Dpto. Álgebra, Análisis Matemático, Geometría y Topología

Trajectories of vector fields asymptotic to formal invariant curves

2026-04-13T19:01:06Z

We prove that a formal curve that is invariant by a C ∞ vector field ξ of Rm has a geometrical realization, as soon as the Taylor expansion of ξ is not identically zero along . This means that there is a trajectory γ ⊂ Rm of ξ which is asymptotic to . This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R3. Ann. Inst. Henri Poincaré 3(2) (1986), 111–183] nearly forty years ago. We also construct an invariant C0 manifold S in some open horn around which is composed entirely of trajectories asymptotic to and contains the germ of any such trajectory. If ξ is analytic, we prove that there exists a trajectory γ asymptotic to which is, moreover, non-oscillating with respect to subanalytic sets.

Radial Foliations in Dimension Three

2026-02-25T09:08:24Z

Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C3, 0), 𝐸, 𝓕) is almost radial when it has a reduction of singularities without singular points; it will be “radial” under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the “open book” situation. We end the paper with a discussion on the general almost radial case.

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

2026-02-25T12:27:53Z

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

The Tjurina Number for Sebastiani–Thom Type Isolated Hypersurface Singularities

2026-02-16T20:05:16Z

In this note, we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this, we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and Tjurina numbers in terms of the singularity index of any isolated hypersurface singularity, not necessarily a join of singularities.

The Tjurina number of a plane curve with two branches and high intersection multiplicity

2026-02-11T20:00:33Z

In this paper we provide a family of reduced plane curves with two branches that have a constant Tjurina number in their equisingularity class, along with a closed formula for it in terms of topological data.

On the image of a curve in a normal surface by a plane projection

2026-02-05T20:00:26Z

We consider a finite analytic morphism φ = (f, g) defined from a complex analytic normal surface (Z, z) to C2. We describe the topology of the image by φ of a reduced curve on (Z, z) by means of iterated pencils defined recursively for each branch of the curve from the initial one . This result generalizes the one obtained in a previous paper for the case in which (Z, z) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of φ, in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.

The miniversal deformation of certain complete intersection monomial curves

2026-02-03T20:00:28Z

The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup—these curves make up a notable family of complete intersection monomial curves. First, we dispense a general decomposition result of a basis B of the miniversal deformation of any complete intersection monomial curve. As a consequence, we explicitly calculate B in the particular case of a monomial curve defined from a free semigroup. This direct computation yields some estimates for the dimension of the moduli space of the family The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup—these curves make up a notable family of complete intersection monomial curves. First, we dispense a general decomposition result of a basis B of the miniversal deformation of any complete intersection monomial curve. As a consequence, we explicitly calculate B in the particular case of a monomial curve defined from a free semigroup. This direct computation yields some estimates for the dimension of the moduli space of the family C.

Limit spectral distribution for non-degenerate hypersurface singularities

2026-04-14T12:27:57Z

We establish Kyoji Saito’s continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito’s notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.

Supersymmetric gaps of a numerical semigroup with two generators

2026-04-14T11:45:22Z

In this paper we introduce the new concepts of supersymmetric and self-symmetric gaps of a numerical semigroup with two generators. Those concepts are based on certain symmetries of the gaps of the semigroup with respect to their Wilf number. We prove that the set of supersymmetric and self-symmetric gaps completely determines the semigroup and we compare this set with the fundamental gaps of the semigroup.

A formula for the conductor of a semimodule of a numerical semigroup with two generators

2026-04-14T11:38:13Z

We provide an expression for the conductor of a semimodule of a numerical semigroup with two generators in terms of the syzygy module and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of the semigroup, as well as a formula for the conductor of the semimodule in terms of its dual semimodule.

A note on a question of Dimca and Greuel

2026-04-14T11:29:43Z

In this note we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities.

The minimal Tjurina number of irreducible germs of plane curve singularities

2026-02-25T12:27:46Z

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution.

Identities in prime rings

2025-12-11T20:00:31Z

Dado un anillo, una identidad polinómica generalizada (GPI) es una identidad polinómica cuyos coeficientes pueden ser tomados del propio anillo. Los anillos primos forman una clase de anillos muy adecuada para tratar problemas relacionados con identidades, como por ejemplo las que surgen de la teoría de Herstein, el estudio de los objetos y estructuras no asociativos construidos a partir de anillos asociativos. En dicho estudio aparece a menudo un tipo especial de GPI, que tiene una única variable y depende solamente de las potencias de un único elemento del anillo. La herramienta estándar para simplificar este tipo de GPI, el lema de Martindale, es potente pero no sistemática. Aquí presento un nuevo método, basado en una traducción del problema al contexto de anillos de polinomios, que produce una simplificación sistemática y considera cuerpos de todas las características al mismo tiempo.

Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution

2025-12-11T20:00:30Z

In this paper we study ad-nilpotent elements of semiprime rings $R$ with involution $*$ whose indices of ad-nilpotence differ on $\Skew(R,*)$ and $R$. The existence of such an ad-nilpotent element $a$ implies the existence of a GPI of $R$, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients $Q_m^s(R)$ of $R$, $a$ remains ad-nilpotent of the original indices in $\Skew(Q_m^s(R),*)$ and $Q_m^s(R)$. There exists an idempotent $e\in Q_m^s(R)$ that orthogonally decomposes $a=ea+(1-e)a$ and either both $ea$ and $(1-e)a$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of $a$ in $\Skew(Q_m^s(R),*)$ is congruent with 0 modulo 4), or $ea$ and $(1-e)a$ have different indices of ad-nilpotence (in this case the index of ad-nilpotence of $a$ in $\Skew(Q_m^s(R),*)$ is congruent with 3 modulo 4). Furthermore we show that $Q_m^s(R)$ has a finite $\mathbb{Z}$-grading induced by a $*$-complete family of orthogonal idempotents and that $eQ_m^s(R)e$, which contains $ea$, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence $n$.

The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

2026-04-08T12:40:35Z

We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

Positive definite functions as uniformly ergodic multipliers of the Fourier algebra

2025-12-04T20:00:35Z

Propiedades ergódicas de funciones definidas positivas en el álgebra de Fourier