Vietnam Journal of Mathematics 34:3 (2006) 317–329
On the Functional Equation P(f)=Q(g)
in Complex Numbers Field
*
Nguyen Trong Hoa
Daklak Pedagogical College, Buon Ma Thuot Province, Vietnam
Received November 9, 2005
Revised March 23, 2006
Abstract. In this paper, we study the existence of non-constant meromorphic so-
lutions
f and g of the functional equation P (f)=Q(g), where P(z) and Q(z) are
given nonlinear polynomials with coefficients in the complex field
C.
2000 Mathematics Subject Classification: 32H20, 30D35.
Keywords: Functional equation, unique range set, meromorphic function, algebraic
curves.
1. Introduction
Let C be the complex number field. In [3], Li and Yang introduced the following
definition.
Definition. A non-constant polynomial P (z) defined over C is called a unique-
ness polynomial for entire (or meromorphic) functions if the condition P (f)=
P (g), for entire (or meromorphic) functions f and g, implies that f ≡ g. P(z ) is
cal led a strong uniqueness polynomial if the condition P (f)=CP (g), for entire
(or meromorphic) functions f and g, and some non-zero constant C, implies
that C =1and f ≡ g.
Recently, there has been considerable progress in the study of uniqueness
polynomials, Boutabaa, Escassut and Hadadd [10] showed that a complex poly-
∗
This work was partially supported by the National Basic Research Program of Vietnam
318 Nguyen Trong Hoa
nomial P is a strong uniqueness polynomial for the family of complex p olyno-

k
)
n
k
,
where n
1
+ ···+ n
k
= n− 1 and α
1
, ,α
k
are distinct zeros of P

. The number
k is called the derivative index of P.
The polynomial P (z) is said to satisfy the condition separating the roots of
P

(separation condition) if P (α
i
) = P (α
j
) for all i = j, i, j =1, 2 ,k.
Here we only consider two nonlinear polynomials of degrees n and m, respec-
tively
P (x)=a
n
x

,
Q

(y)=mb
m
(y − β
1
)
m
1
(y − β
l
)
m
l
,
where n
1
+ +n
k
= n − 1,m
1
+ +m
l
= m − 1,α
1
, ,α
k
are distinct zeros
of P

then k ≥ I and l ≥ J. We obtain the following results.
Theorem 2.1. Let P (x) and Q(y) be nonlinear polynomials of degree n and
m, respectively, n ≥ m. Assume that P(x) − Q(y) has no linear factor, and
I, J,n
i
,m
j
be defined as above. Then there exist no non-constant meromorphic
functions f and g such that P(f)=Q(g) provided that P and Q satisfy one of
the following conditions
(i)

i|α
i
/∈∆
n
i
≥ n − m +3,
(ii)

j|β
j
/∈Λ
m
j
≥ 3.
Corollary 2.2. Let P(x) and Q(y) be nonlinear polynomials of degree n and
m, respectively, n ≥ m. Assume that P (x) − Q(y) has no linear factor. Let
k, l be the derivative indices of P, Q, respectively and ∆, Λ,I,J be defined as
above. Then there exist no non-constant meromorphic functions f and g such

2
are multiplicities of distinct zeros
β
1
,β
2
of Q

, respectively, such that β
1
,β
2
/∈ Λ,
(v) k − I =1and n
1
≥ n − m +3, where n
1
is the multiplicity of zero α
1
of P

such that α
1
/∈ ∆,
(vi) l− J =1and m
1
≥ 3, where m
1
is the multiplicity of zero β
1

)=Q(β
t
),
with t =1, 2. Then there exists no pair of non-constant meromorphic functions
f,g such that P(f)=Q(g) if one of the following conditions is satisfied
(i) m
1
≥ m
2
≥ 3,m
1
≥ n
1
,m
2
≥ n
2
, or
(ii) m
1
≥ n
1
,m
1
> 3,n
2
>m
2
≥ 3,
m

n
1
−m
1
m
2
−3
, or
(iv) n
1
>m
1
≥ m
2
> 3,n
2
>m
2
,
m
1
+1
m
1
≥
n
1
−m
1
m

} and m
1
,m
2
≥ 3.
Remark. In the case n = m = 2, the equation P (f)=Q(g) has some non-
constant meromorphic function solutions. Indeed, in this case we can rewrite
the equation P (f)=Q(g) in the form:
(f − a)
2
=(bg − c)
2
+ d,
where a, b, c, d ∈ C and b = 0. Assume that h is a non-constant meromorphic
function. Let
f =
1
2
(h +
d
h
)+a, g =
1
2b
(−h +
d
h
)+
c
b



