Skip to main content
SpringerLink
Log in

On the Algebraic Geometry of Polynomial Dynamical Systems

  • Chapter
  • First Online:
Book cover Emerging Applications of Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 149))

Abstract

This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks. It is shown that several problems relating to their structure and dynamics, as well as control theory, can be formulated and solved in the language of algebraic geometry.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. ALLEN, J. FETROW, L. DANIEL, S. THOMAS, AND D. JOHN, Algebraic dependency models of protein signal transduction networks from time series data, Journal of Theoretical Biology, 238:317~330, 2006.

    Google Scholar 

  2. G. CALL AND J. SILVERMAN, Canonical height on varieties with morphisrns, Compositio Math., 89:163~205, 1993. [31 O. COLON-REYES, A. JARRAH, R. LAUBENBACHER, AND B. STURMFELS, Monomial dynamical systems over finite fields, Complex Systems, 16(4):333-342, 2006.

    Google Scholar 

  3. O. COLON-REvES, R. LAUBENBACHER, AND B. PAREIGIS, Boolean Monomial Dynamical Systems, Annals of Combinatorics, 8:425-439, 2004.

    Google Scholar 

  4. J. DEEGAN AND E. PACKEL, A new index for simple n-person games, Int. J. Game Theory, 7:113-123, 1978.

    Google Scholar 

  5. E. DIMITORVA, P. VERA-LICOA, J. McGEE, AND R. LAUBNEBAHCER, Discretization of time series data. Submitted, 2007.

    Google Scholar 

  6. E. DIMITROVA, A. JARRAH, B. STIGLER, AND R. LAUBENABCHER, A Groebner Fan-based Method for Biochemical Network, ISSAC Proceedings, pp. 122126, ACM Press, 2007.

    Google Scholar 

  7. D. GRAYSON AND M. STILLMAN, Macaulay 2, a software system for research in algebraic geometry, World Wide Web, http://math.uiuc.edu/Macaulay2.

  8. G.-M. GREUEL, G. PFISTER, AND H. SCHONEMANN, Singular 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://WWW.singular.uni-kl.de.

  9. B. HASSELBLATI' AND J. PROPP, Degree growth of monomial maps, arXiv: Math. DS/0604521 v2, 2006.

    Google Scholar 

  10. [11]. J A. HERNANDEZ-ToLEDO, Linear Finite Dynamical Systems, Communications in Algebra, 33(9):2977-2989, 2005.

    Google Scholar 

  11. A. JARRAH AND R. LAUBENBACHER, Discrete Models of Biochemical Networks:

    Google Scholar 

  12. The Toric Variety of Nested Canalyzing Functions, Algebraic Biology, 2007

    Google Scholar 

  13. H. Anai and K. Horimoto and T. Kutsia, 4545, LNCS, pp. 15~22, Springer.

    Google Scholar 

  14. A. JARRAH, R. LAUBENBACHER, B. STIGLER, AND M. STILLMAN, Reverseengineering

    Google Scholar 

  15. of polynomial dynamical systems, Advances in Applied Mathematics

    Google Scholar 

  16. 39(4):477-489,2007.

    Google Scholar 

  17. A. JARRAH, R. LAUBENBACHER, MIKE STILLMAN, AND P. VERA-LICONA, An efficient algorithm for the phase space structure of linear dynamical systems over finite fields. Submitted, 2007.

    Google Scholar 

  18. A. JARRAH, B. RAPOSA, AND R. LAUBENBACHER, Nested canalyzing, unate cascade, and polynomial functions, Physica D, 233:167-174, 2007.

    Google Scholar 

  19. A. JARRAH, H. VASTANI, K. DUCA, AND R. LAUBENBACHER, An optimal control problem for in vitro virus competition, 43rd IEEE Conference on Decision and Control, 2004~ Invited paper, December.

    Google Scholar 

  20. S. KAUFFMAN, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22:437-467, 1969.

    Google Scholar 

  21. S. KAUFFMAN, C. PETERSON, B. SAMUELSSON, AND C. TROEIN, Random Boolean network models and the yeast transcriptional network, Proc. Natl. Acad. Sci. USA., 100:14796~9, 2003.

    Google Scholar 

  22. S. KAUFFMAN, C. PETERSON, B. SAMUELSSON, AND C. TROEIN, Genetic networks with canalyzing Boolean rules are always stable, PNAS, 101(49):17102-17107, 10.1073/pnas.0407783101, 2004.

  23. R. LAUBENBACHER AND B. STIGLER, A computational algebra approach to the reverse-engineering of gene regulatory networks, Journal of Theoretical Biology, 229:523-537, 2004.

    Google Scholar 

  24. ~--, Design of experiments and biochemical network inference, Algebraic and Geometric Methods in Statistics, Gibilisco P., Riccomagno E., 2007, Cambridge University Press, Cambridge.

    Google Scholar 

  25. L. LrDL AND G. MULLEN, When does a polynomial over a finite field permute the elements of the field? American Mathematical Monthly, 95(3):243-246, 1988.

    Google Scholar 

  26. ---, When does a polynomial over a finite field permute the elements of the field? II American Mathematical Monthly, 100(1):71-74, 1993.

    Google Scholar 

  27. R. LIDL AND H. NIEDERREITER, Finite Fields, Cambridge University Press, 1997, New York.

    Google Scholar 

  28. H. MARCHAND AND M. LEBORGNE, On the optimal control of polynomial dynamical systems over Z/pZ, Fourth Workshop on Discrete Event Systems, IEEE, 1998, Cagliari, Italy.

    Google Scholar 

  29. ---, Partial order control of discrete event systems modeled as polynomial dynamical systems, IEEE International conference on control applications, 1998, Trieste, Italy.

    Google Scholar 

  30. J. PETTIGREW, J.A.G. ROBERTS, AND F. VIVALDI, Complexity of regular invertible p-adic motions, Chaos, 11 :849-857, 2001.

    Google Scholar 

  31. [28J L. REGER AND K. SCHMIDT, Aspects on analysis and synthesis of linear discrete systems over the finite field GF(q), Proc. European Control Conference ECC2003, 2003, Cambridge University Press.

    Google Scholar 

  32. R. THOMAS AND R. D'ARI, Biological Feedback, eRePress, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Virginia Bioinformatics Institute, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Abdul S. Jarrah & Reinhard Laubenbacher

Authors
  1. Abdul S. Jarrah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhard Laubenbacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reinhard Laubenbacher .

Editor information

Editors and Affiliations

  1. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

  2. Department of Mathematics Harvard University, Cambridge, 02138, U.S.A.

    Seth Sullivant

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Jarrah, A.S., Laubenbacher, R. (2009). On the Algebraic Geometry of Polynomial Dynamical Systems. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09686-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation