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.
Miguel Angel Guadarrama-García (Mon,) studied this question.