Skip to main content

Inversion Modulo Zero-Dimensional Regular Chains

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7442))

Abstract

We consider the questions of inversion modulo a regular chain in dimension zero and of matrix inversion modulo such a regular chain. We show that a well-known idea, Leverrier’s algorithm, yields new results for these questions.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abdeljaoued, J., Lombardi, H.: Méthodes matricielles: introduction à la complexité algébrique. Mathématiques & Applications, vol. 42. Springer (2004)

    Google Scholar 

  2. Bostan, A., Flajolet, P., Salvy, B., Schost, É.: Fast computation of special resultants. J. Symb. Comp. 41(1), 1–29 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. Journal of the ACM 25(4), 581–595 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory. Springer (1997)

    Google Scholar 

  5. Csanky, L.: Fast parallel matrix inversion algorithms. SIAM J. Comput. 5(4), 618–623 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dahan, X., Jin, X., Moreno Maza, M., Schost, É.: Change of ordering for regular chains in positive dimension. Theoretical Computer Science 392(1-3), 37–65 (2008)

    Google Scholar 

  7. Dahan, X., Moreno Maza, M., Schost, É., Xie, Y.: On the complexity of the D5 principle. Transgressive Computing, 149–168 (2006)

    Google Scholar 

  8. Dahan, X., Moreno Maza, M., Schost, É., Wu, W., Xie, Y.: Lifting techniques for triangular decompositions. In: ISSAC 2005, pp. 108–115. ACM Press (2005)

    Google Scholar 

  9. Della Dora, J., Discrescenzo, C., Duval, D.: About a New Method for Computing in Algebraic Number Fields. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 289–290. Springer, Heidelberg (1985)

    Google Scholar 

  10. Faddeev, D., Sominskii, I.: Collected problems in higher algebra. Freeman (1949)

    Google Scholar 

  11. von Zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press (1999)

    Google Scholar 

  12. Langemyr, L.: Algorithms for a multiple algebraic extension. In: Effective Methods in Algebraic Geometry. Progr. Math, vol. 94, pp. 235–248. Birkhäuser (1991)

    Google Scholar 

  13. Le Verrier, U.J.J.: Sur les variations séculaires des éléments elliptiques des sept planètes principales : Mercure, Venus, La Terre, Mars, Jupiter, Saturne et Uranus. J. Math. Pures Appli. 4, 220–254 (1840)

    Google Scholar 

  14. Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library. In: Kotsireas, I.S. (ed.) Maple Conference 2005, pp. 355–368 (2005)

    Google Scholar 

  15. Li, X., Moreno Maza, M., Schost, É.: Fast arithmetic for triangular sets: from theory to practice. Journal of Symbolic Computation 44(7), 891–907 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, X., Maza, M.M., Pan, W.: Computations modulo regular chains. In: ISSAC 2009, pp. 239–246. ACM Press (2009)

    Google Scholar 

  17. Li, X., Moreno Maza, M., Pan, W.: Gcd computations modulo regular chains. Technical report, Univ. Western Ontario, 30 pages (2009) (submitted)

    Google Scholar 

  18. Moreno Maza, M., Rioboo, R.: Polynomial GCD Computations over Towers of Algebraic Extensions. In: Giusti, M., Cohen, G., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 365–382. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  19. Preparata, F.P., Sarwate, D.V.: An improved parallel processor bound in fast matrix inversion. Information Processing Letters 7(2), 148–150 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Technical report, Univ. Tübingen, 73 pages (1982)

    Google Scholar 

  21. Shoup, V.: Fast construction of irreducible polynomials over finite fields. Journal of Symbolic Computation 17(5), 371–391 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shoup, V.: Efficient computation of minimal polynomials in algebraic extensions of finite fields. In: ISSAC 1999, pp. 53–58. ACM Press (1999)

    Google Scholar 

  23. Souriau, J.-M.: Une méthode pour la décomposition spectrale et l’inversion des matrices. Comptes rendus des Séances de l’Académie des Sciences 227, 1010–1011 (1948)

    MathSciNet  MATH  Google Scholar 

  24. Vassilevska Williams, V.: Breaking the Coppersmith-Winograd barrier (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Moreno Maza, M., Schost, É., Vrbik, P. (2012). Inversion Modulo Zero-Dimensional Regular Chains. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-32973-9_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32972-2

  • Online ISBN: 978-3-642-32973-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics