This note shows that when studying geometric pro perties, a polynomial system is defined as a line field on a projective space such that its singular set has co dimension at least 2. By this definition, the concept of the degree of a polynomial system does not coincide with the usual one. The usual degenerate polynomial system of degree n+1 should be regarded as a system of degree n . Note that the definition is independent coordinate system. And, by this definition, some geometric properties concerning polynomial vector fields turn out to be evident.
Li Yusheng等人曾给出一个独立数的下界公式:α(G)≥Nfa+1(d),其中fa(x)=∫01(1-t)1/adt/(a+(x-a).t)。为了得到r(H,Kn)的上界,可以考虑建立不含H作为子图的临界图G的独立数的下界。即通过对临界图G及其邻域导出子图Gv的平均次数的分析,得出G的阶(顶点数)N与n之间的不等式关系。再利用函数fa(x)的分析性质得出当n趋于无穷大时,N+1的最小可能渐近表达式,即为r(H,Kn)的渐近上界。主要介绍这种分析方法在解决K+-K,"K+C","K"等图形和完全图Ramsey数渐近上界问题中的应用。
