Computing with D-algebraic power series

  • Joris van der HoevenEmail author
Original Paper


In this paper, we will present several algorithms for computing with D-algebraic power series. Such power series are specified by one or more algebraic differential equations and a sufficient number of initial conditions. The emphasis is not on the efficient computation of coefficients of such power series (various techniques are known for that), but rather on the ability to decide whether expressions involving D-algebraic power series are zero. We will both consider univariate and multivariate series and, besides the usual ring operations and differentiation, we will also consider composition, implicitly determined power series and monomial transformations.


D-algebraic power series Algorithm Zero test Implicit function 

Mathematics Subject Classification

68W30 34A09 34A12 


  1. 1.
    Bostan, A., Chyzak, F., Ollivier, F., Salvy, B., Schost, É., Sedoglavic, A.: Fast computation of power series solutions of systems of differential equations. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 1012–1021, New Orleans, Louisiana, U.S.A. (2007)Google Scholar
  2. 2.
    Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM 25, 581–595 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Della Dora, J., Dicrescenzo, C., Duval, D.: A new method for computing in algebraic number fields. In Goos G., Hartmanis, J., (eds.) Eurocal’85 (2), Volume 174 of Lecture Notes in Computer Science, pp. 321–326. Springer, Berlin (1985)Google Scholar
  4. 4.
    Denef, J., Lipshitz, L.: Power series solutions of algebraic differential equations. Math. Ann. 267, 213–238 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Denef, J., Lipshitz, L.: Decision problems for differential equations. J. Symb. Logic 54(3), 941–950 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fischer, M.J., Stockmeyer, L.J.: Fast on-line integer multiplication. In: Proceedings of the 5th ACM Symposium Theory of Computing, vol. 9, pp. 67–72 (1974)Google Scholar
  7. 7.
    Khovanskii, A.G.: Fewnomials. Translations of Mathematical Monographs, vol. 88. American Mathematical Society, Providence (1991)Google Scholar
  8. 8.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)zbMATHGoogle Scholar
  9. 9.
    Péladan-Germa, A.: Tests effectifs de nullité dans des extensions d’anneaux différentiels. Ph.D. thesis, Gage, École Polytechnique, Palaiseau, France (1997)Google Scholar
  10. 10.
    Risch, R.H.: Algebraic properties of elementary functions in analysis. Am. J. Math. 4(101), 743–759 (1975)zbMATHGoogle Scholar
  11. 11.
    Ritt, J.F.: Differential Algebra. American Mathematical Society, New York (1950)CrossRefzbMATHGoogle Scholar
  12. 12.
    Shackell, J.: A differential-equations approach to functional equivalence. In: Proceedings of the ISSAC ’89, pp. 7–10, Portland, Oregon, ACM, New York. ACM Press (1989)Google Scholar
  13. 13.
    Shackell, J.: Zero equivalence in function fields defined by differential equations. In: Proceedings of the American Mathematical Society, vol. 336, no. 1, pp. 151–172 (1993)Google Scholar
  14. 14.
    van der Hoeven, J.: A new zero-test for formal power series. In: Mora T., (ed.) Proceedings of the ISSAC ’02, pp. 117–122. Lille, France (2002)Google Scholar
  15. 15.
    van der Hoeven, J.: Relax, but don’t be too lazy. JSC 34, 479–542 (2002)MathSciNetzbMATHGoogle Scholar
  16. 16.
    van der Hoeven, J.: Newton’s method and FFT trading. JSC 45(8), 857–878 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    van der Hoeven, J.: Faster relaxed multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC’14), Kobe, Japan, pp. 405–412 (2014)Google Scholar
  18. 18.
    van der Hoeven, J.: Effective power series computations. Technical report, HAL (2014)
  19. 19.
    van der Hoeven, J., Shackell, J.R.: Complexity bounds for zero-test algorithms. JSC 41, 1004–1020 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRSLIX, École polytechniquePalaiseau CedexFrance

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