Polynomial solutions of the Monge-Ampère equation

The question of the existence of polynomial solutions to the Monge-Ampère equation $z_{xx}z_{yy}-z_{xy}^2=f(x,y)$ is considered in the case when $f(x,y)$ is a polynomial. It is proved that if $f$ is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the $x$, $y$-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved.
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