小泊松括号多项式齐次分支的相依性
Dependence of Homogeneous Components of Polynomials with Small Degree of Poisson Bracket
摘要：
设F，G在C[x_1，...，x_n]是n个变量x_1，...，x_n在复数域C上。本文证明了：如果泊松括号[F，G]的次数足够小，则这些多项式的齐次分量存在严格的约束。证明了deg（F）-1次多项式F的齐次分量与deg（F）-2次多项式F的齐次分量之间的关系，以及deg（F）-1次多项式F的齐次分量的整除性的一些结果.此外，我们建议，可能是一个适当的，重新表述的猜想余关于估计的泊松括号程度的两个多项式。
英文摘要：
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.
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