On the Algebraic Geometry of Polynomial Dynamical Systems
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.
Key wordsPolynomial dynamical system inference of biochemical networks control theory computational algebra
Unable to display preview. Download preview PDF.
- 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.. 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, 2007Google 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, ReverseengineeringGoogle Scholar
- 15.of polynomial dynamical systems, Advances in Applied MathematicsGoogle 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