,W(X,Z ):=




XZ
dX dZ




.
Definition. Let C be an algebraic curve in P
2
(C). A 1-form ω on C is said to
be regular if it is the pull-back of a rational 1-form on P
2
(C) such that the set of
poles of ω does not intersect C. A well-defined rational regular 1-form on C is
said to be a 1-form of Wronskian type.
Notice that to solve the functional equation P (f)=Q(g), is similar to find
meromorphic functions f, g on C such that (f(z),g(z)) lies in curve {P (x) −
Q(y)=0}. On the other hand, if C is hyperbolic on C and suppose that f,g
are meromorphic functions such that (f(z),g( z)) ∈ C, where z ∈ C, then f and
g are constant. Therefore, to prove that a functional equation P(f)=Q(g)
has no non-constant meromorphic function solution, it suffices to show that any
irreducible component of the curves {F (X,Y,Z)=0} has genus at least 2, where
F (X,Y,Z) is the homogenization of the polynomial P (x) − Q(y)inP

(X, Z):=Z
n−1
P

(
X
Z
),
Q

(Y,Z):=Z
m−1
Q

(
Y
Z
),
then
∂F
∂X
= P

(X, Z),
∂F
∂Y
= −Z
n−m
Q


n − 1ifn = m
m if n>m.
It is known that (see [4] for details)
W (Y, Z)
∂F
∂X
=
W (Z, X)
∂F
∂Y
=
W (X, Y )
∂F
∂Z
. (4)
Therefore,
W (Y, Z)
P

(X, Z)
=
W (X, Z)
Z
n−m
Q

(Y,Z)
=
W (X, Y )


β
+ higher terms],
where α, β ∈ N
∗
and a
α
,b
β
=0. The α (respectively, β) is the order (also the
multiplicity number) of x at ρ, (respectively, the order of y at ρ) for ϕ and is
denoted by
322 Nguyen Trong Hoa
α := ord
ρ,ϕ
(x) (respectively, β := ord
ρ,ϕ
(y)).
In order to prove the main results, we need the following lemmas.
Lemma 3.1. Let P and Q be two nonlinear polynomials of degrees n and m,
respectively, n ≥ m, and C be a projective curve, defined by (3). If P(α
i
) =
Q(β
j
) for all zeros α
i
of P

and β
j

n = m or n = m + 1 then C is a smooth curve. If n − m ≥ 2 then C is singular
with a unique singular point at (0 : 1 : 0).

Remark 3.2.
(i). We also require that the 1-form, defined by (5), is non trivial when it
restricts to a component of C. This is equivalent to the condition that the nom-
inators are not identically zero when they restrict to a component of C i.e.,
the Wronskians W(X, Y ),W(X, Z ),W(Y,Z) are not identically zero. It means
that the homogeneous polynomial defining C has no linear factors of the forms
aX − bY, aY − bZ, or aX − bZ, with a, b ∈ C if P = Q. Indeed, suppose on the
contrary that, aX −bZ is a factor of the curve C defined by (3). Without loss of
generality, we can take a =0. Since aX − bZ is a factor of F (X,Y,Z), we have
0=F (
b
a
Z,Y,Z)=Z
n
{P (
b
a
Z
Z
) − Q(
Y
Z
)} = Z
n
{P (
b
a


From Lemma 3.1, the only possible singularities of the curve C in P
2
(C)\[Z =
0] are at (α
i
: β
j
:1), where α
1
, ,α
k
are distinct zeros of P

and β
1
, ,β
l
are
distinct zeros of Q

. Assume that the distinct zeros α
1
, , α
k
of P

with mul-
tiplicities n
1

: 1) is a singularity of C}, (7)
Λ:={β
j
| (α
i
: β
j
: 1) is a singularity of C}. (8)
Setting I =#∆,J=#Λ, then we have k ≥ I and l ≥ J. Without loss of
generality, we can take
Λ={β
1
, ,β
J
} and m
1
≥ m
2
≥ ≥ m
J
.
Lemma 3.3. Suppose that Λ,β
t
,m
t
are defined as above. Then, the 1-form
θ :=
W (X, Z)

t|β

(X − α
i
Z)
n
i
W (Y, Z),
is regular on C.
Proof. By (5) and the hypotheses of the Lemma, we have
σ =
Z
n−m

i|α
i
/∈∆
(X − α
i
Z)
n
i
W (Y, Z)
=
pZ
n−m

i|α
i
∈∆
(X − α
i


Proposition 3.5. Assume that n ≥ m, P (x) − Q(y) has no linear factor and
k, l, ∆,J,n
i
,m
j
are defined as above. Then the curve C is Brody hyperbolic if
one of following conditions is satisfied
(i)

i|α
i
/∈∆
n
i
≥ n − m +3.
(ii)

