Skip to main content

Some Introductory Remarks on Computer Algebra

  • Conference paper
European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 202))

  • 1149 Accesses

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.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Amrhein, O. Gloor, R. E. Maeder, Visualizations for mathematics based on a computer algebra system, J. Symbolic Computation 23 (1997), 447–452.

    Article  Google Scholar 

  3. K. Appel, W. Haken, Every planar map is four colorable, AMS, Providence, 1989.

    Book  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  5. C. Babbage, Scribbling books, volume 2 (1836), Science Museum Library, London.

    Google Scholar 

  6. A. Beilinson, Coherent sheaves on P N and problems of linear algebra, Funkt. Anal. Appl. 12 (1978), 214–216.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. E. R. Berlekamp, Factoring polynomials over large finite fields, Math. Comp. 24 (1970), 713–735.

    Article  MathSciNet  Google Scholar 

  9. H. U. Besche, B. Eick, Construction of finite groups, J. Symbolic Computation 27 (1999), 387–404.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. U. Besche, B. EickThe groups of order at most 1000 except 512 and 768, J. Symbolic Computation 27 (1999), 405–413.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25.

    MathSciNet  MATH  Google Scholar 

  12. B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108.

    MathSciNet  MATH  Google Scholar 

  13. M. Bronstein, Integration of elementary functions, J. Symbolic Computation 9 (1990), 117–173.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Bronstein, Symbolic integration I — transcendental functions, Springer-Verlag, Berlin, 1997.

    Book  MATH  Google Scholar 

  15. M. Bronstein, Symbolic integration tutorial, ISSAC’98, downloadable from http://www-sop.inria.fr/cafe/Manuel/cafe/Manuel. Bronstein/bronstein-eng.html

  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. 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. B. Buchberger, F. Winkler (eds.), Gröbner bases and applications, Linz 1999, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  21. J. F. Canny, D. Manocha, Multipolynomial resultant algorithms, J. Symbolic Computation 15 (1997), 99–122.

    MathSciNet  Google Scholar 

  22. D. G. Cantor, H. Zassenhaus, A new algorithm for factoring polynomials over finite fields, Math. Comp. 36 (1981), 587–592.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Cayley, The collected mathematical papers, vols 1-12, Cambridge University Press, Cambridge, 1889.

    Google Scholar 

  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. H. Cohen, A course in computational algebraic number theory (3rd corrected printing), Springer-Verlag, New York, 1996.

    Google Scholar 

  26. H. Cohen, Advanced topics in computational number theory, Springer-Verlag, New York, 2000.

    Book  MATH  Google Scholar 

  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. 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. D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms, second edition, Springer-Verlag, New York, 1997.

    Google Scholar 

  30. J. H. Davenport, On the integration of algebraic functions,Springer-Verlag, New York, 1981.

    Book  MATH  Google Scholar 

  31. W. Decker, L. Ein, F.-O. Schreyer, Construction of surfaces in P 4, J. Algebraic Geometry 2 (1993), 185–237.

    MathSciNet  MATH  Google Scholar 

  32. W. Decker, T. de Jong, Gröbner bases and invariant theory, in [18], 61–89.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Dixmier, D. Lazard, Minimum number of fundamental invariants for the binary form of degree 7, J. Symbolic Computation 6 (1988), 113–115.

    Article  MathSciNet  MATH  Google Scholar 

  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. D. Eisenbud, S. Popescu, Gale duality and free resolutions of ideals of points, Invent. Math. 136 (1999), 419–449.

    Article  MathSciNet  MATH  Google Scholar 

  37. G. Ellingsrud, S.-A. Stromme, The number of twisted cubic curves on the general quintic threefold, Math. Scand. 76 (1995), 5–34.

    MathSciNet  MATH  Google Scholar 

  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. F. Franklin, On the calculation of the generating functions and tables of groundforms for binary quantics, Amer. J. of Math. 3 (1883), 128–153.

    Article  MathSciNet  Google Scholar 

  40. J. von zur Gathen, J. Gerhard, Modern computer algebra, Cambridge University Press, Cambridge, 1999.

    MATH  Google Scholar 

  41. K. Gatermann, Computer algebra methods for equivariant dynamical systems, Springer-Verlag, Berlin, 2000.

    Book  MATH  Google Scholar 

  42. K. Geddes, S. R. Czapor, G. Labahn, Algorithms for computer algebra, Kluwer Academic Publishers, Boston, 1992.

    Book  MATH  Google Scholar 

  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. L. Gonzalez-Vega, T. Recio, Industrial applications of computer algebra: climbing up a mountain, going down a hill, in these proceedings.

    Google Scholar 

  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.

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. H. Grace, A. Young, The algebra of invariants, Cambridge University Press, Cambridge, 1903.

    MATH  Google Scholar 

  48. G.-M. Greuel, Applications of computer algebra to algebraic geometry, singularity theory and symbolic-numerical solving, in these proceedings.

    Google Scholar 

  49. J. Hammond, On the solution of the differential equation of sources, Amer. J. of Math. 5 (1883), 218–227.

    Article  MathSciNet  Google Scholar 

  50. D. Hilbert, Ober die Theorie der algebraischen Formen,Math. Ann. 36 (1890), 473–534.

    Article  MathSciNet  MATH  Google Scholar 

  51. D. HilbertOber die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–373.

    Article  MathSciNet  MATH  Google Scholar 

  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. H. G. Kahrimanian, Analytical differentiation by a digital computer, Master’s thesis, Temple University, Philadelphia, 1953.

