Integration of multivariate rational functions given by straight-line programs

  • Guillermo Matera
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of Q(X1,...,X n ) as input, decides whether they allow a rational potential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.


Rational Function Partial Fraction Great Common Divisor Probabilistic Algorithm Parallel Time 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baur W., Strassen V.: The complexity of partial derivatives. Theoret. Comput. Sci. 22 (1982) 317–330.Google Scholar
  2. 2.
    Berkowitz S.J.: On computing the determinant in small parallel time using a small number of processors. Information Processing Letter 18 (1984) 147–150.Google Scholar
  3. 3.
    Borodin A., von zur Gathen J., Hopcroft J.: Fast parallel matrix and GCD computations. Information and Control 52 (1982) 241–256.Google Scholar
  4. 4.
    Bronstein M.: Formulas for series computations. Applied Algebra in Engineering, Communication and Computing, AAECC 2, Springer-Verlag (1992) 195–206.Google Scholar
  5. 5.
    Davenport J.H.: Intégration Formelle. IMAG Reserch Report No.375 (1983) 1–23.Google Scholar
  6. 6.
    Fitchas N., Giusti M. and Smietanski F.: Sur la complexité du théorème des zéros. Preprint Ecole Polytechnique Palaiseau (1992).Google Scholar
  7. 7.
    von zur Gathen J.: Parallel algorithms for algebraic problems. Proc. 13-th. Conf. MFCS, Springer LN Comput. Sci. 356 (1989) 269–300.Google Scholar
  8. 8.
    Heintz J.: On the computational complexity of polynomials and bilinear mappings. A survey. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 5th Intern. Conf. AAECC-5, Menorca 1987, L. Huguet and A. Poli, eds., Springer LN Comput. Sci. 356 (1989) 269–300.Google Scholar
  9. 9.
    Heintz J. and Schnorr C.P.: Testing polynomials which are easy to compute, in: 12-th Ann. ACM Symp. Theory of Computing (1980) 262–280.Google Scholar
  10. 10.
    Ibarra O.H. and Moran S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J.ACM 30, 1 (1983) 217–228.Google Scholar
  11. 11.
    Kaltofen E.: Greatest common divisors of polynomials given by Straight-line Programs. J.ACM 35 No. 1 (1988) 234–264.Google Scholar
  12. 12.
    Lang S.: Algebra. Adisson-Wesley Publ. Comp., Reading, Massachusetts (1969).Google Scholar
  13. 13.
    Stoss H.J.: On the representation of rational functions of bounded complexity. Theoret. Comput. Sci. 64 (1989) 1–13.Google Scholar
  14. 14.
    Strassen V.: Berechnung und Programm I. Acta Inform.1 (1972) 320–334.Google Scholar
  15. 15.
    Strassen V.: Vermeidung von Divisionen. J. reine u. angew. Math. vol. 264 (1973) 182–202.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Guillermo Matera
    • 1
  1. 1.Departamento de Matemáticas, Fac. de Ciencias ExactasUniv. de Buenos AiresBuenos AiresArgentina

Personalised recommendations