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Pedro Fortuny Ayuso

Publications and source records attributed to Pedro Fortuny Ayuso.

17 recordsLinked to original sources
arXiv2025

Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of Kähler differentials for generic $Γ\in {\mathcal C}$, algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of $Γ$.

math.AG↗
arXiv2024

Constructive solution of Zariski's Moduli Problem for Plane Branches

In this paper we give an explicit solution to Zariski's moduli problem for plane branches. We compute (in an algorithmic way) the set of Kähler differentials of an irreducible germ of holomorphic plane curve. We show that there is a basis of this set whose main elements correspond to dicritical foliations. Indeed, we discuss several concepts of generation for the semimodule of values of Kähler differentials of the curve and provide basis of Kähler differentials, for every of these concepts, whose geometric properties are described. Moreover, we give an algorithmic construction of the bases.

math.AG↗
arXiv2023

Complexity of Puiseux solutions of differential and $q$-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.

math.CV↗
arXiv2023

The Poincaré problem for reducible curves

We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e. they are never contained in non-trivial analytic families of equisingular invariant curves. In this case we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.

math.CV↗
arXiv2020

The local Poincaré 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 $γ$. We provide a lower bound for the vanishing multiplicity of ${\mathcal F}$ at the origin in terms of the equisingularity class of $γ$. 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 $γ$ and provide an explicit bound in this case.

math.AG↗
arXiv2019

Vector flows and the analytic moduli of singular plane branches

We provide a geometric elementary proof of the fact that an analytic plane branch is analytically equivalent to one whose terms corresponding to contacts with holomorphic one-forms -- except for Zariski's $λ$-invariant -- are zero (so called "short parametrizations"). This is the main step missed by Zariski in his attempt to solve the moduli problem.

math.AG↗
arXiv2018

Analytic Moduli of Plane Branches and Holomorphic Flows

We study the behaviour (in the infinitesimal neighbourhood of the singularity) of a singular plane branch under the action of holomorphic flows. The techniques we develop provide a new elementary, geometric and dynamical solution to Zariski's moduli problem for singular branches in $({\mathbb C}^{2},0)$. Furthermore, we study whether elements of the same class of analytic conjugacy are conjugated by a holomorphic flow; in particular we show that there exists an analytic class that is not complete: meaning that there are two elements of the class that are not analytically conjugated by a local diffeomorphism embedded in a one-parameter flow.

math.AG↗
arXiv2014

A von Staudt-type formula for $\displaystyle{\sum_{z\in\mathbb{Z}_n[i]} z^k }$

In this paper we study the sum of powers in the Gaussian integers $\mathbf{G}_k(n):=\sum_{a,b \in [1,n]} (a+b i)^k$. We give an explicit formula for $\mathbf{G}_k(n) \pmod n $ in terms of the prime numbers $p \equiv 3 \pmod 4$ with $p \mid \mid n$ and $p-1 \mid k$, similar to the well known one due to von Staudt for $\sum_{i=1}^n i^k \pmod n$. We apply this formula to study the set of integers $n$ which divide $\mathbf{G}_n(n)$ and compute its asymptotic density with six exact digits: $0.971000\ldots$.

math.NT↗
arXiv2002

Ramifications, Reduction of Singularities, Separatrices

We study the behaviour of sequences of blowing-ups under ramifications, and use the results to give a simple proof of Camacho-Sad's Theorem on the existence of Separatrices for singularities of plane holomorphic foliations. The main result we prove is that for any finite sequence $π$ of blowing-ups, there is a ramification morphism $ρ$ such that the elimination of indeterminations $\tildeπ$ of $π^{-1}\circ ρ$ is a sequence of blowing-ups \emph{with centers at regular points of the exceptional divisors}; moreover, we show that if $π$ is the reduction of singularities of a foliation $\mathcal{F}$, then $ρ$ can be such that $\tildeπ^{\star}ρ^{\star}\mathcal{F}$ has only simple singularities.

math.DS↗
arXiv2001

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.

math.AG↗