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

TL;DR

This paper investigates how small Poisson brackets impose strict structural constraints on the homogeneous components of polynomials, revealing relationships and divisibility properties, and proposing a reformulation of Yu's conjecture.

Contribution

It establishes constraints on homogeneous components of polynomials with small Poisson brackets and proposes a new perspective on Yu's conjecture.

Findings

Constraints on homogeneous components when Poisson bracket degree is small

Relationships between degrees of polynomial components

Proposed reformulation of Yu's conjecture

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.