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Evelia R. García Barroso

Publications and source records attributed to Evelia R. García Barroso.

At least 19 recordsLinked to original sources
arXiv2026

On Milnor and Tjurina numbers of Pairs of Holomorphic Functions Germs

Motivated by the bifurcation formula for the Milnor number of pairs of holomorphic function germs, we introduce, via foliation theory, the Tjurina number of such pairs. We prove that this is an analytic invariant, derive an explicit formula for it, and establish several of its properties, including a bifurcation formula. As an application, we characterize semitame meromorphic function germs in terms of this invariant. We also investigate several properties of the Milnor number of a pair of holomorphic function germs. In particular, we establish a version of Teissier's Lemma for pairs, derive an upper bound for the Milnor number of a pair, and apply these results to generic pencils of algebraic curves.

math.AG↗
arXiv2025

On some indices of foliations and applications

In this paper we establish a relationship between the Milnor number, the $χ$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.

math.CV↗
arXiv2025

Nilpotent Holomorphic Foliations on $(\mathbb{C}^n,\mathbf{0})$

In this paper, we study nilpotent holomorphic foliations in complex dimension $n+1$, at the origin, defined by germs of integrable 1-forms whose linear part is given by \(zdz\). These foliations generalize the classical nilpotent foliations in dimension two. We show that every nilpotent foliation in higher dimensions can be described as the pullback of Takens' normal form, which naturally leads to the existence of cuspidal hypersurfaces as invariant sets. We focus on the case where the separatrix is a quasi-ordinary cuspidal hypersurface, and we provide a characterization of those foliations that are of generalized hypersurface type. Furthermore, we recall the Newton polyhedron of a foliation and prove that, for foliations with a quasi-ordinary cuspidal separatrix, being of generalized hypersurface type is equivalent to the coincidence of the Newton polyhedra of the foliation and its separatrix. This provides a new criterion that extends previous results studied by Fernández-Sánchez and J. Mozo (2006), and later partially generalized by the second-named author.

math.DS↗
arXiv2025

Eggers decomposition of polar curves in positive characteristic

Let $f$ be a power series in two variables with coefficients in an algebraically closed field of positive characteristic. We identify a necessary and sufficient condition, read from the Eggers-Wall tree of $f$, for $\frac{\partial f}{\partial y}$ to admit Eggers decomposition.

math.AG↗
arXiv2025

An upper bound for the GSV-index of a foliation

Let $\mathcal{F}$ be a holomorphic foliation at $p\in\mathbb{C}^2$, and $B$ be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type inequality $3μ_p(\mathcal{F},B)-4τ_p(\mathcal{F},B)+GSV_p(\mathcal{F},B)\leq 0$, where $μ_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, $τ_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}\{x,y\}$ by the ideal generated by the components of any $1$-form defining $\mathcal{F}$ and any equation of $B$, and $GSV_p(\mathcal{F},B)$ is the \textit{Gómez-Mont-Seade-Verjovsky index} of the foliation $\mathcal{F}$ with respect to $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carramiñana, Almirón, Blanco, Melle-Hernández and Genzmer-Hernandes, but it is in line with the idea developed by Wang.

math.CV↗
arXiv2025

On Milnor and Tjurina numbers of foliations

We study the relationship between the Milnor and Tjurina numbers of a singular foliation $\mathcal{F}$, in the complex plane, with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$. For that, we associate with $\mathcal{F}$ a new number called the $χ$-number and we prove that it is a $C^{1}$ invariant for holomorphic foliations. We compute the polar excess number of $\mathcal{F}$ with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of $\mathcal{B}$ and the $χ$-number of $\mathcal{F}$. On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of $\mathcal{F}$. Finally, we state numerical relationships between some classic indices, as Baum-Bott, Camacho-Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.

math.CV↗
arXiv2025

Lotuses as computational architectures

Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve singularity by a sequence of toroidal modifications with respect to suitable local coordinates, one gets a naturally associated lotus, which allows to unify the classical trees used to encode the combinatorial type of the singularity. In this paper we explain how to associate a lotus to each constellation of crosses, which is a finite constellation of infinitely near points endowed with compatible germs of normal crossings divisors with two components, and how this lotus may be seen as a computational architecture. Namely, if the constellation of crosses is associated to an embedded resolution of a complex reduced plane curve singularity $A$, one may compute progressively as vertex and edge weights on the lotus the log-discrepancies of the exceptional divisors, the orders of vanishing on them of the starting coordinates, the multiplicities of the strict transforms of the branches of $A$, the orders of vanishing of a defining function of $A$, the associated Eggers-Wall tree, the delta invariant and the Milnor number of $A$, etc. We illustrate these computations using three recurrent examples. Finally, we describe the changes to be done when one works in positive characteristic.

