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This is a preprint of: “Generalized rational first integrals of analytic differential systems”, Wang
Cong, Jaume Llibre, Xiang Zhang, J
...
251, 2770–2788, 2011
...
1016/j
...
2011
...
016]

GENERALIZED RATIONAL FIRST INTEGRALS
OF ANALYTIC DIFFERENTIAL SYSTEMS
WANG CONG1 , JAUME LLIBRE2 AND XIANG ZHANG1
Abstract
...
The main
results extend some of the previous related ones, for instance the classical Poincar´e’s one [16], the Furta’s one [8], part of Chen et al’s ones
[4], and the Shi’s one [18]
...


1
...
In this paper we want to study the generalized
rational first integrals of the analytic differential systems
...


x˙ = f (x),

A function F (x) of form G(x)/H(x) with G and H analytic functions in
(Cn , 0) is a generalized rational first integral if
⟨f (x), ∂x F (x)⟩ ≡ 0,

x ∈ (Cn , 0),

where ⟨·, ·⟩ denotes the inner product of two vectors in Cn , and ∂x F =
(∂x1 F,
...
As usually, if
G and H are polynomial functions, then F (x) is a rational first integral
...
So
generalized rational first integrals include rational first integrals and analytic
first integrals as particular cases
...
If f (0) = 0,
i
...
x = 0 is a singularity of system (1), the existence of first integrals for
system (1) in (Cn , 0) is usually much involved
...
34A34, 34C05, 34C14
...
Differential systems, generalized rational first integrals,
resonance
...
Let λ =
(λ1 ,
...
We say that the eigenvalues
λ satisfy a Z+ –resonant condition if
( )n
⟨λ, k⟩ = 0,
for some k ∈ Z+ , k ̸= 0,
where Z+ = N ∪ {0} and N is the set of positive integers
...

Poincar´e [16] was the first one in studying the relation between the existence of analytic first integrals and resonance, he obtained the following
classical result (for a proof, see for instance [8])
...

Recall that k (k < n) functions are functionally independent in an open
subset U of Cn if their gradients have rank k in a full Lebesgue measure
subset of U
...

In 2003 Li, Llibre and Zhang [14] extended the Poincar´e’s result to the
case that λ admit one zero eigenvalue and the others are not Z+ –resonant
...

In 2007 the Poincar´e’s result was extended by Shi [18] to the Z–resonant
case
...
In other words, if λ do
not satisfy any Z–resonant condition, then system (1) has no rational first
integrals in (Cn , 0)
...

Our first main result is the following
...
Assume that the differential system (1) satisfies f (0) = 0
and let λ = (λ1 ,
...
Then the number
of functionally independent generalized rational first integrals of system (1)
in (Cn , 0) is at most the dimension of the minimal vector subspace of Rn
containing the set {k ∈ Zn : ⟨k, λ⟩ = 0, k ̸= 0}
...
e
...
1 of Chen, Yi and Zhang [4], and the
Theorem 1 of Shi [18]
...
Here we will

GENERALIZED RATIONAL FIRST INTEGRALS

3

use a different approach to prove Theorem 1
...

We should say that Theorem 1 has some relation with Propositions 3
...
4 of Goriely [10]
...
And in the latter
he provided a necessary condition for the existence of independent analytic
first integrals of a weight homogeneous vector field
...
For studying these cases we consider
semi–quasi–homogeneous systems
...
, fn ) be vector functions
...
, sn ∈ Z \ {0} if for all ρ > 0
fi (ρs1 x1 ,
...
, xn ),

i = 1,
...


The exponents s := (s1 ,
...
e
...
A vector function f is quasi–homogeneous
of weight degree q with weight exponents if each component fi is quasi–
homogeneous of weight degree q, i
...
fi (ρs1 x1 ,
...
, xn )
for i = 1,
...

System (1) is semi–quasi–homogeneous of degree q with the weight exponent s if
(2)

f (x) = fq (x) + fh (x),

where ρE−S fq is quasi–homogeneous of degree q with weight exponent s and
ρE−S fh is the sum of quasi–homogeneous of degree either all larger than q
or all less than q with weight exponent s
...
latter) is called
positively (resp
...
Here we have used
the notations: E is the n × n identity
( matrix, S is the
) n × n diagonal matrix
diag(s1 ,
...
, ρ1−sn
...

