers (see the table): ES-1060, ES-1061 REDUCE 2, 3.2 SCHOONSCHIP FORMAC 73 CAMAL ASHMEDAI AMP

CDC-6500 REDUCE 2 SCHOOSCHIP CLAM SYMBAL

BESM-6 AVTO-ANALITIK UPP SAVAG GRATOS

All Bysteems for the BESM-6 computer are Soviet ones and are described in refs [6,10-16], An implementation of CAS has encountered great difficulties because of unsufficient computer memory, differences between operating systems, adaptation of computer dependent parts of CAS and bo on. IV. Anouther group of works is connected with the development of CAS to extend the field of its application in JINR. Some of such works are following: 1) Improvement for interface between SCHOONSCHIP aand FORTRAN for symbolic-numeric computations [26]. 2) Development of the algorithm for virtual memory control in case of compiled LISP function in order to improve usage of REDUCE for the

5

CDC-6500 computer [27]. 3) Creation of general mathematical packages to extend possibilities of CAS for the following problems: - solving by power series method of an ordinary differential equation of the form (REDUCE) [28]

+ pCxly + qy = 0 , pCx? and q Of the ring (see [7 , 9]). Howe­ ver, the working time of the method from [9] is nonelementary (in Kalmar sense), in particular, it cannot be bounded from above by any finite tower of exponential functions (one can consider the working time on HAM or on any other polynomially equivalent computational model, e.g. Turing machine). The main result of the present paper is the following theorem, in which a quantifier elimination algorithm is designed with an elementary complexity bound (see also [3]). THEOREM. There is an algorithm which for a given formula of the kind (1) of the first-order theory of ordinary differential equati­ ons produces an equivalent to it quantifier-free formula of this theory of the form

(2)

where 6 Z { ,..., within time polynomial in

ar ^differential polynomials, M (N d, )m' c for a suitable

constant 0 Moreover, for the parameters of the polynomials (fl; X X 4^(0; :) ; hold the following bounds: .bsolute value of every (integer) coefficient of a polynomial • is less than 1/ The method from [9j contains two subroutines, transforming a system of differential equations in a certain disjunction of systems. The first subroutine is applied in the case, when informally speaking, for some distinguished indeterminate its derivative of the maximal order occurs at least in two polynomials. As a result of executing the first subroutine each obtained system has at most one polynomial containing this derivative. The second subroutine consists in split­ ting a system and decreasing the order of the distinguished indeter­ minate. Just executing the first subroutine leads in [9] to nonele­ mentary complexity bound. In the present paper transforming (instead of the first subroutine) to a disjunction of systems such that each of then contains at most one polynomial, in which occurs the derivati­ ve of the maximal order of the distinguished indeterminate is going in a quite another way, based on the constructing the greatest common divisor of a family of one-variable polynomials with parametric coef­ ficients (lemma 1 in section 1), apparently, interesting itself. The proof of lemma 1 is similar to the construction from [2], but on the other hand the direct application of the result from [¿J (see also

13

)[4] yields a worse complexity bound than in lemma 1. In section 2 a modification of the subroutine from [9] of splitting a system and decreasing the order of the distinguished indeterminate is exposed, then a quantifier elimination algorithm and its complexity analysis are exhibited. In (91, moreover a quantifier elimination method for the first-

order theory of partially differential equations is described. It would be interesting to clarify, whether there exists such a method with elementary complexity? This problem is connected (see [9]) with estimating in an effective version of Hilbert’s theorem on Idealbasis. 1. Constructing the greatest common divisor of a family of one-variable polynomials with parametric coefficients

We present the main result of this section (lemma 1) in a more general form than it is necessary for the main theorem, namely for the polynomials with the coefficients from a field F finitely gene­ rated over a prime subfield (cf. [1 , 2 , 4 , 5]). Thus F = H(T| ,•••,%) where either H =$ or H = Fp , i.e.

H is a prime subfield, are algebraically independent over the field H , an element is algebraic separable over the field H(Ti........ T.) , let ^) for H=Q|, and as

; ,I6

is the least possible. Define the deg-

ot€ H

is defined as its bit-size in the when H=Fp . Denote by the

maximum of the sizes of all the coefficients (from the field

of the polynomials m

rn

Q,- ■

•

)

H

at the monomials of variables

For the functions ^>0, ^>0, if for a suitable polynomial

P

we write g i an inequality

is valid. Consider some polynomials k0, kK 6 F [ X^ ..., X assume that the following bounds are true:

, YJ

and

14

for every 04*4K • Introduce a notation = the polynomials b^ € F [X^..., Xft] . Denote by* p

Y^ where an algebraic

closure of the field F . LEMMA 1. There is an algorithm which for given polynomials bo ..., bK

b( 'OJ x £

yields such two families of polynomials

F[X4,...,X^], 04 t 4 Nj, that a) quasiprojective varieties Xj,={x&F

:

=

05 q (X) 0} for 1 4 4 N4 form a decomposition of an open (in Zariski topology) set F11' \ { X 6 F : b^ ; (X) = 0 for all 14*4K and j},‘ ’’ b) for each

4 4^

’fclle following two varieties coincide:

{(x,g)GF"'xF=FM,+1: b4(x^)=...= hK(®^)=Oj

bo(x,^o}n(T^ x F) =

= {(X,y)eFM'+'f :^(X,l|)=0} n (T| A F)

