Dependence of Homogeneous Components of Polynomials with Small Degree of Poisson Bracket

Abstract

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.