Assume that ρE−S fq is quasi–homogeneous of weight degree q > 1 with
weight exponent s
...
Any solution c = (c1 ,
...
Denote by B the set of balances
...


4

WANG CONG, JAUME LLIBRE AND XIANG ZHANG

Let dc be the dimension of the minimal vector subspace of Rn containing
the set
{k ∈ Zn : ⟨k, λc ⟩ = 0, k ̸= 0}
...
Assume that system (1) is semi–quasi–homogeneous of weight
degree q with weight exponent s, and f (x) satisfies (2) with f (0) = 0
...

c∈B

Theorem 2 is an extension of Theorem 1 of Furta [8], of Corollary 3
...
In some sense it is also an extension of the
main results of Yoshida [20, 21], where he proved that if a quasi–homogenous
differential system is algebraically integrable, then every Kowalevkaya exponent should be a rational number
...

Now we turn to investigate the existence of generalized rational first integrals of system (1) in a neighborhood of a periodic orbit
...
Recall that a Poincar´e
map associated to a periodic orbit is defined on a transversal section to the
periodic orbit, and its linear part has the eigenvalue 1 along the direction
tangent to the periodic orbit
...
Assume that the analytic differential system (1) has a periodic
orbit with multipliers µ = (µ1 ,
...
Then the number of functionally
independent generalized rational first integrals of system (1) in a neighborhood of the periodic orbit is at most the maximum number of linearly independent vectors in Rn of the set
{k ∈ Zn−1 : µk = 1, k ̸= 0}
...
, xn ) and k = (k1 ,
...
· xknn
...
Assume that x = 0 is a constant solution of (4), i
...

f (t, 0) = 0
...


GENERALIZED RATIONAL FIRST INTEGRALS

5

If H(t, x)is constant and non–zero, then F (t, x) is an analytic first integral
...
If G(t, x) and H(t, x) are both homogeneous polynomials in x
of degrees l and m respectively, we say that G/H is a rational homogeneous
first integral of degree l − m
...
Consider the linear
equation
(6)

x˙ = A(t)x
...
The monodromy operator associated to (6) is the map
P : (Cn−1 , 0) → (Cn−1 , 0) defined by P(x0 ) = x0 (2π)
...
, Fm (t, x) are functionally independent
in S1 ×(Cn , 0) if ∂x F1 (t, x),
...


Theorem 4
...
, µn ) be the eigenvalues of the monodromy
operator (i
...
the characteristic multipliers of (6))
...


We remark that Theorem 4 is an improvement of Theorem 5 of [14] in
two ways: one is from analytic (formal) first integrals to generalized rational
first integrals, and second our result is for more than one first integral
...
The set Ξ in Theorem 4 is called the resonant lattice
...


The rest of this paper is dedicated to prove our main results
...

2
...

Let C(x) be the field of rational functions in the variables of x, and C[x] be
the ring of polynomials in x
...
, Fk (x) ∈
C(x) are algebraically dependent if there exists a complex polynomial P of
k variables such that P (F1 (x),
...
For general definition on
algebraical dependence in a more general field, see for instance [6, p
...


6

WANG CONG, JAUME LLIBRE AND XIANG ZHANG

The first result provides an equivalent condition on functional independence
...
1]
...
The functions F1 (x),
...

Proof
...
By contradiction, if F1 (x),
...
, zk ) of minimal
degree such that
(7)

P (F1 (x),
...


Here minimal degree means that for any polynomial Q(z1 ,
...
, Fk (x)) ̸≡ 0
...
, Fk (x)) ≡ 0 for j = 1,
...

These are equivalent to



∂P
∂Fk (x)
∂F1 (x)
(F1 (x),
...

 ∂x1


∂x1 

  ∂z1


...


...




 ≡ 0
...


...


...


...


...
, Fk (x))

...
, zn ) has the minimal degree and some of the derivatives
∂P
(z1 ,
...
, Fk (x)) ≡ 0,
...
, Fk (x)) ≡ 0,
∂z1
∂zk
cannot simultaneously hold
...
Consequently F1 (x),
...
So we have proved that if
F1 (x),
...

