Algebraic-Geometric and Diophantine Aspects of Liouvillian Integrability in the Theory of Polynomial Vector Fields

• Dana Schlomiuk ^[1] ; Pacôme Tabe ^[1] ; Nicolae Vulpe ^[2]
  1. [1] University of Montreal

     University of Montreal

     Canadá

     
  2. [2] State University of Moldova
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • The goal of this article is to determine within the theory of polynomial differential systems the role of the geometry of their invariant algebraic curves and in particular of their multiplicities as well as of the multiplicities of singularities of the systems located on them, in their Liouvillian integrability. By analyzing in depth the family of Liouvillian integrable systems QSL≥4 of quadratic differential systems with real coefficients and with invariant lines of total multiplicity at least 4 we determine this role. We also show that along with the geometry and in particular the multiplicities of the curves and of singularities, also Diophantine equations play an important role.

    Our theorems give specific reasons for why the class splits in three subclasses with distinct kinds of integrating factors, depending on how the systems can be perturbed into generic ones, i.e. systems with all their invariant algebraic curves as well as their singularities simple (of multiplicity one).Analyzing the limiting process for integrating factors of the perturbed systems as the perturbation parameter tends

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