Skip to main content

Computer algebra: Past and future

  • Conference paper
  • First Online:
EUROCAL '85 (EUROCAL 1985)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 203))

Included in the following conference series:

Abstract

The preceding just touches on some of the highlights of the accomplishments and unsolved problems in computer algebra. A really comprehensive survey would be much too long for the space available here. I close with the following quote, which has been attributed to Albert Einstein and helps, perhaps, to keep a proper prospective on our work: "The symbolic representation of abstract entities is doomed to its rightful place of relative insignificance in a world in which flowers and beautiful women abound."

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5. References

  1. D.S. Arnon, Algorithms for the geometry of semi-algebraic set, Ph.D. Thesis, Tech. Report No. 436, Computer Sciences Dept., Univ. of Wisconsin (1981).

    Google Scholar 

  2. ___, Topologically reliable display of algebraic curves, Proc. of SIGGRAPH '83 ACM, New York (1983), 219–227.

    Google Scholar 

  3. ___, On mechanical quantifier elimination for elementary algebra and geometry: solution of a nontrivial problem, these Proceedings.

    Google Scholar 

  4. ___, G.E. Collins and S. McCallum, Cylindrical algebraic decomposition I: the basic algorithm, SIAM J. on Comput. 13, 4 (1984), 865–877.

    Article  Google Scholar 

  5. J. Ax, On Schanuel's conjectures, Annals of Math. 93 (1971), 252–268.

    Google Scholar 

  6. E.T. Bell, Men of Mathematics, Simon and Schuster, New York (1937).

    Google Scholar 

  7. E.R. Berlekamp, Factoring polynomials over finite fields, Bell Sys. Tech. J. 46, (1967), 1853–1859.

    Google Scholar 

  8. W. Böge, Decision procedures and quantifier elimination for elementary real algebra and parametric polynomial nonlinear optimization, preliminary manuscript (1980).

    Google Scholar 

  9. W.S. Brown, On Euclid's algorithm and the computation of polynomial greatest common divisors, J. ACM 18, 4 (Oct 1971), 478–504.

    Google Scholar 

  10. ___, On the subresultant PRS algorithm, in [Jenk76], 271.

    Google Scholar 

  11. ___, The subresultant PRS algorithm, ACM TOMS 4, (1978), 237–249.

    Article  Google Scholar 

  12. ___, J.P. Hyde and B.A. Tague, The ALPAK system for non-numerical algebra on a digital computer II: rational functions of several variables and truncated power series with rational function coefficients, Bell Sys. Tech. J. 43 (March 1964), 785–804.

    Google Scholar 

  13. ___ and J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18, 4 (vol 1971), 505–514.

    Article  Google Scholar 

  14. B. Buchberger, An algorithm for finding a basis for the residue class ring of a zerodimensional polynomial ideal (German), Ph.D. Thesis Math. Inst., Univ. of Innsbruck, Austria (1965) and Aequationes Math 4, 3 (1970), 374–383.

    Article  Google Scholar 

  15. ___, A theoretical basis for the reduction of polynomials to canonical form, ACM SIGSAM Bull. 10, 3 (1976), 19–29.

    Google Scholar 

  16. ___, Some properties of Gröbner bases for polynomial ideals, ACM SIGSAM Bull. 10, 4 (1976), 19–24.

    Google Scholar 

  17. ___, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, in [Ng79], 3–21.

    Google Scholar 

  18. ___, A note on the complexity of constructing Gröbner bases, in [vHul83], 137–145.

    Google Scholar 

  19. ___, A critical-pair/completion algorithm for finitely generated ideals in rings, Proc. of the Conf. Rekursive Kombinatorik, Springer-Verlag LNCS (to appear).

    Google Scholar 

  20. ___, CAMP: a teaching project in symbolic computation at the University of Linz, SIGSAM Bull. 18, 4(Nov 1984),8–9.

