What question did this study set out to answer?

To establish conditions under which certain real polynomial functions and their parabolic curves do not exist.

April 1, 2026Open Access

On the non-existence of certain real algebraic surfaces

Key Points

To establish conditions under which certain real polynomial functions and their parabolic curves do not exist.
Algebraic analysis of polynomial functions in the real Euclidean plane.
Investigation of Hessian positivity in open sets bordered by smooth curves.
Characterization of special parabolic points based on their indices.
No real polynomial function exists with a Hessian that is positive and only one parabolic point with index +1.
Conditions for graphs to have more special parabolic points with index -1 than with index +1 were identified.

Abstract

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 (in Arnold’s Problems. Springer, Berlin, 2024). 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 + 1. Besides, we find conditions on f so that its graph has more special parabolic points with index -1 - 1 than with index +1.

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On the non-existence of certain real algebraic surfaces

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Abstract

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