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

Let F, G ??? C[x1, . . . , xn] be polynomials in n variables x1, . . . , xn over 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.

Automated summary

This paper investigates the properties of the Poisson bracket of polynomials in n variables and proves that there are strict constraints on the homogeneous components of the polynomials when the degree of the Poisson bracket is small enough. It also establishes a relationship between the homogeneous components of a polynomial F with degrees deg F ???1 and deg F ???2, and presents some results on the divisibility of the homogeneous component of degree deg F ??? 1. Furthermore, a modification of a conjecture regarding the estimation of the degree of the Poisson bracket of two polynomials is proposed.