arXiv subjects

C. Galindo

Publications and source records attributed to C. Galindo.

9 recordsLinked to original sources

arXiv2026

On $δ$-sequences and surfaces at infinity

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $δ$-sequence. This sequence provides geometrical information on $C$ such as the dual graph of the resolution of the singularity of $C$ at infinity. Since different $δ$-sequences can generate the same semigroup, it is an interesting problem to know the geometrical behaviour of curves $C$ sharing the same semigroup $S$. An analogous problem arises in a more general context when considering surfaces at infinity and their $δ$-semigroups. We show how to construct $δ$-sequences, and how to obtain different families that generate the same semigroup $S$, allowing us to study the geometrical content encoded by $S$.

math.AG↗

arXiv2018

On the computation of Darboux first integrals of a class of planar polynomial vector fields

We study the class of planar polynomial vector fields admitting Darboux first integrals of the type $\prod_{i=1}^r f_i^{α_i}$, where the $α_i$'s are positive real numbers and the $f_i$'s are polynomials defining curves with only one place at infinity. We show that these vector fields have an extended reduction procedure and give an algorithm which, from a part of the extended reduction of the vector field, computes a Darboux first integral for generic exponents.

math.DS↗

arXiv2016

Hopf braces and Yang-Baxter operators

This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid in our context. Furthermore, Hopf braces provide the right setting for considering left symmetric algebras as Lie-theoretical analogs of braces.

math.QA↗

arXiv2014

A class of polynomial planar vector fields with polynomial first integral

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm computes a minimal first integral. In addition, we solve the Poincaré problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field.

math.CA↗

arXiv2013

The log-canonical threshold of a plane curve

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in a paper by Kuwata in Amer. J. Math. 121.

math.AG↗

arXiv2008

On the characterization of algebraically integrable plane foliations

We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go.

math.DS↗

arXiv2008

$δ$-sequences and Evaluation Codes defined by Plane Valuations at Infinity

We introduce the concept of $δ$-sequence. A $δ$-sequence $Δ$ generates a well-ordered semigroup $S$ in $\mathbb{Z}^2$ or $\mathbb{R}$. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by $Δ$ from the polynomial ring in two indeterminates to a semigroup $S$ as above. We prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of $δ$-sequences of the different existing types, and so this paper provides the tools to know and use a new large set of codes.

cs.IT↗

arXiv2008

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of $\gr_V R$. The study of their dimensions allows us to relate the Poincaré series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of $V$. Moreover, the Poincaré series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$.

math.AG↗

arXiv2005

Algebraic Integrability of Foliations of the Plane

We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface obtained after blowing-up the set B_F of infinitely near points needed to get the dicritical exceptional divisors of a minimal resolution of the singularities of F. This condition can be detected in several ways, one of them from the proximity relations in B_F and, as a particular case, it holds when the cardinality of B_F is less than 9.

math.DS↗