Lucia Lopez - Academia.edu (original) (raw)

Papers by Lucia Lopez

Discrete & Computational Geometry, 2014

In this note we study the distribution of real inflection points among the ovals of a real non-si... more In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers d and r such that 4 ≤ r ≤ 2d 2 − 2d, there is a non-singular hyperbolic curve of degree 2d in R 2 with exactly r line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree 6. Keywords Maximally inflected hyperbolic real curves and their convex hull • Patchworking of real algebraic curves • Tropical curves

Journal of Science Communication América Latina, 2020

El objetivo principal es investigar el impacto que tienen un conjunto de actividades de comunicac... more El objetivo principal es investigar el impacto que tienen un conjunto de actividades de comunicación de las matemáticas realizadas en espacios públicos de colonias con marginación alta, en el fortalecimiento del tejido social y mejoramiento de la percepción de seguridad. Una plaza de la colonia Chamilpa en Cuernavaca, México es tomada como caso de estudio. Esta colonia se encuentra entre las de mayor marginación de la ciudad. Los resultados proporcionan evidencia empírica para soportar que la comunicación de las matemáticas en espacios públicos aumenta la percepción de seguridad y mejora la cohesión social.

Proceedings of the London Mathematical Society, 2019

We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weig... more We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.

L’Enseignement Mathématique, 2019

A la mémoire de notre ami Jean-Jacques Risler,à qui nous n'avons pas eu le temps de raconter ces ... more A la mémoire de notre ami Jean-Jacques Risler,à qui nous n'avons pas eu le temps de raconter ces incongruités.

Contemporary Mathematics, 2013

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curva... more This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: (1) The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. (2) A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. (3) The fact that this inequality is sharp in the non-singular case.

Journal of Singularities, 2012

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real ... more We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic M-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.

International Journal of Mathematics, 2010

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series ari... more We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

Discrete & Computational Geometry, 2014

In this note we study the distribution of real inflection points among the ovals of a real non-si... more In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers d and r such that 4 ≤ r ≤ 2d 2 − 2d, there is a non-singular hyperbolic curve of degree 2d in R 2 with exactly r line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree 6. Keywords Maximally inflected hyperbolic real curves and their convex hull • Patchworking of real algebraic curves • Tropical curves

Journal of Science Communication América Latina, 2020

El objetivo principal es investigar el impacto que tienen un conjunto de actividades de comunicac... more El objetivo principal es investigar el impacto que tienen un conjunto de actividades de comunicación de las matemáticas realizadas en espacios públicos de colonias con marginación alta, en el fortalecimiento del tejido social y mejoramiento de la percepción de seguridad. Una plaza de la colonia Chamilpa en Cuernavaca, México es tomada como caso de estudio. Esta colonia se encuentra entre las de mayor marginación de la ciudad. Los resultados proporcionan evidencia empírica para soportar que la comunicación de las matemáticas en espacios públicos aumenta la percepción de seguridad y mejora la cohesión social.

Proceedings of the London Mathematical Society, 2019

We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weig... more We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.

L’Enseignement Mathématique, 2019

A la mémoire de notre ami Jean-Jacques Risler,à qui nous n'avons pas eu le temps de raconter ces ... more A la mémoire de notre ami Jean-Jacques Risler,à qui nous n'avons pas eu le temps de raconter ces incongruités.

Contemporary Mathematics, 2013

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curva... more This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: (1) The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. (2) A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. (3) The fact that this inequality is sharp in the non-singular case.

Journal of Singularities, 2012

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real ... more We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic M-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.

International Journal of Mathematics, 2010

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series ari... more We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.