math.AG↗
arXiv2023

On $p$-Frobenius of affine semigroups

The aim of this paper is to study the $p$-Frobenius vector of affine semigroups $S\subset \mathbb N^q$; that is, the maximum element, with respect to a graded monomial order, with at most $p$ factorizations in $S$. We produce several algorithms to compute these vectors. Finally, we study how the $p$-Frobenius vectors behave when considering gluings of $S$ with $\mathbb N^q$.

math.AC↗
arXiv2022

A note on the plane curve singularities in positive characteristic

Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $μ(f)$, the delta invariant $δ(f)$ and the number $r(f)$ of its irreducible components. Put $\bar μ(f)=2δ(f)-r(f)+1$. If $p=0$ then $\bar μ(f)=μ(f)$ (the Milnor formula). If $p>0$ then $μ(f)$ is not an invariant and $\bar μ(f)$ plays the role of $μ(f)$. Let $\mathcal N_f$ be the Newton polygon of $f$. We define the numbers $μ(\mathcal N_{f})$ and $r(\mathcal N_{f})$ which can be computed by explicit formulas. The aim of this note is to give a simple proof of the inequality $\bar μ(f)-μ(\mathcal N_{f})\geq r(\mathcal N_{f})- r(f)\geq 0$ due to Boubakri, Greuel and Markwig. We also prove that $\bar μ(f)=μ(\mathcal N_{f})$ when $f$ is non-degenerate.

math.AG↗
arXiv2022

On Briançon-Skoda theorem for foliations

We generalize Mattei's result relative to the Briançon-Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.

math.CV↗
arXiv2021

Factorizations of the same length in abelian monoids

Let $\mathcal S \subseteq \mathbb Z^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in $\mathcal S$ having at least two factorizations of the same length, namely the ideal $\mathcal L_{\mathcal S}$. To this end, we work with a certain (lattice) ideal associated to the monoid $\mathcal S$. Our study can be seen as a new approach generalizing \cite{chapman:2011}, which only studies the case of numerical semigroups. When $\mathcal S$ is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal $\mathcal L_{\mathcal S}$ when $\mathcal S$ is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that $\mathcal L_{\mathcal S}$ is a principal ideal; (3) we classify the computational problem of determining the largest integer not in $\mathcal L_{\mathcal S}$ as an $\mathcal{NP}$-hard problem.

math.AC↗
arXiv2020

On characterizations of nondicritical generalized curve foliations

We characterize nondicrital generalized curve foliations with fixed reduced separatrix. Moreover, we give suficient conditions when a plane analytic curve is its reduced separatrix. For that, we introduce a distinguished expression for a given 1-form, called {\it Weierstrass form}. Then, using Weierstrass forms, we characterize the nondicritical generalized curve foliations: first, for foliations with monomial separatrix using toric resolution; second, for foliations with reduced separatrix, using the $GSV$-index. In this last case the characterization, which is our main result, could be interpreted in function of a polar of the foliation and a polar of its reduced separatrix.

math.AG↗
arXiv2020

The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of $(S, C)$. One may define it by looking either at the Newton-Puiseux series associated to $C$ relative to a generic local coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of $(C,o)$ by successive toric modifications.

math.AG↗
arXiv2019

Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism $(\ell, f)$, where $\ell$ is a smooth curve and $f$ is an irreducible curve (branch) of multiplicity less than five or a branch that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

math.AG↗
arXiv2019

Contact exponent and the Milnor number of plane curve singularities

We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].

math.AG↗
arXiv2018

On the Milnor formula in arbitrary characteristic

The Milnor formula $μ=2δ-r+1$ relates the Milnor number $μ$, the double point number $δ$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality $μ\geq 2δ-r+1$ in arbitrary characteristic and showed that the equality $μ=2δ-r+1$ characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic $p$. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. (2010) 25) or if $p$ is greater than the intersection number of the singularity with its generic polar (Nguyen H.D., Annales de l'Institut Fourier, Tome 66 (5) (2016)). Then we improve our result on the Milnor number of irreducible singularities (Bull. London Math. Soc. 48 (2016)). Our considerations are based on the properties of polars of plane singularities in characteristic $p$.

math.AG↗