Rocio Blanco - Academia.edu (original) (raw)

Papers by Rocio Blanco

arXiv

We study monomial ideals, always given by a unique monomial, like a reasonable first step to esti... more We study monomial ideals, always given by a unique monomial, like a reasonable first step to estimate in general the number of blow ups of Villamayor's algorithm of resolution of singularities. To resolve a monomial ideal < X a 1 1 •. .. • X an n > is interesting due to its equivalence with the particular toric problem < Z c − X a 1 1 •. .. • X an n >. In the special case, when all the exponents ai are greater than or equal to the critical value c, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case will be called "minimal codimension case". Partial sums of Catalan numbers (starting 1, 2, 5,. . .) are 1, 3, 8, 22,. .. , and count, besides this new application, the number of paths starting from the root in all ordered trees with n + 1 edges. Catalan numbers appear in many combinatorial problems, counting the number of ways to insert n pairs of parenthesis in a word of n + 1 letters, plane trees with n + 1 vertices,. . ., etc. In the case of higher codimension, still unresolved, we give an example to state the foremost troubles. Computation of examples has been helpful in both cases to study the behaviour of the invariant function. Computations have been made in Singular (see [7]) using the desing package by G. Bodnár and J. Schicho, see [4].

manuscripta mathematica

We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated ... more We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials.

Eprint Arxiv 1109 1655, Sep 1, 2011

Over the last decade, implementations of several desingularization algorithms have appeared in va... more Over the last decade, implementations of several desingularization algorithms have appeared in various contexts. These differ as widely in their methods and in their practical efficiency as they differ in the situations in which they may be applied. The purpose of this article is to give a brief overview over a selection of these approaches and the applicability of the respective implementations in Singular.

Quarterly Journal of Mathematics, 2012

In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ... more In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial equations without any restriction, including monomials and p-th powers, where p is the characteristic of the base field.

Ijm, 2009

Computation of examples has been helpful in both cases to study the behaviour of the resolution i... more Computation of examples has been helpful in both cases to study the behaviour of the resolution invariant. Computations have been made in Singular (see \cite{sing}) using the \emph{desing} package by G. Bodnár and J. Schicho, see \cite{lib}.

Harmony of Gröbner Bases and the Modern Industrial Society - The Second CREST-CSBM International Conference, 2012

Lecture Notes in Computer Science, 2010

Page 1. A New Desingularization Algorithm for Binomial Varieties in Arbitrary Characteristic Rocı... more Page 1. A New Desingularization Algorithm for Binomial Varieties in Arbitrary Characteristic Rocıo Blanco ⋆ Universidad de Castilla-La Mancha. Departamento de Matemáticas, EU de Magisterio, Edificio Fray Luis de León, Avda ...

Asian Journal of Mathematics, 2011

Given a variety X over a field k one wants to find a desingularization, which is a proper and bir... more Given a variety X over a field k one wants to find a desingularization, which is a proper and birational morphism X ′ → X, where X ′ is a regular variety and the morphism is an isomorphism over the regular points of X. If X is embedded in a regular variety W , there is a notion of embedded desingularization and related to this is the notion of log-resolution of ideals in O W . When the field k has characteristic zero it is well known that the problem of resolution is solved. The first proof of the existence of resolution of singularities is due to H. Hironaka in his monumental work [Hir64] (see also ). If characteristic of k is positive the problem of resolution in arbitrary dimension is still open. See [Hau10] for recent advances and obstructions (see also [Hau03]). The proof by Hironaka is existential. There are constructive proofs, always in characteristic zero case, see for instance [VU89], [VU92], [BM97], we refer to [Hau03] for a complete list of references. Those constructive proofs give rise to algorithmic resolution of singularities, that allows to perform implementation at the computer [BS00], [FKP04].

The Quarterly Journal of Mathematics, 2012

Mathematische Nachrichten, 2012

This paper is devoted to give all the technical constructions and definitions that will lead to t... more This paper is devoted to give all the technical constructions and definitions that will lead to the construction of an algorithm of resolution of singularities for binomial ideals.

Journal of Symbolic Computation, 2011

We present a new method to achieve an embedded desingularization of a toric variety. Let W be a r... more We present a new method to achieve an embedded desingularization of a toric variety. Let W be a regular toric variety defined by a fan Σ and X ⊂ W be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms X ′ ⊂ W ′ are regular and X ′ has normal crossing with the exceptional divisor. *

International Journal of Mathematics, 2009

We study monomial ideals, always locally given by a monomial, like a reasonable first step to est... more We study monomial ideals, always locally given by a monomial, like a reasonable first step to estimate in general the number of monoidal transformations of Villamayor's algorithm of resolution of singularities. The resolution of a monomial ideal [Formula: see text] is interesting due to its identification with the particular toric problem [Formula: see text]. In the special case, when all the exponents ai are greater than or equal to the critical value c, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case will be called "minimal codimensional case". Partial sums of Catalan numbers (starting 1,2,5,…) are 1,3,8,22,…. These partial sums are well known in Combinatorics and count the number of paths starting from the root in all ordered trees with n + 1 edges. Catalan numbers appear in many combinatorial problems, counting the number of ways to insert n pairs of parenthesis in a word of n...