Necessity
...

For any v1 ,
...
, vr ) the minimal field containing v1 ,
...

Recall that for a finitely generated field extension K of the field C, denoted by K/C, the transcendence degree of K over C is defined to be the
smallest integer m such that for some y1 ,
...
, ym ), the field of complex coefficient rational functions in y1 ,
...
149])
...
, yn )
if every element, saying k, of K is algebraic over C(y1 ,
...
e
...
+ pl ∈ C(y1 ,
...
, ym ) for j = 1,
...
29])
...
, ym } is called a transcendence base of K over C (see for instance [6, p
...
A finitely generated
field extension K of C is separably generated if there is a transcendence base
{z1 ,
...
, zm ) (see for instance [11, p
...
Let K be a field extension of a
field L
...
An algebraic extension K/L is separable if for every α ∈ K, the minimal
polynomial of α over L is separable
...

Since F1 ,
...
, Fk ) is a separably generated and finitely generated field extension of C of transcendence
degree k
...
, Fk )
...
8A], which states that if k is an algebraically closed field, then any finitely
generated field extension K of k is separably generated
...
, k) on C(F1 ,
...

Since F1 ,
...
, Fk ) of transcendence degree
e 1,
...
Hence there exist n derivations D
on C(F1 ,
...
, k
...
, n}
...
, Fk ) satisfy
where dsj ∈ C(x)
...
, k}
...
, ∇x Fk have the rank k, and consequently F1 ,
...


We remark that Lemma 5 has a relation in some sense with the result of
Bruns in 1887 (see [7]), which stated that if a polynomial differential system
of dimension n has l (1 ≤ l ≤ n − 1) independent algebraic first integrals,
then it has l independent rational first integrals
...
4 of Goriely [10]
...
For a rational or
a generalized rational function F (x) = G(x)/H(x) in (Cn , 0), we denote by
F 0 (x) the rational function G0 (x)/H 0 (x)
...
Then we have
(
)(
)−1
∞
∞
∑
G(x)
G0 (x) ∑ Gi (x)
H i (x)
F (x) =
=
+
1+
H(x)
H 0 (x)
H 0 (x)
H 0 (x)
i=1

=

(10)

i=1

∞
∑
Ai (x)
+
,
H 0 (x)
B i (x)

G0 (x)

i=1

where

Ai (x)

and

B i (x)

are homogeneous polynomials
...


In what follows we will say that deg Ai (x) − deg B i (x) is the degree of
Ai (x)/B i (x), and G0 (x)/H 0 (x) is the lowest degree term of F (x) in the
expansion (10)
...

Lemma 6
...
, Fm (x) =
,
H1 (x)
Hm (x)

be functionally independent generalized rational functions in (Cn , 0)
...
, zm ) for i = 2,
...
, Fm (x)),
...
, Fm (x)) are functionally
0 (x)
independent generalized rational functions, and that F10 (x), Fe20 (x),
...

Proof
...
In order that this paper is self–contained, we
provide a proof here (see also the idea of the proof of Lemma 2
...

0 (x) are functionally independent, the proof is done
...
, Fm
Without loss of generality we assume that F10 (x),
...
, Fk+1
0 (x) are algebraically
dependent
...
, Fk+1

GENERALIZED RATIONAL FIRST INTEGRALS

9

dependent
...
, zk+1 ) such that
0
P (F10 (x),
...


The fact that F10 (x),
...

∂zk+1
Since F1 (x),
...
,
Fk+1 (x) are functionally independent
...
,Fk+1 (x))
∂(F1 (x),
...
,xn )
i ,
...
We denote by D this last determinant, and
denote by D0 the determinant of the square matrix M 0 :=

0
∂(F10 (x),
...
,xik+1 )
...

Define

µ(F1 ,
...

∂Gi
∂Hi
Since ∂Fi /∂xj = Hi ∂xj − Gi ∂xj /(Hi )2 has the lowest degree larger than
∂G0

∂H 0

or equal to d(Fi ) − 1 (the former happens if Hi0 ∂xji − G0i ∂xji ≡ 0), it follows
from the definition of D that
µ(F1 ,
...


