Values Of Singular Point And Integrability Conditions For Several Types Of Polynomial Differential Autonomous Systems On The Complex Field

Planar polynomial differential autonomous systems of the center-focus of research has attracted much attention in recent years.The classic successive function method and the formal series method involve a lot of calculations of integrability or solving equations.It is difficult to find an effective algorithm even if making use of the computer.In the second chapter, the complex plane polynomial differential system was described as First of all,by the algebraic equivalent the calculation of focal quantities was transformed into the singular point quantities of system (1) from the real domain to the corresponding complex domain and the algebraic recurrence formula of the singular point quantities was given.Its advantage was that just the coefficients of polynomial system (1) as symbols completed limited arithmetic.So,editing and simplification of the singular point quantities achieved simply on the computer.Second,we discussed the rotation invariants (or Lie-invariants) and the singular point quantities in the structure theorem.Finally,we introduced the use of invariant algebraic curves to quest integrating factors of system (1).In the third chapter a class of cubic polynomial differential system by was studied in the complex domain.By the algebraic recursion formula for computing singular point quantities,the first five singular point values of system (2) at origin were derived. When the first five singular values of the system (2) were zero,by simplification of the singular point quantities,five sets of conditions were obtained.Then,when these conditions were true, integrating factors of system (2) were found by the invariant algebraic curves of system (2).Finally,the sufficiency and necessity of these conditions were proved when system (2) were integrable.In the fourth chapter we studied the two types of quartic polynomial differential autonomous systems on the complex field.The necessary and sufficient conditions were found when the singular point quantities of the two types of quartic systems were equal to zero.Finally,the integrability of these two types of systems was solved.