    Google Scholar 

  21. ___, Gröbner bases: an algorithmic method in polynomial ideal theory, in Recent Trends in Multidimensional Systems Theory, N.K. Bose (ed.), D. Riedel Publ. Co. (1985), to appear.

    Google Scholar 

  22. ___, G.E. Collins and R. Loos (eds), Computer Algebra: Symbolic and Algebraic Computation, 2nd ed. (1983), Springer-Verlag, Vienna.

    Google Scholar 

  23. ___ and R. Loos, Algebraic simplification, in [BCL83], 11–43.

    Google Scholar 

  24. J.A. Campbell, J.G. Kent and R.J. Moore, Experiments with a symbolic programming system for complex analysis, BIT 16 (1976), 241–256.

    Google Scholar 

  25. G.W. Cherry, Algorithms for integrating elementary functions in terms of logarithmic integrals and error functions, Ph.D. Thesis, Univ. of Delaware (August 1983).

    Google Scholar 

  26. ___, Integration in finite terms with special functions: the logarithmic integral, SIAM J. on Comput. (to appear).

    Google Scholar 

  27. ___, Integration in finite terms with special functions: the error function, J. Symbolic Computation, (to appear).

    Google Scholar 

  28. P.J. Cohen, Decision procedures for real and p-adic fields, Comm. Pure and Applied Math. 22 (1969), 131–151.

    Google Scholar 

  29. G.E. Collins, PM, a system for polynomial manipulation, Comm. ACM 9, 8 (1966), 578–589.

    Google Scholar 

  30. ___, Subresultants and reduced polynomial remainder sequences, J. ACM 14 (Jan 1967), 128–142.

    Google Scholar 

  31. ___, The SAC-2 polynomial GCD and resultant system, U. of Wisconsin Comp. Sci. Tech. Report #145 (Feb. 1972).

    Google Scholar 

  32. ___, The computing time of the Euclidean algorithm, SIAM J. on Comput. 3, 1 (1974), 1–10.

    Google Scholar 

  33. ___, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Proc. of Second GI Conf. on Automata Theory and Formal Languages, Springer-Verlag LNCS 33 (1975), 134–183.

    Google Scholar 

  34. ___, Factoring univariate integral polynomials in polynomial average time, in [Ng79], 317–329.

    Google Scholar 

  35. ___, Quantifier elimination for real closed fields: a guide to the literature, in [BCL83], 79–81.

    Google Scholar 

  36. ___ and A.G. Akritas, Polynomial real root isolation using Descartes rule of signs, in [Jenk76], 272–275.

    Google Scholar 

  37. ___ and R. Loos, Real zeros of polynomials, in [BCL83], 83–94.

    Google Scholar 

  38. J.H. Davenport, On the Integration of Algebraic Functions, Springer-Verlag LNCS 102, (1981).

    Google Scholar 

  39. J.D. Dixon, The number of steps in the Euclidean algorithm, J. Number Theory 2 (1970), 414–422.

    Google Scholar 

  40. E. Engeler, Scientific computation: the integration of symbolic, numeric and graphic computation, these Proceedings (1985).

    Google Scholar 

  41. M.J. Fischer and M.O. Rabin, Super exponential complexity of Presburger arithmetic, MIT MAC Tech Memo 43 (1974).

    Google Scholar 

  42. J. Fitch, Proc. 1984 European Symp. on Symbolic and Algebraic Manipulation, Springer-Verlag LNCS 174 (1984).

    Google Scholar 

  43. R.W. Floyd (ed.), Proceedings of the ACM symposium on symbolic and algebraic manipulation, Comm. ACM 9, 8 (Aug 1966), 547–643.

    Google Scholar 

  44. R. Gebauer and H. Kredel, An algorithm for constructing Gröbner bases of polynomial ideals, SIGSAM Bull. 18, 1 (1984), 9.

    Google Scholar 

  45. R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. 75, 1 (1978), 40–42.

    Google Scholar 

  46. S.J. Harrington, A new symbolic integration system for Reduce, Computer J. 22 (1979),127–131.

    Google Scholar 

  47. H.A. Heilbronn, On the average length of a class of finite continued fractions, in Number Theory and Analysis, Paul Turán (ed.), Plenum, New York (1969), 87–96.

