arXiv subjects

Laura Ortiz-Bobadilla

Publications and source records attributed to Laura Ortiz-Bobadilla.

7 recordsLinked to original sources

arXiv2026

Neighborhoods of singular elliptic curves and the Riemann-Hilbert correspondence at infinity

In his thesis work, the first author studies neighborhoods of singular elliptic curves of type $I_0^*$ (Kodaira's classification) and provides the analytic classification. This is related via a double covering construction to the analytic invariants found by Touzet, Voronin and the second author for the analytic classification of neighborhoods of smooth elliptic curves. After recalling these constructions, we explain in this note how this construction is conjecturally related to the Riemann-Hilbert correspondence at infinity for Painlev{é} VI case when considering neighborhood of Okamoto divisor.

math.CV↗

arXiv2026

Curves of tangencies of foliation pairs and normalizing transformations

In this work we give a complete description of the collection of curves of tangencies induced by germs of foliation pairs -- non dicritical and dicritical -- given by analytic differential equations with degenerated non dicritical and dicritical singularities, satisfying some genericity assumptions. To this purpose we use local models and analytic normalizing transformations. Moreover, for each natural number $k$ we obtain $k$-normal forms for the normalizing transformations. These normal forms are used to give parametrizations, up to a finite jet, of the branches of the curves of tangencies. We also prove that under natural genericity assumptions any germ of analytic curve having pairwise transversal smooth branches is realized as curve of tangencies of a -- non dicritical and dicritical -- foliation pair.

math.DS↗

arXiv2025

Noetherianity and length of Melnikov functions

We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+εη=0$, with $γ(t)\subset H^{-1}(t)$ a family of cycles. The \emph{Poincaré first return map} is of the form $P(t)=t+\sum_j ε^j M_j^γ(t).$ The functions $M_j^γ$ are called \emph{Melnikov functions} and are given by \emph{iterated integrals of orbit length} at most $j$. We show that, for each $k\in\mathbb{N}$, there exists a \emph{universal Noetherianity index} $n_{\scriptscriptstyle H,γ}(k)$, independent of the deformation $η$, such that, if $M_j^γ\equiv0$, for $j=1,\ldots,n_{ H,γ}(k)$, then $M_j^γ$ is of orbit length $j-k$, for any Melnikov function $M_j^γ$. We call the smallest index with this property just the \emph{Noetherianity index} $ν_{\scriptscriptstyle H,γ}(k)$. In order to prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt-Raudenbush differential algebra theorem. We calculate the universal Noetherianity index $n_{H,γ}(k)$ in various nontrivial examples.

math.CA↗

arXiv2024

Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions

Let $F\in\mathbb{C}[x,y]$ be a polynomial, $γ(z)\in π_1(F^{-1}(z))$ a non-trivial cycle in a generic fiber of $F$ and let $ω$ be a polynomial $1$-form, thus defining a polynomial deformation $dF+εω=0$ of the integrable foliation given by $F$. We study different invariants: the orbit depth $k$, the nilpotence class}$n$, the derivative length}$d$ associated with the couple $(F,γ)$. These invariants bound the length $\ell$ of the first nonzero Melnikov function of the deformation $dF+εω$ along $γ$. We study in detail a simple example of a polynomial $F$ given as product of four lines. We show how these invariants vary depending on the relative position of the four lines and relate it also to the length of the corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by the study of this example.

math.CA↗

arXiv2019

Infinite Orbit depth and length of Melnikov functions

In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and their small perturbations: $dF+εω=0$. The first nonzero Melnikov function $M_μ=M_μ(F,γ,ω)$ of the Poincaré map along a loop $γ$ of $dF=0$ is given by an iterated integral. In a previous work (see arXiv 1703.03837), we bounded the length of the iterated integral $M_μ$ by a geometric number $k=k(F,γ)$ which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system $F$ and its orbit $γ$ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations $dF+εω$ with arbitrary high length first nonzero Melnikov function $M_μ$ along $γ$. We construct deformations $dF+εω=0$ whose first nonzero Melnikov function $M_μ$ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions $M_μ$.

math.DS↗

arXiv2019

Godbillon-Vey sequence and Francoise algorithm

We consider foliations given by deformations $dF+εω$ of exact forms $dF$ in $\mathbb{C}^2$ in a neighborhood of a family of cycles $γ(t)\subset F^{-1}(t)$. In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function $Δ$ along $γ$ of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation integrability of a form $ω$. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.

math.DS↗

arXiv2017

Bounding the length of iterated integrals of the first nonzero Melnikov function

We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb{C}[x,y]$ and the first nonzero term $M_μ$ of the displacement function $Δ(t,ε)=\sum_{i=μ}M_i(t)ε^i$ along a cycle $γ(t)\in F^{-1}(t)$. It is known that $M_μ$ is an iterated integral of length at most $μ$. The bound $μ$ depends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term $M_μ$ depending only on the topology of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_μ$ to be given by an abelian integral i.e. by an iterated integral of length $1$. We conjecture that our bound is optimal.

math.CA↗