论文标题
分析差异系统的本地第一积分的规律性和收敛性
Regularity and convergence of local first integrals of analytic differential systems
论文作者
论文摘要
庞加莱证明，在分析自主差异系统的非共振奇异性附近，正式的第一积分不存在。在一个零特征值和其他非谐音的谐振情况下，关于本地第一积分的规律性和收敛性仍然存在一个开放的问题。在这里，我们提供了这个问题的答案。该系统始终在奇点附近具有局部$ c^\ infty $ FIRST积分。在通过具有相同线性部分在奇异性的分析差分系统形成的任何有限维空间中，所有系统都具有局部分析第一积分，或者仅在多极性子集中的系统中具有局部分析性第一积分。
Poincaré proved nonexistence of formal first integrals near a nonresonant singularity of analytic autonomous differential systems. In the resonant case with one zero eigenvalue and others nonresonant, there remains an open problem on regularity and convergence of local first integrals. Here we provide an answer to this problem. The system has always a local $C^\infty$ first integral near the singularity when it is nonisolated. In any finite dimensional space formed by analytic differential systems having the same linear part at the singularity, either all the systems have local analytic first integrals or only the systems in a pluripolar subset have local analytic first integrals.