    Google Scholar 

  48. L.E. Heindel, Integer arithmetic algorithms for polynomial real zero determination, J. ACM 18, 4 (1971), 533–548.

    Google Scholar 

  49. G. Hermann, The question of finitely many steps in polynomial ideal theory (German), Math. Annal. 95 (1926), 736–788.

    Google Scholar 

  50. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Annals of Math. 79 (1964), 109–326.

    Google Scholar 

  51. G. Huet and D.C. Oppen, Equations and rewrite rules, in Formal Language Theory: Perspectives and Open Problems, R.V. Book (ed.), Academic Press, New York (1980), 349–405.

    Google Scholar 

  52. R.D. Jenks (ed.), 1976 ACM Symposium on Symbolic and Algebraic Computation, ACM, New York, NY (August 1976).

    Google Scholar 

  53. D.E. Jordon, R.Y. Kain and L.C. Clapp, Symbolic factoring of polynomials in several variables, Comm. ACM 8, 9 (August 1966), 638–643.

    Google Scholar 

  54. S.C. Johnson, Tricks for improving Kronecker's polynomial factoring algorithm, Bell Telephone Labs Tech. Report.

    Google Scholar 

  55. E. L. Kaltofen, Factorization of polynomials, in [BCL83], 95–113.

    Google Scholar 

  56. ___, On the complexity of factoring polynomials with integer coefficients, Ph.D. Thesis, Rensselaer Polytechnic Inst. (1982).

    Google Scholar 

  57. ___, A polynomial-time reduction from bivariate to univariate integral polynomial factorization, Proc. 23rd Symposium on Foundations of Computer Science, IEEE (1982), 57–64.

    Google Scholar 

  58. ___, A polynomial reduction from multivariate to bivariate polynomial factorization, Proc. 14th Annual ACM Symp. on the Theory of Comput. (1982), 261–266.

    Google Scholar 

  59. A. Kandri-Rody and D. Kapur, On the relationship between Buchberger's Gröbner basis algorithm and the Knuth-Bendix completion procedure, TIS Report No. 83CRD286, General Electric R & D Ctr. (1983).

    Google Scholar 

  60. ___, Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings, in [Fitc84], 195–206.

    Google Scholar 

  61. ___, Computing the Gröbner basis of polynomial ideals over integers, Proc. of 3rd Macsyma Users' Conf., Schenectady, NY (1984).

    Google Scholar 

  62. M. Karr, Summation in finite terms, J. ACM 28, 2 (1981), 305–350.

    Google Scholar 

  63. D.E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, Mass. (1969).

    Google Scholar 

  64. J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, manuscript (1979).

    Google Scholar 

  65. S. Landau, Factoring polynomials over algebraic number fields, SIAM J. on Comput. 14, 1 (1985), 184–195.

    Google Scholar 

  66. ___ and G. Miller, Solvability by radicals is in polynomial time, Proc. 15th Annual ACM Symp. on Theory of Comp. (1983), 140–151.

    Google Scholar 

  67. D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, in [vHul83], 146–156.

    Google Scholar 

  68. M. Lauer, Canonical representatives for residue classes of a polynomial ideal, in [Jenk76], 339–345.

    Google Scholar 

  69. ___, Computing in homomorphic images, in [BCL83], 139–168.

    Google Scholar 

  70. A.K. Lenstra, H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Annal. 261 (1982), 515–534.

    Google Scholar 

  71. V.E. Lewis (ed.), Proc. 1977 Macsyma Users' Conf., MIT, Cambridge, MA (1977).

    Google Scholar 

  72. R. Loos, Toward a formal implementation of computer algebra, Proc. 1974 European Symposium on Symbolic and Algebraic Manipulation, ACM SIGSAM Bull 8, 3 (1974), 9–16.