Furthermore µ(F1 ,
...
We note that
d(D0 ) = d(D) if and only if det(D0 ) ̸= 0, and this is equivalent to the
0 (x)
...
, Fk+1
have µ(F1 ,
...
Set
Fbk+1 (x) = P (F1 (x),
...


First we claim that the functions F1 (x),
...
Indeed, define
b := det(∂(F1 (x),
...
, xi , xi )
...
Fk+1 (x) that
(
)
bk+1 (x)) ∂(F1 ,
...


...

,
F
(x),
F
1
k
b = det
D
∂(F1 ,
...
, xik , xik+1 )
(11)

= D

∂P
(F1 ,
...

∂zk+1

10

WANG CONG, JAUME LLIBRE AND XIANG ZHANG

The first equality is obtained by using the derivative of composite functions,
and the second equality follows from the fact that
(
)
∂(F1 (x),
...
, Fk+1 )
...
, Fk , Fk+1 )
∂zk+1

Since (∂P/∂zk+1 )(z1 ,
...
, Fk+1
k+1 )(F1 (x),
...
This proves that F1 (x),
...
The claim follows
...
, Fk , Fbk+1 ) < µ(F1 ,
...

Indeed, writing the polynomial P as the summation
∑
αk+1
P (z) =
pα z α , z α = z1α1 ·
...
, αk+1 ) ∈ (Z+ )k+1
...
, d(Fk+1 )⟩ : pα ̸= 0, αk+1 ̸= 0}
...
, Fk+1 (x)) contains
0 (x)) ≡ 0, so the lowest degree part in the expanFk+1 and P (F10 (x),
...
This implies the lowest degree of Fb(x)
is larger than that of P (z)
...
, Fk+1 ) = d(D) + ν − d(Fk+1 ),
∂zk+1
where the last equality holds because the partial derivative of P with respect
to zk+1 is such that P loses one Fk+1 , and so the total degree loses the degree
of Fk+1
...
, Fk , Fbk+1 ) = d(D)

k
∑
j=1

d(Fj ) − d(Fbk+1 )

= µ(F1 ,
...
, Fk+1 )
...

By the two claims, from the functionally independent generalized ratio0 (x) benal functions F1 (x),
...
, Fk0 (x), Fk+1
ing functionally dependent, we get functionally independent generalized rational functions F1 (x),
...
, Fk , Fbk+1 )
< µ(F1 ,
...

0 (x) are functionIf µ(F1 ,
...
, Fk0 (x), Fbk+1
ally independent
...


GENERALIZED RATIONAL FIRST INTEGRALS

11

0
If µ(F1 ,
...
, Fk0 (x), Fbk+1
are also functionally dependent
...
, zk+1 ) such that

F1 (x),
...
, Fk+1 (x)),

are functionally independent and µ(F1 ,
...
The last equality
0
implies that the rational functions F10 ,
...
Furthermore, the generalized rational functions
F1 (x),
...
, Fm (x),

are functionally independent, because Fek+1 (x) involves only F1 ,
...
, Fk , Fek+1 are functionally independent, and also F1 ,
...
This can also be obtained by direct calculations
as follows
(
)
∂(F1 (x),
...
, Fm (x))
det
∂(x1 ,
...


...

∂x1
∂xn



...


...


...





∂Fk
∂Fk

...


...
∂z1 ∂xn +
...
+ ∂zk+1 ∂x1


∂Fk+1
∂Fk+2



...


...






...


...

(
)
∂(F1 (x),
...

∂zk+1
∂(x1 ,
...
Otherwise we can continue the above
procedure, and finally we get the functionally independent generalized rational functions F1 (x),
...
, Fk+1 (x)),
...
, Fm (x)) such that their lowest order rational func0 (x),
...
, m are polynomials in F1 ,
...

The proof of the lemma is completed
...
A
rational monomial is by definition the ratio of two monomials, i
...
of the
form xk /xl with k, l ∈ (Z+ )n
...
A rational function is homogeneous if its denominator
and numerator are both homogeneous polynomials
...

In system (1) we assume without loss of generality that A is in its Jordan
normal form and is a lower triangular matrix
...
The vector field associated to (1) is written in
X = X1 + Xh := ⟨Ax, ∂x ⟩ + ⟨g(x), ∂x ⟩
...
If F (x) = G(x)/H(x) is a generalized rational first integral of
the vector field X defined by (1), then F 0 (x) = G0 (x)/H 0 (x) is a resonant
rational homogeneous first integral of the linear vector field X1 , where we
assume that F 0 is non–constant, otherwise if F 0 (x) ≡ a ∈ C, then we
consider (F − a)0 , which is not a constant
...
1 of [2], for a different proof see for example [14])
...
Let Hnm be the linear space of complex coefficient homogeneous
polynomials of degree m in n variables
...


{⟨k, λ⟩ − c : k ∈ (Z+ )n , |k| = k1 +
...


Proof of Lemma 7
...

That F (x) is a first integral in a neighborhood of 0 ∈ Cn is equivalent to
⟨∂x F (x), f (x)⟩ ≡ 0,

x ∈ (Cn , 0)
...
e
...

H 0 (x)

This shows that F 0 (x) is a rational homogeneous first integral of the linear
system associated with (1)
...
From the equality (12) we can
assume without loss of generality that G0 (x) and H 0 (x) are relative prime
...


GENERALIZED RATIONAL FIRST INTEGRALS

13

So there exists a constant c such that
⟨
⟩
⟨
⟩
∂x G0 (x), Ax − cG0 (x) ≡ 0,
∂x H 0 (x), Ax − cH 0 (x) ≡ 0
...
Recall from Lemma 8 that Lc has respectively the spectrums on
Hnl
Sl := {⟨l, λ⟩ − c : l ∈ (Z+ )n , |l| = l},
and on Hnm
Sm := {⟨m, λ⟩ − c : m ∈ (Z+ )n , |m| = m}
...
Separate G (x) in two parts G (x) = G01 (x) + G02 (x)
l and G0 ∈ Hl
...


Lc G01 (x) ≡ 0 and

Lc G02 (x) ≡ 0
...
e
...

This proves that G0 (x) = G01 (x), i
...
each monomial, say xl , of G0 (x)
satisfies ⟨l, λ⟩ − c = 0
...
This implies that ⟨l − m, λ⟩ = 0
...

2
Having the above lemmas we can prove Theorem 1
...
Let
F1 (x) =

G1 (x)
Gm (x)
,
...

Since the polynomial functions of Fi (x) for i = 1,
...


...

,
F
(x)
=
,
m
0 (x)
Hm
H10 (x)

are functionally independent
...
, Fm
first integrals of the linear vector field X1 , that is, these first integrals are
rational functions in the variables given by resonant rational monomials
...
The semi–simple matrix As
is similar to a diagonal matrix
...
e
...
, λn )
...
Separate X1 = Xs + Xn
...
e
...
Then
0 (x) are also first integrals of
Xs (xm ) = ⟨λ, m⟩xm = 0)
...
, Fm
Xs
...
In addition, we can show that the
number of functionally independent resonant rational monomials is equal
to the maximum number of linearly independent vectors in Rn of the set
{k ∈ Zn : ⟨k, λ⟩ = 0}
...




3
...

The negative case can be studied similarly, and its details are omitted
...
A rational function G(x)/H(x) is rational quasi–homogeneous
with weight exponent s if G(x) and H(x) are both quasi–homogeneous with
weight exponent s
...
For a generalized
rational function F (x) = G(x)/H(x) we denote by F (q) (x) the rational
quasi–homogeneous function G(q) (x)/H (q) (x)
...
Its
proof can be obtained in the same way as that of Lemma 6, where we replace
the usual degree by the weight degree
...

Lemma 9
...
, Fm (x) =
,
H1 (x)
Hm (x)

be functionally independent generalized rational functions in (Cn , 0) with Gi
and Hi semi–quasi–homogeneous for i = 1,
...
Then there exist polynomials Pi (z1 ,
...
, m such that F1 (x), Fe2 (x) = P2 (F1 (x),
...
, Fem (x) = Pm (F1 (x),
...
, Fem (x) are functionally independent rational quasi–homogeneous functions
...
Since system (1) is semi–quasi–homogeneous
of degree q > 1, we take the change of variables
x → ρS x,

t → ρ−(q−1) t,

where ρS = diag (ρs1 ,
...
System (1) is transformed into
x˙ = fq (x) + feh (x, ρ),

(13)

∑ ie
where feh (x, ρ) =
ρ fq+i (x) and ρE−S feq+i (x) is quasi–homogeneous of
i≥1

weight degree q + i
...
,
ρl H(ρS x)

is a generalized rational first integral of the semi–quasi–homogeneous system (13), where the dots denote the sum of the higher order rational quasi–
homogeneous functions
...


