Miguel Angel Guadarrama-García

arXiv2025

On the non-existence of certain real algebraic surfaces

In this note is given an algebraic solution to the problem 1997-6 proposed by D. A. Panov in the list of Arnold's problems \cite{Arnld2b}. In particular, it is shown that there does not exist a real polynomial function $f$ on the real euclidean plane, whose Hessian is positive in an open set bordered by smooth connected curve, and the parabolic curve of the graph of $f$ has only one special parabolic point with index $+1$. Besides, we find conditions on $f$ so that its graph has more special parabolic points with index -1 than with index +1.

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↗