Adriana Ortiz-Rodríguez

National Autonomous University of Mexico | UNAM · Institute of Mathematics

The global qualitative behaviour of fields of principal directions for the graph of a real-valued polynomial function f on the plane is studied. We determine and analyse the projective extension of these fields and show that they are defined by an analytic quadratic form on the whole unit 2-sphere. We prove that every umbilic point at infinity of t...

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 Poincar\'e-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...

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 l...

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 ide...

Consider a polynomial f∈ℝ[x,y] whose Hessian curve is compact and the unbounded connected component of its complement is hyperbolic. We study the fields of asymptotic directions on this component. Thus, we determine an index formula for the field of asymptotic directions involving the number of connected components of the Hessian curve constituting...

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...

Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian...

We study some realization problems related to the Hessian polynomials. In particular, we solve the Hessian curve realization problem for degrees zero, one, two, and three and the Hessian polynomial realization problem for degrees zero, one, and two.

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...

Let f be a real polynomial of degree n⩾3 in two variables. It is known that its hessian is a real polynomial in two variables of degree at most 2n−4. In 1876, A. Harnack prove that the number of connected components of an algebraic plane curve of degree m embedded in RP2 is at most (m−1)(m−2)/2+1. So, by A. Harnack, the number of compact connected...

In this Note we give a class of real polynomials of degree n⩾3 in two variables such that the parabolic curves of theirs graphs have at least (n−1)(n−2)/2 connected components diffeomorphic to the circle and exactly n(n−2) special parabolic points. These polynomials are of the form f=l1⋯ln where li, i=1,…,n, are generic real affine functions on the...