Let c0 be a balance
...
Define
F q0 (u0 , u) = ul−m
F (q) (c0 + u)
...

We claim that F q0 (u0 , u) is a first integral of
(16)

u′0 = −

1
u0 ,
q−1

u′ = Ku + f q (u)
...
The claim follows
...
, Fr (x) =

...
, Fr(q) (x) =

Gr (x)
(q)

Hr (x)

,

are functionally independent, and they are rational quasi–homogeneous first
integrals of (14)
...
, r, have weight degree li and
mi , respectively
...
, Frq0 (u0 , u) = ul0r −mr Fr(q) (c0 + u),
are functionally independent rational quasi–homogeneous first integrals of
(16)
...
, λ0n )
...
Then for k = (k1 ,
...
, (q − 1)kn )⟩
...
Hence we get from Theorem 1
that equation (16) has at most dc0 functionally independent first integrals
...

c∈B

This completes the proof of the theorem
...
Proof of Theorem 3
Assume that system (1) has the maximal number, say r, of functionally
independent generalized rational first integrals in a neighborhood U of the
given periodic orbit, denoted by
F1 (x) =

Gr (x)
G1 (x)
,
...
Let P (x) be the Poincar´e map
defined in a neighborhood of the periodic orbit
...
, r
are also first integrals of P (x)
...

We now turn to study the maximal number of functionally independent
generalized rational first integrals of the Poincar´e map
...
, Fr0 (x) are functionally independent
...
Set
(17)

P (x) = Bx + Ph (x),

where Ph (x) is the higher order terms of P (x)
...
Let Bs
be the semi–simple part of B
...
, r, are first integrals
of P (x), we have
(18)

Gi (P (x))
Gi (x)
≡
,
Hi (P (x))
Hi (x)

x ∈ U
...


From the last equality, we can assume without loss of generality that G0i (x)
and Hi0 (x) are relatively prime
...
Equation (19) can be written as
(20)

Hi0 (Bx)
G0i (Bx)
≡
,
G0i (x)
Hi0 (x)

x ∈ U
...


18

WANG CONG, JAUME LLIBRE AND XIANG ZHANG

For completing the proof of Theorem 3 we need the following result (see
for instance, Lemma 11 of [14])
...
Let Hnm be the complex linear space of homogeneous polynomials of degree m in n variables, and let µ be the n–tuple of eigenvalues of B
...


Then the set of eigenvalues of Lc is {µk − c : k ∈ (Z+ )n , |k| = m}
...
Using
the notations given in Lemma 10 we write equations (21) as
Lci (G0i )(x) ≡ 0 and Lci (Hi0 )(x) ≡ 0
...

The above proof shows that the ratio of any two monomials in the numerator and denominator of Fi0 (x), i ∈ {1,
...

Hence each of the ratios is a first integral of the vector field Bs x, and consequently Fi0 (x), i = 1,
...
In addition, we can
check easily that the maximal number of functionally independent elements
of {xk : µk = 1, k ∈ Zn , k ̸= 0} is equal to the dimension of the minimal
linear subspace in Rn containing the set {k ∈ Zn : µk = 1}
...

5
...
Its proof can be got in the same way as the proof of
e the field of complex coefficient
Lemma 6, where we replace the field C by C(t)
generalized rational functions in t
...

Lemma 11
...
, Fm (t, x) =
,
F1 (t, x) =
H1 (t, x)
Hm (t, x)

(t, x) ∈ S1 × (Cn , 0),

be functionally independent generalized rational functions and 2π periodic
in t
...
, zm ) for i = 2,
...
, Fm (t, x)),
...
,
Fm (t, x)) are functionally independent generalized rational functions, and
0 (t, x) are functionally independent rational hothat F10 (t, x), Fe20 (t, x),
...