    Google Scholar 

  73. ___, Generalized polynomial remainder sequences, in [BCL83], 115–137.

    Google Scholar 

  74. E.W. Mayr and A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Advances in Math 46 (1982), 305–329.

    Google Scholar 

  75. F. Mora and H.M. Möller, The computation of the Hilbert function, in [vHul83], 157–167.

    Google Scholar 

  76. J. Moses, Symbolic integration, Ph.D. Thesis, M.I.T. (1967), also Project MAC Tech. Report TR-47.

    Google Scholar 

  77. ___, The integration of a class of special functions with the Risch algorithm, SIGSAM Bull., No. 13 (December 1969), 13–27.

    Google Scholar 

  78. J. Moses and D.Y.Y. Yun, The EZGCD algorithm, Proc. ACM National Conf., (Aug 1973), 159–166.

    Google Scholar 

  79. F. Müller, Ein exakter Algorithmus zur nichtlinearen Optimierung für beliebige Polynome mit mehreren Veranderlichen, Verlag Anton Hain, Meisenheim am Glan (1978).

    Google Scholar 

  80. D.R. Musser, Algorithms for polynomial factorization, Ph.D. Thesis, Dept. of Computer Sciences, Univ. of Wisconsin (1971).

    Google Scholar 

  81. ___, Multivariate polynomial factorization, J. ACM 22, (1976), 291–308.

    Google Scholar 

  82. E.W. Ng (ed.), Proc. 1979 European Symposium on Symbolic and Algebraic Manipulation, Spring-Verlag LNCS 72 (1979).

    Google Scholar 

  83. J.R. Pinkert, An exact method for finding the roots of a complex polynomial, ACM TOMS 2 (1976), 351–363.

    Google Scholar 

  84. J.W. Porter, Mathematika 22 (1975), 20–28.

    Google Scholar 

  85. M. Pohst and D.Y.Y. Yun. On solving systems of algebraic equations via ideal bases and elimination theory, in [Wang81], 206–211.

    Google Scholar 

  86. D. Richardson, Some unsolvable problems involving elementary functions of a real variable, J. Symb. Logic 33 (1968), 511–520.

    Google Scholar 

  87. R.H. Risch, On the integration of elementary functions which are built up using algebraic operations, System Development Corp. Tech. Report SP-2801/002/00 (1968).

    Google Scholar 

  88. ___, The problem of integration in finite terms, Trans. AMS 139, (May 1969), 167–189.

    Google Scholar 

  89. ___, Further results on elementary functions, IBM Research Tech Report RC 2402 (March 1969).

    Google Scholar 

  90. ___, The solution of the problem of integration in finite terms, Bull. AMS 76, (1970), 605–608.

    Google Scholar 

  91. ___, Algebraic properties of the elementary functions of analysis, Amer. J. Math. 101 (1979), 743–759.

    Google Scholar 

  92. H. Rolletschek, The Euclidean algorithm for Gaussian integers, in [vHu183], 12–23.

    Google Scholar 

  93. M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. of Math. 65, 2 (1976), 485–492.

    Google Scholar 

  94. M. Rothstein, Aspects of symbolic integration and simplification of exponential and primitive functions, Ph.D. Thesis, Univ. of Wisconsin (1976).

    Google Scholar 

  95. B.D. Saunders, An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations, in [Wang81], 105–108.

    Google Scholar 

  96. S. Schaller, Algorithmic aspects of polynomial residue class rings, Ph.D. Thesis., Dept. of Comp. Sci., Univ. of Wisconsin (1979).

    Google Scholar 

  97. A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. 60 (1954), 365–374.

    Google Scholar 

  98. M.F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations, Amer. J. Math. 103, 4 (1980), 661–682.

    Google Scholar 

  99. ___, Solving homogeneous linear differential equations in terms of second order linear differential equations, preprint.

    Google Scholar 

  100. ___, B.D. Saunders and B.F. Caviness, An extension of Liouville's theorem on integration in finite terms, SIAM J. on Comput. (to appear).

