arXiv subjects

Adriana Ortiz-Rodríguez

Publications and source records attributed to Adriana Ortiz-Rodríguez.

6 recordsLinked to original sources

arXiv2026

On an Arnold's Conjecture Concerning the Space of Hyperbolic Homogeneous Polynomials

The set of homogeneous polynomials of degree $D$ is a topological space that contains the subset $Hyp(D)$ constituted only by hyperbolic polynomials. In 2002, V. I. Arnold conjectured in \cite{arn0} that the number of connected components of $Hyp (D)$ increases, as $D$ increases, at least as a linear function of $D$. In this paper we prove that this conjecture is true. We determine the exact number of connected components of $Hyp (D)$ and we provide a representative for each component. The proof is constructive; our approach uses homotopy invariance of the index of a curve and properties of homogeneous polynomials. We also describe some geometrical properties of the hyperbolic polynomials that we provide.

math.DG↗

arXiv2021

Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincaré-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of $f$. Moreover, we study the projective extension of these fields and prove, under generic conditions, that every umbilic point at infinity of these extensions is isolated, has index equal to 1/2 and its topological type is a Lemon.

math.DG↗

arXiv2017

On the geometric structure of certain real algebraic surfaces

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of both fields of asymptotic lines and the Poincaré index of its singular points when the surface is generic. Thus, we exhibit an index formula for the field of asymptotic lines involving the number of connected components of the projective Hessian curve of $f$ and the number of the special parabolic points. As an application of this investigation, we obtain upper bounds, respectively, for the number of special parabolic points having an interior tangency and when they have an exterior tangency.

math.DG↗

arXiv2016

On some combinatorial formulae coming from Hessian Topology

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a topological property of certain spaces whose elements are graphs of a class of hyperbolic polynomials. These identities are proven by different methods in combinatorics.

math.CO↗

arXiv2013

On the Arnold's conjecture on hyperbolic homogeneous polynomials

The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree $n$. The proof is constructive and provides models. Our approach uses index properties at isolated singularities of hyperbolic quadratic differential forms and combinatorial properties of recurrent functions.

math.DG↗

arXiv2006

Real Hessian Curves

We give some real polynomials in two variables of degrees 4, 5, and 6 whose hessian curves have more connected components than had been known previously. In particular, we give a quartic polynomial whose hessian curve has 4 compact connected components (ovals), a quintic whose hessian curve has 8 ovals, and a sextic whose hessian curve has 11 ovals.

math.AG↗