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.