In the proof of Theorem 4 we also need the Floquet’s Theorem
...


GENERALIZED RATIONAL FIRST INTEGRALS

19

Floquet’s Theorem There exists a change of variables x = B(t)y periodic
of period 2π in t, which transforms the linear periodic differential system
(6) into the linear autonomous one
y˙ = Λy,

Λ is a constant matrix
...
, n, where λ = (λ1 ,
...

Now we can prove Theorem 4
...
, Fm (t, x) =

...


...

,
F
(t,
x)
=
,
m
0 (t, x)
Hm
H10 (t, x)

are functionally independent
...
Therefore system (22) has the
functionally independent generalized rational first integrals
e 1 (t, y)
G
G1 (t, B(t)y)
Fe1 (t, y) =
=
,
...

e
H
m (t, B(t)y)
Hm (t, x)

e 0 (t, y)
e 0 (t, y)
G
G
0
Fe10 (t, y) = 1
,
...
We can assume without loss of generale 0 (t, y) and H
e 0 (t, x) have respectively degrees li and mi , and are
ity that G
i
i
relatively prime for i = 1,
...

We expand Fei (t, y), i = 1,
...
, m,

i
...
, Fei0 (t, y), i = 1,
...


20

WANG CONG, JAUME LLIBRE AND XIANG ZHANG

Equations (23) are equivalent to
(
⟨
⟩)
e 0 (t, y) ∂t G
e 0 (t, y) + ∂y G
e 0 (t, y), Λy
H
i
i
i
(
⟨
⟩)
(25)
e 0 (t, y) ∂t H
e 0 (t, y) + ∂y H
e 0 (t, y), Λy ,
≡G
i
i
i

i = 1,
...


So there exist constants, say ci , such that
⟨
⟩
e 0i (t, y) + ∂y G
e 0i (t, y), Λy − ci G
e 0i (t, y) ≡ 0,
(26)
∂t G
and

⟨
⟩
0
0
e
e
e 0 (t, y) ≡ 0
...
e
...
, n} such that pi = qi for i = 1,
...

Then Υk is a base of the set of homogeneous polynomials of degree k with
the given order
...

(
)
li + n − 1
0
e
We denote by Gi (t) the vector of dimension
formed by
n−1
e 0 (t, y)
...


Using these notations equations (26) and (27) can be written as
e 0i (t) + (Ll − ci )G
e 0i (t) ≡ 0,
∂t G
i

They have solutions

e 0 (0),
e 0 (t) = exp ((ci E1i − Ll )t) G
G
i
i
i

e i0 (t) + (Lm − ci )H
e i0 (t) ≡ 0
...
In order that
e 0 (t) and H
e 0 (t) be 2π periodic, we should have
G
i
i
e 0i (0) = 0,
(exp ((ci E1i − Lli )2π) − E1i ) G

(28)
and

e i0 (0) = 0
...
, λn ) be the eigenvalues of Λ
...


GENERALIZED RATIONAL FIRST INTEGRALS

21

In order that equations (28) and (29) have nontrivial solutions we must have
exp(⟨l, λ⟩2π) = exp(ci 2π)

for all l ∈ (Z+ )n , |l| = li ,

and
exp(⟨m, λ⟩2π) = exp(ci 2π)
It follows that
exp(⟨l − m, λ⟩2π) = 1

for all m ∈ (Z+ )n , |m| = mi
...
e
...


The above proof shows that the ratio of any two monomials in the denominator and numerator of each Fei0 (t, y) for i ∈ {1,
...

Hence working in a similar way to the proof of Theorem 1 we get that m is
at most the dimension of Ξ
...

Acknowledgements
We sincerely thank the referee for his/her valuable suggestions and comments in which who mentioned the existence of a proof on Lemma 6 given
in [22] and [1]
...
The third author is partially supported by NNSF of
China grant 10831003 and Shanghai Pujiang Programm 09PJD013
...
Baider, R
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Churchill, D
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Rod and M
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Singer, On the infinitesimal
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germs of analytic vector fields, J
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Garc´ıa, H
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nada, P
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