Differential elimination and biological modelling. This paper describes applications of a computer algebra method, differential elimination, to applied mathematics problems mostly borrowed from biology. The two considered applications are related to parameter estimation and model reduction problems. In both cases, differential elimination can be viewed as a preparation to numerical treatments. Those numerical treatments are, at least partly, sketched in this paper in order to put some light on the real limitations of the applications. Together with the applications, the paper introduces two implementations of the differential elimination algorithms: the diffalg package, which is embedded in the MAPLE computer algebra software, and the BLAD libraries which are standalone open source C libraries. The diffalg package is designed to be manipulated interactively and can be used very quickly and easily by casual readers. The BLAD libraries are designed to provide differential elimination for scientific software independent of any computer algebra system. They are probably better suited than diffalg to the development of software dedicated to the described applications. Using the BLAD libraries implies however to write a C program. For this reason, in this paper, examples are illustrated with diffalg rather than with BLAD.

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  1. Rueda, Sonia L.: Differential elimination by differential specialization of Sylvester style matrices (2016)
  2. Boulier, François; Lemaire, François: Finding first integrals using normal forms modulo differential regular chains (2015)
  3. Wongvanich, N.; Hann, C.E.; Sirisena, H.R.: Robust global identifiability theory using potentials -- application to compartmental models (2015)
  4. Rueda, Sonia L.: Linear sparse differential resultant formulas (2013)
  5. Bächler, Thomas; Gerdt, Vladimir; Lange-Hegermann, Markus; Robertz, Daniel: Algorithmic Thomas decomposition of algebraic and differential systems (2012)
  6. Boulier, François; Lefranc, Marc; Lemaire, François; Morant, Pierre-Emmanuel: Model reduction of chemical reaction systems using elimination (2011)
  7. Meshkat, Nicolette; Anderson, Chris; DiStefano, Joseph J.III: Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input-output equations (2011)
  8. Yoshida, Hiroshi: A condition for regeneration of a cell chain inspired by the Dachsous-Fat system (2011)
  9. Boulier, François; Lemaire, François; Moreno Maza, Marc: Computing differential characteristic sets by change of ordering (2010)
  10. Rueda, Sonia L.; Sendra, J.Rafael: Linear complete differential resultants and the implicitization of linear DPPEs (2010)
  11. Meshkat, Nicolette; Eisenberg, Marisa; DiStefano, Joseph J. III: An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner bases (2009)
  12. Boulier, François; Lefranc, Marc; Lemaire, François; Morant, Pierre-Emmanuel: Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits (2008)
  13. Boulier, François; Lemaire, François: Differential algebra and system modeling in cellular biology (2008)
  14. Boulier, François: Differential elimination and biological modelling (2007)