Computer Algebra: Past and Future

This survey reviews some of the important accomplishments in computer algebra since 1966 and indicates some directions for future research and development.

[1]  Stuart Carl Schaller.  Algorithmic aspects of polynomial residue class rings , 1979 .

[2]  Shunro Watanabe,et al.  A technique for solving ordinary differential equations using Riemann's P-functions , 1981, SYMSAC '81.

[3]  J. Dixon.  The number of steps in the Euclidean algorithm , 1970 .

[4]  Erwin Engeler,et al.  Scientific Computation: The Integration of Symbolic, Numeric and Graphic Computation , 1985, European Conference on Computer Algebra.

[5]  K. S. Kölbig.  Explicit Evaluation of Certain Definite Integrals Involving Powers of Logarithms , 1985, J. Symb. Comput..

[6]  J. Hyde,et al.  The Alpak system for nonnumerical algebra on a digital computer — II: Rational functions of several variables and truncated power series with rational-function coefficients , 1964 .

[7]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[8]  George E. Collins.  Factoring univariate integral polynomial in polynomial average time , 1979, EUROSAM.

[9]  B. F. Caviness,et al.  A Structure Theorem for Exponential and Primitive Functions , 1979, SIAM J. Comput..

[10]  Stephen M. Watt.  A System for Parallel Computer Algebra Programs , 1985, European Conference on Computer Algebra.

[11]  R. Risch.  The problem of integration in finite terms , 1969 .

[12]  Erich Kaltofen,et al.  A Generalized Class of Polynomials that are Hard to Factor , 1983, SIAM J. Comput..

[13]  David Y. Y. Yun,et al.  On solving systems of algebraic equations via ideal bases and elimination theory , 1981, SYMSAC '81.

[14]  H. Hironaka.  Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II , 1964 .

[15]  Anthony C. Hearn,et al.  Analytic computation of some integrals in fourth order quantum electrodynamics , 1974 .

[16]  Joel Moses.  The integration of a class of special functions with the Risch algorithm , 1969, SIGS.

[17]  Dennis Soule Arnon.  Algorithms for the geometry of semi-algebraic sets , 1981 .

[18]  Bruno Buchberger,et al.  Algorithm 628: An algorithm for constructing canonical bases of polynomial ideals , 1985, TOMS.

[19]  James Ax,et al.  On Schanuel's Conjectures , 1971 .

[20]  B. D. Saunders,et al.  Fast parallel computation of hermite and smith forms of polynomial matrices , 1987 .

[21]  Michael Rothstein.  Aspects of symbolic integration and simplification of exponential and primitive functions. , 1976 .

[22]  E. Berlekamp.  Factoring polynomials over finite fields , 1967 .

[23]  B. David Saunders.  An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations , 1981, SYMSAC '81.

[24]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

[25]  M. H. Protter,et al.  THE SOLUTION OF THE PROBLEM OF INTEGRATION IN FINITE TERMS , 1970 .

[26]  Bruno Buchberger,et al.  Some properties of Gröbner-bases for polynomial ideals , 1976, SIGS.

[27]  Grete Hermann,et al.  Die Frage der endlich vielen Schritte in der Theorie der Polynomideale , 1926 .

[28]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[29]  Rüdiger G. K. Loos.  Toward a formal implementation of computer algebra , 1974, SIGS.

[30]  George E. Collins.  PM, a system for polynomial manipulation , 1966, CACM.

[31]  Franz Winkler.  An algorithm for constructing detaching bases in the ring of polynominals over a field , 1983, EUROCAL.

[32]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[33]  Fritz Schwarz.  An Algorithm for Determining Polynomial First Integrals of Autonomous Systems of Ordinary Differential Equations , 1985, J. Symb. Comput..

[34]  B. F. Caviness,et al.  An Extension of Liouville's Theorem on Integration in Finite Terms , 1985, SIAM J. Comput..

[35]  A. Seidenberg.  A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA , 1954 .

[36]  James R. Slagle,et al.  A Heuristic Program that Solves Symbolic Integration Problems in Freshman Calculus , 1963, JACM.

[37]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[38]  M. Rosenlicht,et al.  On Liouville’s theory of elementary functions , 1976 .

[39]  George Szekeres,et al.  A Canonical Basis for the Ideals of a Polynomial Domain , 1952 .

[40]  James R. Pinkert,et al.  An Exact Method for Finding the Roots of a Complex Polynomial , 1976, TOMS.

[41]  Joachim von zur Gathen,et al.  Parallel algorithms for algebraic problems , 1983, SIAM J. Comput..

[42]  M. Fischer,et al.  SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC , 1974 .

[43]  Arjen K. Lenstra,et al.  Factoring polynominals over algebraic number fields , 1983, EUROCAL.

[44]  John ffitch,et al.  The Bath concurrent LISP machine , 1983, EUROCAL.

[45]  Michael F. Singer,et al.  Solving homogeneous linear differential equations in terms of second order linear differential equations , 1985 .

[46]  Bruno Buchberger,et al.  A theoretical basis for the reduction of polynomials to canonical forms , 1976, SIGS.

[47]  A. Slisenko.  Complexity problems in computational theory , 1981 .

[48]  Erich Kaltofen,et al.  Polynomial-Time Reductions from Multivariate to Bi- and Univariate Integral Polynomial Factorization , 1985, SIAM J. Comput..

[49]  David R. Musser,et al.  Multivariate Polynomial Factorization , 1975, JACM.

[50]  Franz Winkler,et al.  On the Complexity of the Groebner-Bases Algorithm over K[x, y, z] , 1984, EUROSAM.

[51]  W. S. Brown.  On the subresultant PRS algorithm , 1976, SYMSAC '76.

[52]  Erich Kaltofen,et al.  A polynomial reduction from multivariate to bivariate integral polynomial factorization. , 1982, STOC '82.

[53]  Richard Y. Kain,et al.  Symbolic factoring of polynomials in several variables , 1966, CACM.

[54]  Bruno Buchberger.  CAMP: a teaching project in symbolic computation at the University of Linz , 1984, SIGS.

[55]  Barry M. Trager.  Integration of simple radical extensions , 1979, EUROSAM.

[56]  David R. Stoutemyer.  A preview of the next IBM-PC version of muMATH , 1985 .

[57]  G. E. Collins,et al.  Quantifier Elimination for Real Closed Fields: A Guide to the Literature , 1983 .

[58]  James H. Davenport.  The Parallel Risch Algorithm (I) , 1982, EUROCAM.

[59]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[60]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[61]  Bruno Buchberger,et al.  A note on the complexity of constructing Gröbner-Bases , 1983, EUROCAL.

[62]  Joachim von zur Gathen,et al.  Factoring Sparse Multivariate Polynomials , 1983, J. Comput. Syst. Sci..

[63]  Paul S. Wang.  EVALUATION OF DEFINITE INTEGRALS BY SYMBOLIC MANIPULATION , 1971 .

[64]  D. Musser.  Algorithms for polynomial factorization. , 1971 .

[65]  Markus Lauer.  Canonical representatives for residue classes of a polynomial ideal , 1976, SYMSAC '76.

[66]  M. Singer.  Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations , 1981 .

[67]  Dennis S. Arnon,et al.  Topologically reliable display of algebraic curves , 1983, SIGGRAPH.

[68]  Erich Kaltofen,et al.  A polynomial-time reduction from bivariate to univariate integral polynomial factorization , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[69]  Leo P. Harten,et al.  DOE-MACSYMA: Progress Report , 1985, European Conference on Computer Algebra.

[70]  Dennis S. Arnon.  On Mechanical Quantifier Elimination for Elementary Algebra and Geometry: Solution of a Non-Trivial Problem , 1985, European Conference on Computer Algebra.

[71]  Bruce W. Char,et al.  A Tutorial Introduction to Maple , 1986, J. Symb. Comput..

[72]  Yasumasa Kanada,et al.  Parallelism in algebraic computation and parallel algorithms for symbolic linear systems , 1981, SYMSAC '81.

[73]  Erich Kaltofen,et al.  Factorization of Polynomials , 1983 .

[74]  Guy W. Cherry.  Integration in Finite Terms with Special Functions: The Logarithmic Integral , 1986, SIAM J. Comput..

[75]  Bruno Buchberger.  The Parallel L-Machine for Symbolic Computation , 1985, European Conference on Computer Algebra.

[76]  George E. Collins,et al.  Cylindrical Algebraic Decomposition I: The Basic Algorithm , 1984, SIAM J. Comput..

[77]  G. Rw.  Decision procedure for indefinite hypergeometric summation , 1978 .

[78]  D. McIlroy.  Algebraic Simplification , 1966, CACM.

[79]  Paul J. Cohen,et al.  Decision procedures for real and p‐adic fields , 1969 .

[80]  Nikolaos Glinos,et al.  Operational Calculus Technique for Solving Differential Equations , 1984, EUROSAM.

[81]  Deepak Kapur,et al.  Algorithms for Computing Groebner Bases of Polynomial Ideals over Various Euclidean Rings , 1984, EUROSAM.

[82]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[83]  James H. Davenport,et al.  On the Integration of Algebraic Functions , 1979, Lecture Notes in Computer Science.

[84]  James H. Davenport,et al.  Integration in finite terms , 1984, SIGS.

[85]  W. S. Brown.  The Subresultant PRS Algorithm , 1978, TOMS.