    Google Scholar 

  101. J.R. Slagle, A heuristic program that solves symbolic integration problems in freshman calculus, symbolic automatic integrator (SAINT), Ph.D. Thesis, Mass. Inst. of Tech. (1961).

    Google Scholar 

  102. D. Spear, A constructive approach to commutative ring theory, in [Lewi77], 369–376.

    Google Scholar 

  103. G. Szekeres, A canonical basis for the ideals of a polynomial domain, Am. Math. Monthly 59, 6 (1952), 279–386.

    Google Scholar 

  104. A. Tarski, A decision method for elementary algebra and geometry, 2nd revised ed., Univ. of California Press (1951).

    Google Scholar 

  105. B.M. Trager, Algebraic factoring and rational function integration, in [Jenk76], 219–226.

    Google Scholar 

  106. ___, Integration of simple radical extensions, in [Ng79], 408–414.

    Google Scholar 

  107. B. L. van der Waerden, Modern Algebra, Vol. I, Frederick Ungar, New York (1953).

    Google Scholar 

  108. J.A. van Hulzen, Computer Algebra: Proceedings of Eurocal '83, Springer-Verlag LNCS 162, (1983).

    Google Scholar 

  109. J. von zur Gathen, Factoring sparse multivariate polynomials, Proc. of the 24th Annual IEEE Symp. on the Foun. of Comp. Sci., (1983), 172–179.

    Google Scholar 

  110. P.S. Wang, Preserving sparseness in multivariate polynomial factorization, in [Lewi77], 55–61.

    Google Scholar 

  111. ___, An improved multivariate polynomial factoring algorithm, Math. Comp. 32 (1978), 1215–1231.

    Google Scholar 

  112. ___, Analysis of the p-adic construction of multivariate correction coefficients in polynomial factorization: iteration vs. recursion, in [Ng79], 291–300.

    Google Scholar 

  113. ___ (ed.) Proc. 1981 ACM Symp. on Symbolic and Algebraic Computation (1981), ACM, New York.

    Google Scholar 

  114. ___ and L.P. Rothschild, Factoring multivariate polynomials over the integers, Math. Comp. 29 (1975), 935–950.

    Google Scholar 

  115. L.H. Williams, Algebra of polynomials in several variables for a digital computer, J. ACM, 9, 1 (Jan 1962), 29–40.

    Google Scholar 

  116. F. Winkler, An algorithm for constructing detaching bases in the ring of polynomials over a field, in [vHul83], 168–179.

    Google Scholar 

  117. _____, On the complexity of the Gröbner-basis algorithm over K[x,y,z], in [Fitc84], 184–194.

    Google Scholar 

  118. ___, B. Buchberger, F. Lichtenberger and H. Rolletschek, An algorithm for constructing canonical bases of polynomial ideals, ACM TOMS 11, 1 (1985), 66–78.

    Google Scholar 

  119. D.Y.Y. Yun, The Hensel lemma in algebraic manipulation, PhD Thesis, M.I.T., MAC-TR-138 (Nov 1974).

    Google Scholar 

  120. H. Zassenhaus, On Hensel factorization I, J. Num. 1 (1969), 287–292.

    Google Scholar 

  121. G. Zacharias, Generalized Gröbner bases in commutative polynomial rings, Bachelor Thesis, MIT (1978).

    Google Scholar 

  122. R.E. Zippel, Probabilistic algorithms for sparse polynomials, Ph.D. Thesis, MIT (1979).

    Google Scholar 

  123. _____, Probabilistic algorithms for sparse polynomials, in [Ng79], 216–226.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Bruno Buchberger

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Caviness, B.F. (1985). Computer algebra: Past and future. In: Buchberger, B. (eds) EUROCAL '85. EUROCAL 1985. Lecture Notes in Computer Science, vol 203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15983-5_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-15983-5_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15983-4

  • Online ISBN: 978-3-540-39684-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics