Saddle points of parabolic polynomials

Let $G(t,x)$ be the Green's function of a parabolic differential operator $\frac\partial{\partial t}+P\bigl(\frac1i\frac\partial{\partial x}\bigr)$. In a previous article of the authors (Mat. Sb. (N.S.) 91(133) (1973), 520–522) estimates for $G$ are obtained by means of a convex function $\nu_P$ invariantly defined by $P$, and the saddle points are distinguished under the assumption that $\nu_P$ is smooth. In the present paper the question of the existence of a finite number of saddle points is studied without assuming the smoothness of $\nu_P$; an example of a polynomial $P$ is constructed for which the function $\nu_P$ is not smooth. It is shown that for almost all polynomials $P$ the function $\nu_P$ is strictly convex almost everywhere.
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