[86]  W. Böge,et al.  Quantifier Elimination for Real Closed Fields , 1985, AAECC.

[87]  Jean E. Sammet.  Proceedings of the ACM Symposium on Symbolic and Algebraic Manipulation: March 29-21, 1966 Washington, D. C. , 1966, CACM.

[88]  J. A. Campbell,et al.  Experiments with a symbolic programming system for complex analysis , 1976 .

[89]  Erich Kaltofen,et al.  On the complexity of factoring polynomials with integer coefficients , 1982 .

[90]  Paul S. Wang.  An improved multivariate polynomial factoring algorithm , 1978 .

[91]  David Y. Y. Yun,et al.  The EZ GCD algorithm , 1973, ACM Annual Conference.

[92]  Robert H. Risch.  ALGEBRAIC PROPERTIES OF THE ELEMENTARY FUNCTIONS OF ANALYSIS. , 1979 .

[93]  Dima Grigoriev,et al.  Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields , 1984, MFCS.

[94]  G. E. Collins,et al.  Real Zeros of Polynomials , 1983 .

[95]  Heinrich Rolletschek.  The Euclidean algorithm for Gaussian integers , 1983, EUROCAL.

[96]  Michael F. Singer,et al.  Elementary first integrals of differential equations , 1983 .

[97]  Bruce W. Char,et al.  Using Lie transformation groups to find closed form solutions to first order ordinary differential equations , 1981, SYMSAC '81.

[98]  W. S. Brown,et al.  On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors , 1971, JACM.

[99]  Michael Karr.  Theory of Summation in Finite Terms , 1985, J. Symb. Comput..

[100]  H. Michael Möller,et al.  The computation of the Hilbert function , 1983, EUROCAL.

[101]  Arjen K. Lenstra,et al.  Factorization of Polynomials , 1983 .

[102]  Joel Moses,et al.  Symbolic integration: the stormy decade , 1966, CACM.

[103]  Gary L. Miller,et al.  Solvability by Radicals is in Polynomial Time , 1985, J. Comput. Syst. Sci..

[104]  Shunro Watanbe,et al.  An Experiment Toward a General Quadrature for Second Order Linear Ordinary Differential Equations by Symbolic Computation , 1984, EUROSAM 1984.

[105]  Paul S. Wang.  Analysis of the p-adic construction of multivariate correction coefficiencts in polynomial factorization: Iteration vs. recursion , 1979 .

[106]  Michael Karr,et al.  Summation in Finite Terms , 1981, JACM.

[107]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[108]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[109]  Robert S. Sutor.  The Scratchpad II Computer Algebra Language and System , 1985, European Conference on Computer Algebra.

[110]  J. Porter.  On a theorem of Heilbronn , 1975 .

[111]  Erich Kaltofen.  A Polynomial-Time Reduction from Bivariate to Univariate Integral Polynomial Factorization , 1982, FOCS.

[112]  D. A. Spear.  A constructive approach to commutative ring theory , 1977 .

[113]  George E. Collins.  The Computing Time of the Euclidean Algorithm , 1974, SIAM J. Comput..

[114]  Paul S. Wang,et al.  Factoring multivariate polynomials over the integers , 1973, SIGS.

[115]  R. Loos.  Generalized Polynomial Remainder Sequences , 1983 .

[116]  Bruno Buchberger,et al.  A critical-pair/completion algorithm for finitely generated ideals in rings , 1983, Logic and Machines.

[117]  Steven J. Harrington.  A New Symbolic Integration System in Reduce , 1979, Comput. J..

[118]  Bruno Buchberger,et al.  A criterion for detecting unnecessary reductions in the construction of Groebner bases , 1979, EUROSAM.

[119]  Michael F. Singer.  Functions satisfying elementary relations , 1977 .

[120]  Leland H. Williams.  Algebra of Polynomials in Several Variables for a Digital Computer , 1962, JACM.

[121]  H. Zassenhaus.  On Hensel factorization, I , 1969 .

[122]  Jerald J. Kovacic,et al.  An Algorithm for Solving Second Order Linear Homogeneous Differential Equations , 1986, J. Symb. Comput..

[123]  David Y. Y. Yun,et al.  The Hensel Lemma in Algebraic Manipulation , 1973, Outstanding Dissertations in the Computer Sciences.

[124]  Joseph F. Traub,et al.  On Euclid's Algorithm and the Theory of Subresultants , 1971, JACM.

[125]  H. Heilbronn.  On the Average Length of a Class of Finite Continued Fractions , 1969 .

[126]  Guy W. Cherry.  Integration in Finite Terms with Special Functions: the Error Function , 1985, J. Symb. Comput..

[127]  Susan Landau,et al.  Factoring Polynomials Over Algebraic Number Fields , 1985, SIAM J. Comput..