, and be­

sides the leading coefficient is distinguished from zero everywhere on The running time of the algorithm can be estimated by a certain polynomial in K, . ¿Finally, the following bounds on the parameters of the polynomials are fulfilled:

REMARK. 1) The property b) shows that one can consider

9(|,

as

a kind of the greatest common divisor of the polynomials k^,..., bK (under the condition h,g =f Q ) considered in a variable Y on the quasiprojective variety 2) The properties a), b) are still correct if to replace

F

by

an arbitrary algebraically closed field containing F . Proof of Lemma 1. Por any -! 4_i 4 K , consider a quasiprojective variety = {x£ Fft: j (X)=...= b4)0 (X) =

Obviously and 4 J

U

-------- ^¿£4 (x)=-- - ^,4+4 ( U,”«'"

cF[X,,...,X„U„UJ 1}

Then

Dt=1-A where wh"9 E® (X„...,XW)

’««»J 5*< 5H m . Introduce quasiprojective varieties ■v'5+4 'X%)efMuwE™w}.

_0

sold 1 * 5+1 0$«^ s+ s'(s; c) that the CATFACT package constructs. (Note that CATFACT gives the unfolding functions u(c) automatically by

effecting the reduction to normal form.) At the present time this package has been implemented in both REDUCE 3.2 and REDUCE 3.3 for unfoldings of arbitrary rank cuspoid singularities by

Colvin (1988), who has improved and extended algorithms developed by Millington (1985) (see

also Millington and Wright, 1985). A future extension of CATFACT, not being pursued at present, may also include the develop­

ment of normal form recognition and reduction algorithms for equivariant singularities (singularities possessing symmetries) and for boundary catastrophes.

80

REFERENCES Arnold, V. I., 1974 ‘Critical points of smooth functions and their normal forms’, Usp. Mat. Nauk 29 pp 11-49. (Translated as Russ. Math. Surveys 29 pp 10-50). Bruce, J. W., du Plessis, A. A. and Wall, C. T. C., 1987 ‘Unipotency and Determinacy’, Invent. Math. 88 pp 521-554.

Buchberger, B., 1985 ‘Grôbner Bases : An algorithmic method in polynomial ideal theory’, in Multidimensional systems theory: progress, directions, and open problems in multidimen­ sional systems (Edited by N. K. Bose) Dordrecht, Holland: D. Reidel.

Colvin, A. P., 1988 Private communication. Cowell, R. G., 1989 ‘Application of ordered standard bases to catastrophe theory’ (Submitted to Proc. LMS.) Cowell, R. G. and Wright, F. J., 1989a ‘Truncation criteria and algorithm for the reduction to normal form of catastrophe unfoldings I: Singularities with zero rank’ (Submitted to Phil. Trans. R. Soc. Lond.) Cowell, R. G. and Wright, F. J., 1989b ‘Truncation criteria and algorithm for the reduction to normal form of catastrophe unfoldings II: Singularities with non-zero rank’ (Submitted to Phil. Trans. R. Soc. Lond.) Gibson, C. G., 1979 Singular points of smooth mappings, London: Pitman.

Millington, K., 1985 ‘Using computer algebra to determine equivalences in catastrophe theory’ (Ph.D. Thesis, University of London).

Millington, K. and Wright, F. J., 1985 ‘Algebraic computations in elementary catastrophe theory,’ EUROCAL’85 Lecture Notes in Computer Science 204 (Berlin, Heidelberg: Springer) pp 116-125. Poston, T. and Stewart, I. N., 1978 Catastrophe Theory and its Applications, London: Pitman. Stewart, I. N., 1981 ‘Applications of catastrophe theory to the physical sciences’, Physica 2D, pp 245-305. Stewart, I. N., 1982 ‘Catastrophe theory in Physics’, Rep. Prog. Phys. 45 pp 185-221.

Thom, R., 1972 Stabilité Structurelle et Morphogénèse. Reading, Mass.: Benjamin. (English translation by D. H. Fowler, 1975, Stuctural Stability and Morphogenesis. Reading, Mass.: Benjamin.) Wright, F. J. and Cowell, R. G., 1987 ‘Computer algebraic tools for applications of catastrophe theory’, in The Physics of Structure Formation : Theory and Simulation. Eds. W. Giittinger and G. Dangelmayr. Berlin: Springer, pp 402-415.

Wright, F. J. and Dangelmayr, G., 1985 ‘Explicit iterative algorithms to reduce a univariate catastrophe to normal form’, Computing 35 pp 73-83.

Zeeman, E. C., 1977 Catastrophe Theory. Reading, Mass.: Addison-Wesley.

COMPUTER ALGEBRA APPLICATION FOR INVESTIGATING INTEGRABILITY OF NONLINEAR EVOLUTION SYSTEMS V.P.Gerdt Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research Head Post Office, P.O. Box 79, Moscow, USSR B. A. Shabat, S.I. Svinolupov Bashkir Branch of the USSR Academy of Sciences Ufa, USSR

A.Yu. Zharkov Saratov State University Astrakhanskaya 83, Saratov, USSR

At present an intensive work on classification of integrable non­ linear partial differential equations with two independent variables is carried out. In a number of the cases ([1],[5],[9]) the formulation of effective criteria of integrability has been achieved and the complete lists of integrable systems have been obtained. For example, in [4],[5],[9] the complete list of integrable systems of the Schrodinger type

u =u +/