| The study of the integrability of differential equations is a topic that people always interested in. Early to 1841, Liouville proved the nonintegrability of Riccati equation by using the thought of Galois theory. Since the work of Liouville, there are prodigiouss developments in the study of the integrability of differential equations , some new theories, such as Lie group theory,single valued group theory and differential algebra theory, were applied in the study of integrability and non-integrability of differential equations. For many years, mathematicians and physical scientists developed many method to study the integrability of differential equations, such as Painleve test,Carleman emb-dding,Lie symmetries method,Prelle-Singer procedure and the method of Compatible vector fields and so on. But to prove a given system is integrable, we need to find out enough many first integrals of system. Even to the simple polynomial systems, find out the polynomial first integral.Consider a system of polynomial ordinary differential equationswhere i = 1,…, n, f = (f1,…, fn)∈k[X]n. About the existence problem of first integrals for the system, people have been studied intensively for a long time. The problem is known to be difficult even for n = 2.In 1996, Nowicki gave some necessary conditions for nonlinear polynomial ordinary differential equations system exists polynomial and rational first nitegrals. In this paper, we give the necessary condition of nonlinear polynomial ordinary differential equation exists polynomial first integral by using the theory and method of differential algebra.In chapter 2, we introduce some preliminary definition and results of differential algebra.Definition 1. Definedthen d : k[X]→k[X] is a linear mapping obviously andd is said a derivation of k[X].Definition 2. The ring of constants of derivation dis called the kernel of d in k[X]. The field of constants of derivation dis called the kernel of d in k(X).Section 2 introduces the proposition of derivations in polynomial rings. The main proposition is:Proposition 1. (1) If f1,…,fn∈k[X], then there exists a unique derivation d of k[X] such that d(x1) = f1,…, d(xn) = fn, where derivation d is defined by (2.1).(2) If d is a derivation of k[X], and f∈k[X], then d(f) =.Proposition 2. Let d be a derivation of k[X]. If f and g are nonzero relatively prime polynomials from k[X] , then d(f/g) = 0 if and only if d(f) = pf and d(g) = pg for some p∈k[X].Proposition 3. Let d be a derivation of k[X] and k (?) k'. Denote by d' the derivation of k'[X] such that d'(xi) = d(xi) for i = 1,…,n. Then k[X]d = k if and only if k'[X]d' = k'.Proposition 4. If f∈k[X]dlin, then all homogeneous components of f belong also to k[X]dlin.Proposition 5. Let f is a nonzero homogeneous polynomial from k[X] satisfying the equality dlin(f) = pf for some p∈k. Then there exist nonneg-ative integers k1,…, kn such thatSection 3 introduces several examples of derivations with trivial ring of constants.Champer 3 introduces the main results of this paper and the proof of the theorem.Consider a system of polynomial ordinary differential equationsx = f{x), x = (x1,…,xn)∈Cn, (3.1)where i =dx/(dt) ,f(x) = (f1(x),…,fn(x))∈k[X]n. Let f (0) = 0, thenwhere i = 1,…, n, aij k, fi(x) = o(|| x ||2).Let A = (aij)n×n, denote byλ1,…,λn the n eigenvalues of the matrix A.The main results are following:Theorem If the eigenvaluesλ1,…,λn of the matrix A are not satisfied with any nonresonant conditionthe system (3.1) does not admit any polynomial first integral.
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