  In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type
theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on
the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case
is known as the Birch theorem) and complex polynomials. We also consider a
stronger conjecture on the homogeneous polynomial fields in the canonical
bundle over real and complex Grassmannians. This conjecture is much stronger
and false in general, but it is proved in the cases of $d=2$ (for $k$'s of
certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and
any $k$). Corollaries for the John ellipsoid of projections or sections of a
convex body are deduced from the case $d=2$ of the polynomial field conjecture.
