On the geometric structure of certain real algebraic surfaces

TL;DR

This paper investigates the affine and projective geometric structures of real algebraic surfaces defined by polynomials, providing criteria for parabolic curve properties, analyzing asymptotic lines, and deriving bounds on special parabolic points.

Contribution

It introduces new criteria for parabolic curve compactness, extends asymptotic line analysis to the projective plane, and formulates an index relation involving the Hessian curve and parabolic points.

Findings

Criteria for parabolic curve compactness

Index formula for asymptotic lines

Upper bounds for special parabolic points

Abstract

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 Poincar\'e 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.