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Differential Galois theory and Lie-Vessiot systems

  • Autores: David Blázquez Sanz
  • Directores de la Tesis: Juan José Morales Ruiz (dir. tes.)
  • Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2008
  • Idioma: inglés
  • Tribunal Calificador de la Tesis: Jean Pierre Ramis (presid.), Amadeu Delshams i Valdés (secret.), Enmanuel Paul (voc.), Jesús Muñoz (voc.), Pere Pascual Gainza (voc.)
  • Materias:
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  • Resumen
    • The main subject of this thesis is the study of ordinary differential equations that admit superposition laws; we call them Lie-Vessiot systems, These equations are non-autonomous vector fields whose general solution is expressed as a function of a finite number of particular solutions and a finite number of arbitrary constants. They are natural generalizations of systems of linear ordinary differential equations, for whom the principle of linear superposition holds. We study the Lie-Vessiot systems both in the complex analytic and differential algebraic context.

      In the complex analytic context we analyze the underlying structure of Lie's superposition theorem (Lie-Scheffers theorem in some places). The original result of Lie is local; we find its global equivalent: the characterization of differential equations admitting superposition laws in terms of Lie group actions (Theorem 2.2). We use Vessiot's notion of automorphic system, which is a differential equation defined on a Lie group, and logarithmic derivative in relation to Lie-Vessiot systems. With these tools we are able to study the reduction of Lie-Vessiot systems by means of Lie's reduction method. We perform the first study of complex analytic global aspects of Lie's reduction. It leads us to a new purely geometric-analytic approach to differential Galois theory. We define the complex analytic Galois group of an automorphic system. It is a refinement of Picard-Vessiot group. The complex analytic Galois group is Zariski dense in the algebraic differential Galois group of Picard-Vessiot theory (Theorem 2.18). We also study the infinitesimal symmetries of automorphic and Lie-Vessiot systems. We ob- tain the relation of the Galois group with the Lie algebra of symmetries (Theorem 2.24). In the algebraic differential context, we develop an algebraic Galois theory for automorphic systems. In order to do this, we perform a systematic study of the relationship of differential schemes and schemes with derivation (Theorem 3.13). Then we study algebraic automorphic and Lie-Vessiot systems, and analyze their geometry (Theorem 4.9). We study the extensions of differential fields that allow us to solve such systems. We proof the existence and uniqueness -up to differential isomorphism- of a smallest splitting extension for the automorphic system, which we call the Galois extension (Definition 4.2.3). It is a generalization of the Picard-Vessiot extension, and it is an strongly normal extension in the general case. Finally, we study the algebraic reduction and integration of automorphic systems. We find an algebraic version of Lie's reduction method that we call Lie-Kolchin reduction (Theorem 5.5). We explore different applications. As a corollary we find an extension of Kolchin's reduction theorem (Theorem 5.9) to the theory of automorphic systems. We also find an easy proof of Drach-Kolchin theorem on the Weierstrass' P function (Theorem 5.10). We obtain some results on the linearization of automorphic systems, and an extension of Liouville's theorem (Theorem 5.14) that related the integrability of an automorphic system by Liouvillian functions to the existence of an algebraic solution of certain related Lie-Vessiot system. As final application we extend a classical result of Darboux (Theorem 5.15) on the equations of rigid motions.


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