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Mat. Sb. (N.S.), 1974, Volume 94(136), Number 3(7), Pages 385–406 (Mi msb3688)  

Saddle points of parabolic polynomials

S. G. Gindikin, M. V. Fedoryuk


Abstract: Let $G(t,x)$ be the Green's function of a parabolic differential operator $\frac\partial{\partial t}+P(\frac1i\frac\partial{\partial x})$. 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.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1974, 23:3, 362–381

Bibliographic databases:

UDC: 517.43
MSC: Primary 35B40, 35K30; Secondary 26A51
Received: 26.04.1973

Citation: S. G. Gindikin, M. V. Fedoryuk, “Saddle points of parabolic polynomials”, Mat. Sb. (N.S.), 94(136):3(7) (1974), 385–406; Math. USSR-Sb., 23:3 (1974), 362–381

Citation in format AMSBIB
\Bibitem{GinFed74}
\by S.~G.~Gindikin, M.~V.~Fedoryuk
\paper Saddle points of parabolic polynomials
\jour Mat. Sb. (N.S.)
\yr 1974
\vol 94(136)
\issue 3(7)
\pages 385--406
\mathnet{http://mi.mathnet.ru/msb3688}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=393851}
\zmath{https://zbmath.org/?q=an:0306.35055}
\transl
\jour Math. USSR-Sb.
\yr 1974
\vol 23
\issue 3
\pages 362--381
\crossref{https://doi.org/10.1070/SM1974v023n03ABEH001722}


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