The Quasi-homogeneous Decomposition And Monodromy Problem Of Polynomial Systems

In the study of the qualitative theory of the plane differential system, one of the importantsubjects is to determine the type of an isolated singular point of the system, and to give somediscriminating criterions, where a classic problem is whether the isolated singular point is ofmonodromy type(i.e., when the isolated singular point is of focus-centre type). Based on thefact that if the analytic system has an orbit tending to its singular point then it can only be eithera spiral orbit or an orbit which goes to singular point with a fixed direction, it follows that theisolated singular point of the analytic system is of monodromy type if and only if there is notany orbit of the system tending to (or leaving out of) the isolated singular point with a fixeddirection. When the isolated singular point is not strong degenerate(i.e., the linearized matrix ofthe system at the isolated singular point is not identically zero), the monodromy problem hasbeen basically solved. But if the isolated singular point is strong degenerate, even if the systemis analytic, then it is still an open problem to solve the monodromy problem.In most classic monographs of qualitative theory of the analytical system, the behaviors oforbits in a neighborhood of an isolated strong singular point are usually studied bydecomposing the system to the sum of homogeneous polynomial systems and then makingtypical domains according to its characteristic(or exceptional) directions. But in fact, thismethod is tedious and it is sometime impossible to solve the monodromy problem because ofneeding infinite computations. Recent years, many authors started to study the behaviors oforbits in a neighborhood of an isolated strong singular point by decomposing the system to thesum of quasi-homogeneous polynomial systems according to the bounded edges of its Newtonfigure.One work in this dissertation is to give a decomposition of an analytic system to the sumof quasi-homogeneous polynomial systems subject to a fixed weight vector(i.e., a certainbounded edge of its Newton figure), which is visual and computing easy, by the fact that thesets of the quasi-homogeneous polynomials and the quasi-homogeneous polynomial systemsare both linear spaces under the usual additions and scalar multiplications, and then studyingtheir dimensions and bases. In the meantime, some examples are given to show that this methodis effective and computable.Another work in this dissertation is to generalize the characteristic equation, the characteristic or exceptional direction, the characteristic orbit, the typical domains, which areintroduced to study the classic problems of a strong degenerate singular point of an analyticsystem by the homogeneous decomposition in planar qualitative theory, to thequasi-characteristic equation, the quasi-characteristic or quasi-exceptional direction, thequasi-characteristic orbit, the quasi-typical domains. And then give their some properties.Especially, the estimation about the number of the quasi-characteristic directions is given. In themeantime, the author also studies the qualitative behaviors of the orbits in a neighborhood of anisolated strong singular point of an analytic system using the above concepts and theirproperties.Finally, a brief summary and some problems for future works are given.