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Computational Complexity of Sparse Real Algebraic Function Interpolation

  • Dima Grigoriev
  • Marek Karpinski
  • Michael Singer
Part of the Progress in Mathematics book series (PM, volume 109)

Abstract

We analyze the computational complexity of the problem of interpolating real algebraic functions given by a black box for their evaluations, extending the results of [GKS 90b, GKS 91b] on interpolation of sparse rational functions.

Keywords

Irreducible Component Betti Number Algebraic Function Minimal Polynomial Zariski Topology 
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.

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References

  1. [BCR 87]
    Bochnak, J., Coste, M., and Roy, M. F., Géométrie algebrique réelle, Springer-Verlag, 1987.Google Scholar
  2. [BT 88]
    Ben-Or, M., Tiwari, P., A deterministic algorithm for sparse multivariate polynomial interpolation, Proc. STOC ACM, 1988, pp. 301–309.Google Scholar
  3. [BSS 88]
    Blum, L., Shub, M., and Smale, S., On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines, Proc. IEEE FOCS, 1988, pp. 387–397.Google Scholar
  4. [D 80]
    Dold, A., Lectures on algebraic topology, Springer-Verlag, 1980.Google Scholar
  5. [ES 52]
    Eilenberg, S., and Steenrod, N., Foundation of algebraic topology, Princeton, 1952.Google Scholar
  6. [GKO 91]
    Grigoriev, D., Karpinski, M., and Odlyzko, A., Nondivisibility of sparse polynomials is in NP under the Extended Riemann Hypothesis, Preprint Max-Planck-Institute für Mathematik N6, 1991; to appear in J. AAECC 3 (1992).Google Scholar
  7. [GKS 90a]
    Grigoriev. D., Karpinski, M., and Singer, M., Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput., 1990, v. 19, N6, pp.1059–1063.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [GKS 90b]
    Grigoriev. D., Karpinski, M., and Singer, M., Interpolation of sparse rational functions without knowing bounds on exponents, Proc. IEEE FOCS, 1990, pp. 840–847.Google Scholar
  9. [GKS 91a]
    Grigoriev. D., Karpinski, M., and Singer, M., The interpolation problem for k-sparse sums of eigenfunctions of operators, Adv. Appl. Math., 1991, v. 12, pp.76–81.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [GKS 91b]
    Grigoriev, D., Karpinski, M., and Singer, M., Computational complexity of sparse rational interpolation, submitted to SIAM J. Comput.Google Scholar
  11. [GV88]
    Grigoriev, D., Vorobjov, N. (Jr.), Solving systems of polynomial inequalities in subexponential time, J. Symb. Comput., 1988, v. 5, pp. 37–64.CrossRefGoogle Scholar
  12. [HRS 90]
    Heintz, J., Roy, M. F., Solerno, P., Sur la complexité du principe de Tarski-Seidenberg, Bull. Soc. Math. France, 1990, 118, pp. 101–126.zbMATHMathSciNetGoogle Scholar
  13. [K 73]
    Kolchin, E., Differential algebra and algebraic groups, Academic Press, 1973.Google Scholar
  14. [K 91]
    Khovanskii, A., Fewmonomials, Transl. Math. Monogr., AMS 88, 1991.Google Scholar
  15. [M 64]
    Milnor, J., On the Betti numbers of real varieties, Proc. AMS 15, 1964, pp. 275–280.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 1993

Authors and Affiliations

  • Dima Grigoriev
    • 1
    • 2
  • Marek Karpinski
    • 2
    • 3
  • Michael Singer
    • 4
  1. 1.Steklov Mathematical InstituteSt. PetersburgRussia
  2. 2.Dept. of Computer ScienceUniversity of BonnBonn 1Germany
  3. 3.International Computer Science InstituteBerkeleyUSA
  4. 4.Dept. of MathematicsNorth Carolina State UniversityRaleighUSA

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