Advertisement

Algorithms for the Functional Decomposition of Laurent Polynomials

  • Stephen M. Watt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5625)

Abstract

Recent work has detailed the conditions under which univariate Laurent polynomials have functional decompositions. This paper presents algorithms to compute such univariate Laurent polynomial decompositions efficiently and gives their multivariate generalization.

One application of functional decomposition of Laurent polynomials is the functional decomposition of so-called “symbolic polynomials.” These are polynomial-like objects whose exponents are themselves integer-valued polynomials rather than integers. The algebraic independence of X, X n , \(X^{n^2/2}\), etc, and some elementary results on integer-valued polynomials allow problems with symbolic polynomials to be reduced to problems with multivariate Laurent polynomials. Hence we are interested in the functional decomposition of these objects.

Keywords

Weight Vector Composition Factor Laurent Polynomial Apply Algorithm Triangular System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ritt, J.: Prime and composite polynomials. Trans. American Math. Society 23(1), 51–66 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Engstrom, H.T.: Polynomial substitutions. American Journal of Mathematics 63(2), 249–255 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Levi, H.: Composite polynomials with coefficients in an arbitrary field of characteristic zero. American Journal of Mathematics 64(1), 389–400 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barton, D.R., Zippel, R.E.: A polynomial decomposition algorithm. In: Proc. 1976 ACM Symposium on Symbolic and Algebraic Computation, pp. 356–358. ACM Press, New York (1976)Google Scholar
  5. 5.
    Kozen, D., Landau, S.: Polynomial decomposition algorithms. J. Symbolic Computation 22, 445–456 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zippel, R.E.: Rational function decomposition. In: Proc. ISSAC 2001, pp. 1–6. ACM Press, New York (1991)Google Scholar
  7. 7.
    Kozen, D., Landau, S., Zippel, R.: Decomposition of algebraic functions. J. Symbolic Computation 22(3), 235–246 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    von zur Gathen, J., Gutierrez, J., Rubio, R.: Multivariate polynomial decomposition. Applied Algebra in Engineering, Communication and Computing 14, 11–31 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zieve, M.E.: Decompositions of Laurent polynomials (2007) Preprint: arXiv.org:0710.1902v1 Google Scholar
  10. 10.
    Watt, S.M.: Making computer algebra more symbolic. In: Proc. Transgressive Computing 2006: A conference in honor of Jean Della Dora, pp. 43–49 (2006)Google Scholar
  11. 11.
    Watt, S.M.: Two families of algorithms for symbolic polynomials. In: Kotsireas, I., Zima, E. (eds.) Computer Algebra 2006: Latest Advances in Symbolic Algorithms – Proceedings of the Waterloo Workshop, pp. 193–210. World Scientific, Singapore (2007)CrossRefGoogle Scholar
  12. 12.
    Watt, S.M.: Symbolic polynomials with sparse exponents. In: Proc. Milestones in Computer Algebra 2008: A conference in honour of Keith Geddes’ 60th birthday, Stonehaven Bay, Trinidad and Tobago, University of Western Ontario, pp. 91–97 (2007) ISBN 978-0-7714-2682-7Google Scholar
  13. 13.
    Weispfenning, V.: Gröbner bases for binomials with parametric exponents. Technical report, Universität Passau, Germany (2004)Google Scholar
  14. 14.
    Yokoyama, K.: On systems of algebraic equations with parametric exponents. In: Proc. ISSAC 2004, pp. 312–319. ACM Press, New York (2004)Google Scholar
  15. 15.
    Pan, W., Wang, D.: Uniform gröbner bases for ideals generated by polynomials with parametric exponents. In: Proc. ISSAC 2006, pp. 269–276. ACM Press, New York (2006)Google Scholar
  16. 16.
    Watt, S.: Functional decomposition of symbolic polynomials. In: Proc. International Conference on Computatioanl Sciences and its Applications (ICCSA 2008), pp. 353–362. IEEE Computer Society, Los Alamitos (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stephen M. Watt
    • 1
  1. 1.University of Western OntarioLondon, OntarioCanada

Personalised recommendations