    Google Scholar 

  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. E. KaltofenPolynomial factorization, in B. Buchberger et al. (eds.), Computer algebra, 95–113, Springer-Verlag, Wien, 1982.

    Google Scholar 

  56. E. KaltofenPolynomial factorization 1982–1986, in I. Simon (ed.), Computers in mathematics, 285–309, Marcel Dekker, New York, 1990.

    Google Scholar 

  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. S. Katz, On the finiteness of rational curves on quintic threefolds, Composito Math. 60 (1986), 151–162.

    Google Scholar 

  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. L. Kronecker, Grundzuge einer arithmetischen Theorie algebraischer Grössen, J. Reine Angew. Math. 92 (1882), 1–122.

    MATH  Google Scholar 

  61. J. P. S. Kung, G.-C. Rota, The Invariant Theory of Binary Forms, Bull. Am. Math. Soc. 10 (1984), 27–85.

    Article  MathSciNet  MATH  Google Scholar 

  62. L. A. Lambe (ed.), Special issue, J. Symbolic Computation 23 (1997), 445–623.

    Article  Google Scholar 

  63. P. L. Larcombe, On Lovelace, Babbage and the origins of computer algebra,in [99].

    Google Scholar 

  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. A. K. Lenstra, H. W. Lenstra, Jr. (eds), The development of the number field sieve, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  66. A. K. Lenstra, H. W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.

    Article  MathSciNet  MATH  Google Scholar 

  67. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.

    Article  MathSciNet  MATH  Google Scholar 

  68. H. W. Lenstra, Jr., Flags and lattice basis reduction, in these proceedings.

    Google Scholar 

  69. A. Lorenzini, The minimal resolution conjecture, J. Algebra 156 (1993), 5–35.

    Article  MathSciNet  MATH  Google Scholar 

  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. W. F. Meyer, Invariantentheorie, in Encyklopädie der mathematischen Wissenschaften, Erster Band,Teubner, Leipzig, 1898–1904.

    Google Scholar 

  72. H. M. Möller, Gröbner bases and numerical analysis, in [18], 159–178.

    Google Scholar 

  73. J. F. Nolan, Analytical differentiation on a digital computer, Master’s thesis, MIT, Cambridge, 1953.

    Google Scholar 

  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. P. J. Olver, Classical invariant theory, Cambridge University Press, Cambridge, 1999.

    Book  MATH  Google Scholar 

  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. 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. C. Pomerance, The quadratic sieve factoring algorithm, in T. Beth et al. (eds.), Advances in cryptology, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  79. S. Popescu, On smooth surfaces of degree > 11 in P4, PhD thesis, Universität des Saarlandes, Saarbrücken, 1993.

    Google Scholar 

  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. 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. R. RischFurther results on elementary functions, Report RC-2402, IBM Corp., Yorktown Heights, 1969.

    Google Scholar 

  83. R. Risch, The problem of integration in finite terms, Trans. A. M. S. 139 (1969), 167–189.

    Google Scholar 

  84. R. Risch, The solution of the problem of integration in finite terms, Bull. A. M. S. 76 (1970), 605–608.

    Article  MathSciNet  MATH  Google Scholar 

  85. R. L. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm ACM 21 (1978), 120–126.

    Article  MathSciNet  MATH  Google Scholar 

  86. P. RoelseFactoring high-degree polynomials over F2 with Niederreiter’s algorithm on the IBM SP2, Math. Comp. 68 (1999), 869–880.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. F. T. von Schubert, De inventione divisorum, Nova Acta Academiae Scientiarum Imperalis Petropolitanae 11 (1793), 172–182.

    Google Scholar 

  90. I. E. Shparlinski, Finite fields: theory and computations,Kluwer Academic Publishers, Dordrecht, 1999.

    Book  MATH  Google Scholar 

  91. M. F. Singer, Formal solutions of differential equations, J. Symbolic Computation 10 (1990), 59–94.

    Article  MathSciNet  MATH  Google Scholar 

  92. L. SmithPolynomial Invariants of Finite Groups, A.K. Peters, Wellesley, 1995.

    MATH  Google Scholar 

  93. B. Sturmfels, Algorithms in invariant theory, Springer-Verlag, Wien, 1993.

    MATH  Google Scholar 

  94. D. D. Swade, Der mechanische Computer des Charles Babbage, Spektrum der Wissenschaft, April 1993.

    Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  97. B. Trager, Integration of algebraic functions, PhD thesis, MIT, Boston, 1984.

    Google Scholar 

  98. S. M. Watt, H. Stetter (eds.), Special issue on Symbolic numeric algebra for polynomials, J. Symbolic Computation 26 (1998), 649–652.

    Article  MathSciNet  MATH  Google Scholar 

  99. M. Wester (ed.), Computer algebra systems. A practical guide,Wiley, Chichester, 1999.

    MATH  Google Scholar 

  100. H. Zassenhaus, On Hensel factorization, J Number Theory 1 (1969), 291–311.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this paper

Cite this paper

Decker, W. (2001). Some Introductory Remarks on Computer Algebra. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8266-8_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9496-8

  • Online ISBN: 978-3-0348-8266-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics