  Let F,G in C[x_1,...,x_n] be two polynomials in n variables x_1,...,x_n over
the complex numbers field C. In this paper, we prove that if the degree of the
Poisson bracket [F,G] is small enough then there are strict constraints for
homogeneous components of these polynomials. We also prove that there is a
relationship between the homogeneous components of the polynomial F of degrees
deg(F)-1 and deg(F)-2 as well some results about divisibility of the
homogeneous component of degree deg(F)-1.
  Moreover we propose, possibly an appropriate, reformulation of the conjecture
of Yu regarding the estimation of the Poisson bracket degree of two
polynomials.
