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On the non-existence of certain real algebraic surfaces

Miguel Angel Guadarrama-García

TL;DR

This work resolves Panov's problem by showing a nonexistence result for polynomials of degree $n\ge 3$ under a specified Hessian positivity and parabolic-curve constraint, using a Poincaré-Hopf type theorem on the extended asymptotic-direction field over $\mathbb{RP}^{2}$. It introduces a comprehensive framework linking Hessian curves, projective Hessian curves, and asymptotic direction fields to the Euler characteristics of auxiliary regions $B^{\pm}$ and to the counts of special parabolic points, enabling both nonexistence results and concrete comparison inequalities. The paper provides precise index formulas, such as $\sum_{\xi} \text{Ind}(\tilde{\mathbb{X}})=\chi(B^{\epsilon})+\frac{P_{-}-P_{+}}{2}$, that connect local singularity data to global topology, and uses these to derive an affirmative condition for the second Panov question under a topological bound involving $k$ and $\chi(B^{\pm})$. Overall, the results advance the understanding of real algebraic surfaces in affine and projective settings, clarifying when certain configurations of special parabolic points can or cannot occur and highlighting the role of Hessian-related invariants in constraining surface geometry.

Abstract

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.

On the non-existence of certain real algebraic surfaces

TL;DR

This work resolves Panov's problem by showing a nonexistence result for polynomials of degree under a specified Hessian positivity and parabolic-curve constraint, using a Poincaré-Hopf type theorem on the extended asymptotic-direction field over . It introduces a comprehensive framework linking Hessian curves, projective Hessian curves, and asymptotic direction fields to the Euler characteristics of auxiliary regions and to the counts of special parabolic points, enabling both nonexistence results and concrete comparison inequalities. The paper provides precise index formulas, such as , that connect local singularity data to global topology, and uses these to derive an affirmative condition for the second Panov question under a topological bound involving and . Overall, the results advance the understanding of real algebraic surfaces in affine and projective settings, clarifying when certain configurations of special parabolic points can or cannot occur and highlighting the role of Hessian-related invariants in constraining surface geometry.

Abstract

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 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 has only one special parabolic point with index . Besides, we find conditions on so that its graph has more special parabolic points with index -1 than with index +1.

Paper Structure

This paper contains 8 sections, 6 theorems, 20 equations, 2 figures.

Table of Contents

  1. The Panov Problem and its Solutions
  2. A classification of the points of a surface $S^{n}_{f}$
  3. On the number of special parabolic points
  4. The Hessian curve of $f$
  5. The projective Hessian curve of $f$
  6. Mutual position of the sets $\mathbb{H}_{f}$ and $\mathbb{E}_{f}$
  7. Projection into the Poincaré sphere
  8. Proof of the main Theorems

Key Result

Corollary 1

There are no polynomials $f\in \mathbb{R}[x,y]$ of degree $n\geq 3$, such that all the real factors of $f_{n}$ are simple, $S^{n}_{f}$ is a generic surface, the parabolic curve of $S^{n}_{f}$ is connected, compact, has one single special parabolic point with index $+1$, and the set of hyperbolic poi

Figures (2)

  • Figure 1: Special parabolic points of index -1 (at left) and +1 (at center and right).
  • Figure 2: Topological type of singular points at infinity.

Theorems & Definitions (6)

  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Theorem 3: GdrrmOrtz1
  • Corollary 2: GdrrmOrtz1
  • Theorem 4: GdrrmOrtz1