The Valuative Theory of Foliations

We describe the valuations following infinitely near singular points of a (singular) holomorphic foliation in the complex plane. They appear to be those satisfying a generalization of L'Hopital's rule. With them, we characterize dicritical vector fields, generic simple singularities, the existence of non- convergent solutions, etc. The construction is generalizable to dimension n.Comment: 16 pages, uses package: diagram.te

Power Series Solutions of Non-Linear q-Difference Equations and the Newton-Puiseux Polygon

Adapting the Newton-Puiseux Polygon process to nonlinear q-difference equations of any order and degree, we compute their power series solutions, study the properties of the set of exponents of the solutions and give a bound for their $q-$Gevrey order in terms of the order of the original equation

The local Poincar\'e problem for irreducible branches

Let ${\mathcal F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\mathbb C}^{2}$ that has a germ of irreducible holomorphic invariant curve $\gamma$. We provide a lower bound for the vanishing multiplicity of ${\mathcal F}$ at the origin in terms of the equisingularity class of $\gamma$. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of $\mathcal{F}$ can be bounded in terms of that of $\gamma$ and provide an explicit bound in this case.Comment: 18 pages, 5 figures, accepted for publication in Revista Matem\'atica Iberoamerican

Role of DGKα and DGKζ in the control of lipid metabolism in breast cancer: implications for therapeutic intervention

Tesis doctoral inédita, leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Biología Molecular. Fecha de lectura: 21-09-2012. Tesis doctoral con mención europe

Complexity of Puiseux solutions of differential and qq-difference equations of order and degree one

We relate the complexity of both differential and $q$-difference equations of order one and degree one and their solutions. Our point of view is to show that if the solutions are complicated, the initial equation is complicated too. In this spirit, we bound from below an invariant of the differential or $q$-difference equation, the height of its Newton polygon, in terms of the characteristic factors of a solution. The differential and the $q$-difference cases are treated in a unified way.Comment: 27 pages, 2 figure