fl-PHDTHESIS.bib

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@phdthesis{Lemaire02,
  author = {Fran{\c{c}}ois Lemaire},
  title = {{Contribution {\`a} l'algorithmique en alg\`ebre
                  diff{\'e}rentielle}},
  school = {Universit\'e Lille I},
  address = {59655, Villeneuve d'Ascq, France},
  month = {January},
  year = 2002,
  url = {http://tel.archives-ouvertes.fr/tel-00001363/fr/},
  abstract = { This thesis is dedicated to the study of nonlinear
                  partial differential equations systems.  The chosen
                  approach is differential algebra.  Given a system of
                  differential equations, we seek information about
                  its solutions. To do so, we first compute particular
                  systems (called differential regular chains) such
                  that the union of their solutions coincide with the
                  solutions of the initial system.  This thesis mainly
                  presents new results in symbolic computation.
                  Chapter 2 clarifies the link between regular chains
                  and differential regular chains.  Two new algorithms
                  (given in chapters 4 and 5) improve existing
                  algorithms computing these differential regular
                  chains.  These algorithms involve purely algebraic
                  techniques which help reduce expression swell and
                  help avoid unnecessary computations.  Previously
                  intractable problems have been solved using these
                  techniques.  An algorithm computing the normal form
                  of a differential polynomial modulo a differential
                  regular chain is described in chapter 2.  The last
                  results deal with analysis.  The solutions we
                  consider are formal power series. Chapter 3 gives
                  sufficient conditions for a solution to be analytic.
                  The same chapter presents a counter-example to a
                  conjecture dealing with the analyticity of formal
                  solutions.  }
}