  In this paper non-asymptotic exponential estimates are derived for the tail
distribution of polynomial martingale differences in terms unconditional tails
distributions of summands. Applications are considered in the theory of
polynomials on independent random variables, to the theory of U-statistics,
multiply martingale series and in the theory of weak compactness measures on
the Banach spaces. Partially supported by the Israel Ministry of Absorbtion
Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.
