Dependence between homogeneous components of polynomials with small degree of Poisson bracket

Let $F,G\in \mathbb {C}[x_1,\dots ,x_n]$ be polynomials in $n$ variables $x_1,\dots ,x_n$ over $\mathbb {C}$. 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 a modification of the conjecture of Yu regarding the estimation of the degree of the Poisson bracket of two polynomials.