Advertisement

Some Introductory Remarks on Computer Algebra

  • Wolfram Decker
Conference paper
Part of the Progress in Mathematics book series (PM, volume 202)

Abstract

Computer algebra is a relatively young but rapidly growing field. In this introductory note to the mini-symposium on computer algebra organized as part of the third European Congress of Mathematics, I will not even attempt to address all major streams of research and the many applications of computer algebra. I will concentrate on a few aspects, mostly from a mathematical point of view, and I will discuss a few typical applications in mathematics. I will present a couple of examples which underline the fact that computer algebra systems provide easy access to powerful computing tools. And, I will quote from and refer to a couple of survey papers, textbooks and web-pages which I recommend for further reading.

Keywords

Elliptic Curf Computer Algebra Symbolic Computation Computer Algebra System Introductory Remark 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Abo, W. Decker, N. Sasakura, An elliptic conic bundle in P4 arising from a stable rank-3 vector bundle, Math. Z. 229 (1998), 725–741.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    B. Amrhein, O. Gloor, R. E. Maeder, Visualizations for mathematics based on a computer algebra system, J. Symbolic Computation 23 (1997), 447–452.CrossRefGoogle Scholar
  3. [3]
    K. Appel, W. Haken, Every planar map is four colorable, AMS, Providence, 1989.zbMATHCrossRefGoogle Scholar
  4. [4]
    A. Aure, W. Decker, K. Hulek, S. Popescu, K. Ranestad, Syzygies of abelian and bielliptic surfaces in P 4, International J. Math. 8 (1997), 849–919.MathSciNetzbMATHGoogle Scholar
  5. [5]
    C. Babbage, Scribbling books, volume 2 (1836), Science Museum Library, London.Google Scholar
  6. [6]
    A. Beilinson, Coherent sheaves on P N and problems of linear algebra, Funkt. Anal. Appl. 12 (1978), 214–216.MathSciNetCrossRefGoogle Scholar
  7. [7]
    E. R. Berlekamp, Factoring polynomials over finite fields,Bell System Tech. J. 46 (1967), 1853–1859.MathSciNetGoogle Scholar
  8. [8]
    E. R. Berlekamp, Factoring polynomials over large finite fields, Math. Comp. 24 (1970), 713–735.MathSciNetCrossRefGoogle Scholar
  9. [9]
    H. U. Besche, B. Eick, Construction of finite groups, J. Symbolic Computation 27 (1999), 387–404.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    H. U. Besche, B. EickThe groups of order at most 1000 except 512 and 768, J. Symbolic Computation 27 (1999), 405–413.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25.MathSciNetzbMATHGoogle Scholar
  12. [12]
    B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108.MathSciNetzbMATHGoogle Scholar
  13. [13]
    M. Bronstein, Integration of elementary functions, J. Symbolic Computation 9 (1990), 117–173.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    M. Bronstein, Symbolic integration I — transcendental functions, Springer-Verlag, Berlin, 1997.zbMATHCrossRefGoogle Scholar
  15. [15]
    M. Bronstein, Symbolic integration tutorial, ISSAC’98, downloadable from http://www-sop.inria.fr/cafe/Manuel/cafe/Manuel. Bronstein/bronstein-eng.html
  16. [16]
    B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassen-rings nach einem nulldimensionalen Polynomideal,PdH thesis, Lepold-FranzensUniversität, Innsbruck, 1965.Google Scholar
  17. [17]
    B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes mathematicae 4 (1970), 374–383, English translation by M. Ambramson and R Lumbert in [18], 535–545.Google Scholar
  18. [18]
    B. Buchberger, F. Winkler (eds.), Gröbner bases and applications, Linz 1999, Cambridge University Press, Cambridge, 1999.Google Scholar
  19. [19]
    P. Candelas, X. C. de la Ossa, P. S. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B. 359 (1991), 21–74.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    J. Cannon, D. Holt, Foreword of the guest editors to the Special issue on computational algebra and number theory: Proceedings of the first MAGMA conference, J. Symbolic Computation 24 (1997), 233–234.CrossRefGoogle Scholar
  21. [21]
    J. F. Canny, D. Manocha, Multipolynomial resultant algorithms, J. Symbolic Computation 15 (1997), 99–122.MathSciNetGoogle Scholar
  22. [22]
    D. G. Cantor, H. Zassenhaus, A new algorithm for factoring polynomials over finite fields, Math. Comp. 36 (1981), 587–592.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    A. Cayley, The collected mathematical papers, vols 1-12, Cambridge University Press, Cambridge, 1889.Google Scholar
  24. [24]
    A. M. Cohen, J. H. Davenport, J. P. Heck, An overview of computer algebra, in A.M. Cohen (ed.), Computer algebra in industry, Wiley, Chichester, 1993.Google Scholar
  25. [25]
    H. Cohen, A course in computational algebraic number theory (3rd corrected printing), Springer-Verlag, New York, 1996.Google Scholar
  26. [26]
    H. Cohen, Advanced topics in computational number theory, Springer-Verlag, New York, 2000.zbMATHCrossRefGoogle Scholar
  27. [27]
    H. Cohen, H. W. Lenstra, Jr., Heuristics on class groups, in D.V. Chudnovsky et al. (eds.), Number theory, New York 1982, Springer-Verlag, Berlin, 1984.Google Scholar
  28. [28]
    H. Cohen, H. W. Lenstra, Jr., Heuristics on class groups of number fields, in H. Jager (ed.), Number theory, Noordwijkerhout, 19–83, Springer-Verlag, Berlin, 1984.Google Scholar
  29. [29]
    D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms, second edition, Springer-Verlag, New York, 1997.Google Scholar
  30. [30]
    J. H. Davenport, On the integration of algebraic functions,Springer-Verlag, New York, 1981.zbMATHCrossRefGoogle Scholar
  31. [31]
    W. Decker, L. Ein, F.-O. Schreyer, Construction of surfaces in P 4, J. Algebraic Geometry 2 (1993), 185–237.MathSciNetzbMATHGoogle Scholar
  32. [32]
    W. Decker, T. de Jong, Gröbner bases and invariant theory, in [18], 61–89.Google Scholar
  33. [33]
    W. Decker, F.-O. Schreyer, Non-general type surfaces in P 4 : some remarks on bounds and constructions, J. Symbolic Computation 29 (2000), 545–582.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    J. Dixmier, D. Lazard, Minimum number of fundamental invariants for the binary form of degree 7, J. Symbolic Computation 6 (1988), 113–115.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    B. Eick, E. A. O’BrienThe groups of order 512, in B.H. Matzat et al. (eds.), Algorithmic algebra and number theory, 379–380, Springer-Verlag, Berlin, 1999.Google Scholar
  36. [36]
    D. Eisenbud, S. Popescu, Gale duality and free resolutions of ideals of points, Invent. Math. 136 (1999), 419–449.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    G. Ellingsrud, S.-A. Stromme, The number of twisted cubic curves on the general quintic threefold, Math. Scand. 76 (1995), 5–34.MathSciNetzbMATHGoogle Scholar
  38. [38]
    R. W. Floyd (ed.), Proc. of ACM Symposium on Symbolic and Algebraic Manipulation (SYMSAM’66), Washington D.C., Comm ACM 9 (1966), 574–643.Google Scholar
  39. [39]
    F. Franklin, On the calculation of the generating functions and tables of groundforms for binary quantics, Amer. J. of Math. 3 (1883), 128–153.MathSciNetCrossRefGoogle Scholar
  40. [40]
    J. von zur Gathen, J. Gerhard, Modern computer algebra, Cambridge University Press, Cambridge, 1999.zbMATHGoogle Scholar
  41. [41]
    K. Gatermann, Computer algebra methods for equivariant dynamical systems, Springer-Verlag, Berlin, 2000.zbMATHCrossRefGoogle Scholar
  42. [42]
    K. Geddes, S. R. Czapor, G. Labahn, Algorithms for computer algebra, Kluwer Academic Publishers, Boston, 1992.zbMATHCrossRefGoogle Scholar
  43. [43]
    G. Gonnet, A study of iteration formulas for root finding, where mathematics, computer algebra and software engineering meet, in these proceedings.Google Scholar
  44. [44]
    L. Gonzalez-Vega, T. Recio, Industrial applications of computer algebra: climbing up a mountain, going down a hill, in these proceedings.Google Scholar
  45. [45]
    P. Gordan, Neuer Beweis des Hilbertschen Satzes über homogene Funktionen, Nachrichten König. Ges. der Wiss. zu Gött., 1899, 240–242, English translation by M. Abramson in SIGSAM Bulletin 32 Number 2 (1998), 47–48.CrossRefGoogle Scholar
  46. [46]
    R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40–42.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    H. Grace, A. Young, The algebra of invariants, Cambridge University Press, Cambridge, 1903.zbMATHGoogle Scholar
  48. [48]
    G.-M. Greuel, Applications of computer algebra to algebraic geometry, singularity theory and symbolic-numerical solving, in these proceedings.Google Scholar
  49. [49]
    J. Hammond, On the solution of the differential equation of sources, Amer. J. of Math. 5 (1883), 218–227.MathSciNetCrossRefGoogle Scholar
  50. [50]
    D. Hilbert, Ober die Theorie der algebraischen Formen,Math. Ann. 36 (1890), 473–534.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    D. HilbertOber die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–373.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    A. Hirschowitz, C. Simpson, La résolution minimale de l’idéal d’un arrangement général d’un grand nombre de points dans P“ i, Invent. Math. 126 (1996), 467–503.Google Scholar
  53. [53]
    H. G. Kahrimanian, Analytical differentiation by a digital computer, Master’s thesis, Temple University, Philadelphia, 1953.Google Scholar
  54. [54]
    E. Kaltofen, Polynomial-time reductions from multivariate to bi-and univariate integral polynomial factorization, SIAM J. Comp. 14 (1985), 469–489.Google Scholar
  55. [55]
    E. KaltofenPolynomial factorization, in B. Buchberger et al. (eds.), Computer algebra, 95–113, Springer-Verlag, Wien, 1982.Google Scholar
  56. [56]
    E. KaltofenPolynomial factorization 1982–1986, in I. Simon (ed.), Computers in mathematics, 285–309, Marcel Dekker, New York, 1990.Google Scholar
  57. [57]
    E. Kaltofen, Polynomial factorization 1987–1991, in D.V. Chudnovsky, R D Jenks (eds.), Proceedings of LATIN’92, Sao Paulo, 294–313, Springer-Verlag, New York, 1992.Google Scholar
  58. [58]
    S. Katz, On the finiteness of rational curves on quintic threefolds, Composito Math. 60 (1986), 151–162.Google Scholar
  59. [59]
    W. Krandick, S. Rump (eds.), Special issue on Validated numerical methods and computer algebra, J. Symbolic Computation 24 (1997), 649–803.Google Scholar
  60. [60]
    L. Kronecker, Grundzuge einer arithmetischen Theorie algebraischer Grössen, J. Reine Angew. Math. 92 (1882), 1–122.zbMATHGoogle Scholar
  61. [61]
    J. P. S. Kung, G.-C. Rota, The Invariant Theory of Binary Forms, Bull. Am. Math. Soc. 10 (1984), 27–85.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    L. A. Lambe (ed.), Special issue, J. Symbolic Computation 23 (1997), 445–623.CrossRefGoogle Scholar
  63. [63]
    P. L. Larcombe, On Lovelace, Babbage and the origins of computer algebra,in [99].Google Scholar
  64. [64]
    A. K. Lenstra, H. W. Lenstra, Jr., Algorithms in number theory, in J. van Leeuwen (ed.), Algorithms and complexity„ Volume A, 673–716, Elsevier, Amsterdam, 1990.Google Scholar
  65. [65]
    A. K. Lenstra, H. W. Lenstra, Jr. (eds), The development of the number field sieve, Springer-Verlag, Berlin, 1993.zbMATHGoogle Scholar
  66. [66]
    A. K. Lenstra, H. W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.MathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    H. W. Lenstra, Jr., Flags and lattice basis reduction, in these proceedings.Google Scholar
  69. [69]
    A. Lorenzini, The minimal resolution conjecture, J. Algebra 156 (1993), 5–35.MathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    L. F. Menabrea, Sketch of the analytical engine invented by Charles Babbage. With notes upon the memoir by the translator, Ada Augusta, Countess of Lovelace, in P. Morrison and E. Morrison (eds.), Charles Babbage and his calculating engines, part II, Dover Publications, New York, 1961.Google Scholar
  71. [71]
    W. F. Meyer, Invariantentheorie, in Encyklopädie der mathematischen Wissenschaften, Erster Band,Teubner, Leipzig, 1898–1904.Google Scholar
  72. [72]
    H. M. Möller, Gröbner bases and numerical analysis, in [18], 159–178.Google Scholar
  73. [73]
    J. F. Nolan, Analytical differentiation on a digital computer, Master’s thesis, MIT, Cambridge, 1953.Google Scholar
  74. [74]
    A. M. Odlyzko, H. J. J. to Riele, Disproof of the Mertens conjecture, J. Reine Angew. Math. 357 (1985), 138–160.Google Scholar
  75. [75]
    P. J. Olver, Classical invariant theory, Cambridge University Press, Cambridge, 1999.zbMATHCrossRefGoogle Scholar
  76. [76]
    S. R. Petrick (ed.), Proc. of the Second Symposium on Symbolic and Algebraic Manipulation (SYMSAM’71), Los Angeles, ACM Press, New York, 1971.Google Scholar
  77. [77]
    C. Pomerance, Analysis and comparism of some factoring algorithms, in H.W. Lenstra, Jr., R. Tijdeman (eds.), Computational methods in number theory, 89–139, Mathematical Centre Tracts 154/155 Math. Centrum, Amsterdam, 1982.Google Scholar
  78. [78]
    C. Pomerance, The quadratic sieve factoring algorithm, in T. Beth et al. (eds.), Advances in cryptology, Springer-Verlag, Berlin, 1985.Google Scholar
  79. [79]
    S. Popescu, On smooth surfaces of degree > 11 in P4, PhD thesis, Universität des Saarlandes, Saarbrücken, 1993.Google Scholar
  80. [80]
    S. Popescu, K. Ranestad, Surfaces of degree 10 in the projective fourspace via linear systems and linkage, J. Algebraic Geometry 5 (1996), 13–76.Google Scholar
  81. [81]
    R. Risch, On the integration of elementary functions which are built up using algebraic operations, Report SP-2801/002/00, System Development Corp., Santa Monica, 1968.Google Scholar
  82. [82]
    R. RischFurther results on elementary functions, Report RC-2402, IBM Corp., Yorktown Heights, 1969.Google Scholar
  83. [83]
    R. Risch, The problem of integration in finite terms, Trans. A. M. S. 139 (1969), 167–189.Google Scholar
  84. [84]
    R. Risch, The solution of the problem of integration in finite terms, Bull. A. M. S. 76 (1970), 605–608.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [85]
    R. L. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm ACM 21 (1978), 120–126.MathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    P. RoelseFactoring high-degree polynomials over F2 with Niederreiter’s algorithm on the IBM SP2, Math. Comp. 68 (1999), 869–880.MathSciNetzbMATHCrossRefGoogle Scholar
  87. [87]
    F.-O. SchreyerSmall fields in constructive algebraic geometry, 221–228, in Moduli of vector bundles, Sanda 1994, 221–228, Dekker, New York, 1996.Google Scholar
  88. [88]
    F.-O. Schreyer, F. Tognoli, Constructions and investigations with small finite fields, to appear in D. Eisenbud et al. (eds.), Mathematical computations with Macaulay2, Springer-Verlag, New York.Google Scholar
  89. [89]
    F. T. von Schubert, De inventione divisorum, Nova Acta Academiae Scientiarum Imperalis Petropolitanae 11 (1793), 172–182.Google Scholar
  90. [90]
    I. E. Shparlinski, Finite fields: theory and computations,Kluwer Academic Publishers, Dordrecht, 1999.zbMATHCrossRefGoogle Scholar
  91. [91]
    M. F. Singer, Formal solutions of differential equations, J. Symbolic Computation 10 (1990), 59–94.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [92]
    L. SmithPolynomial Invariants of Finite Groups, A.K. Peters, Wellesley, 1995.zbMATHGoogle Scholar
  93. [93]
    B. Sturmfels, Algorithms in invariant theory, Springer-Verlag, Wien, 1993.zbMATHGoogle Scholar
  94. [94]
    D. D. Swade, Der mechanische Computer des Charles Babbage, Spektrum der Wissenschaft, April 1993.Google Scholar
  95. [95]
    J. J.Sylvester, Tables of the generating functions and groundforms for the binary quantics of the first ten orders, Amer. J. Math. 2 (1879), 223–251.MathSciNetCrossRefGoogle Scholar
  96. [96]
    J. J. Sylvester, Tables of the generating functions and groundforms for the binary duodecimic, with some general remarks, and tables of the irreducible syzygies of certain quantics, Amer. J. Math. 4 (1881), 41–61.MathSciNetCrossRefGoogle Scholar
  97. [97]
    B. Trager, Integration of algebraic functions, PhD thesis, MIT, Boston, 1984.Google Scholar
  98. [98]
    S. M. Watt, H. Stetter (eds.), Special issue on Symbolic numeric algebra for polynomials, J. Symbolic Computation 26 (1998), 649–652.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    M. Wester (ed.), Computer algebra systems. A practical guide,Wiley, Chichester, 1999.zbMATHGoogle Scholar
  100. [100]
    H. Zassenhaus, On Hensel factorization, J Number Theory 1 (1969), 291–311.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Wolfram Decker
    • 1
  1. 1.Department of MathematicsUniverität des SaarlandesSaarbrückenGermany

Personalised recommendations