j|β
j
/∈Λ
m
j
≥ 3.
Proof. By Lemma 3.3, set
ϑ := Z

j|β
j
/∈Λ

{R
i
θ| i =1, 2, ,
(p + 1)(p +2)
2
}
are linearly independent and are global regular 1-forms of Wronskian type on
the curve C. Thus, the genus g
C
of C is
g
C
≥
(p + 1)(p +2)
2
.
Therefore, C is Brody hyperbolic if p ≥ 1, that means,

j|β
j
/∈Λ
m
j
≥ 3.
By Lemma 3.4, we set
ς := Z

i|α
i
/∈∆

n

t=n
i
+1
(x − α
i
)
t
,
Q(y) − Q(β
j
)=
m

t=m
j
+1
(y − β
j
)
t
,
with P (α
i
)=Q(β
j
), hence
F (X,Y,Z)=Z
n

n−m
m

t=m
j
+1
(Y − β
j
Z)
t
.
Using Puiseux expansion of F (X,Y,Z)atρ
ij
=(α
i
: β
j
: 1), we have
(n
i
+ 1)ord
ρ
ij
,F
(X − α
i
Z)=(m
j
+ 1)ord
ρ

i
1
Z) −
α
i
2
−α
i
1
β
j
2
−β
j
1
(Y − β
j
1
Z)ifβ
j
1
= β
j
2
(Y − β
j
2
Z) −
β
j

, 1) = L
12
(α
i
2
,β
j
2
, 1) = 0,
and
ord
ρ
t
,F
L
12
≥ min{ord
ρ
t
,F
(X − α
i
t
Z), ord
ρ
t
,F
(Y − β
j
t

Z)ifm
j
t
≥ n
i
t
,
(10)
for t =1, 2.
Now, assume that P satisfies the separation condition. Then J ≥ I and for
every β
j
∈ Λ, there exists a unique value α
i
j
∈ ∆ such that (α
i
j
: β
j
: 1) is
singular point of C (these α
i
j
can be equal to each other). Therefore,
Γ={(α
i
j
: β
j

the following conditions is satisfied
(i) m
1
≥ m
2
≥ 3,m
1
≥ n
1
,m
2
≥ n
2
, or
(ii) m
1
≥ n
1
,m
1
> 3,n
2
>m
2
≥ 3,
m
2
+1
m
2

m
2
−3
, or
(iv) n
1
>m
1
≥ m
2
> 3,n
2
>m
2
,
m
1
+1
m
1
≥
n
1
−m
1
m
2
−3
and
m

. Indeed, assume on the contrary that β
1
= β
2
. Since ρ
1
= ρ
2
, we obtain
α
1
= α
2
. Hence P(α
1
)=Q(β
1
)=Q(β
2
)=P (α
2
), which is a contradiction. Let
L := (X − α
1
Z) −
α
2
− α
1
β

t
≥ n
t
,
(12)
for t =1, 2. The rational 1-forms
ω
1
:=
L
m
1
+m
2
−3
(Y − β
1
Z)
m
1
−1
(Y − β
2
Z)
m
2
W (X, Z),
ω
2
:=

t
: β
t
: 1) (for
t =1, 2), since P satisfies the separation condition, we have for every u = t then
326 Nguyen Trong Hoa
(α
u
: β
t
:1)/∈ C, with t =1, 2, respectively. ω
i
,i=1, 2 are regular at ρ
t
if the
1-forms
χ
11
:=
L
m
1
+m
2
−3
(Y − β
1
Z)
m
1

m
1
W (X, Z),
χ
22
:=
L
m
1
+m
2
−3
(Y − β
2
Z)
m
2
−1
W (X, Z),
are regular at ρ
t
with t =1, 2. From (12), we have
ord
ρ
1
,F
L
m
1
+m

–m
1
)
m
1
+1
ord
ρ
1
,F
(X–α
1
Z)ifm
1
<n
1
,
ord
ρ
2
,F
L
m
1
+m
2
−3
(Y − β
2
Z)

ord
ρ
2
,F
(X − α
2
Z)ifm
2
<n
2
,
ord
ρ
1
,F
L
m
1
+m
2
−3
(Y − β
1
Z)
m
1
≥

(m
2

Z)ifm
1
<n
1
,
ord
ρ
2
,F
L
m
1
+m
2
–3
(Y –β
2
Z)
m
2
–1
≥

(m
1
− 2)ord
ρ
2
,F
(Y − β

Thus, the 1- form χ
11
is regular at ρ
1
if one of the following conditions is satisfied
(r
1
) m
1
≥ n
1
and m
2
≥ 2, or
(r
2
) m
1
<n
1
and (m
1
+ 1)(m
2
− 2) ≥ (m
1
− 1)(n
1
− m
1

− m
2
).
The 1-form χ
21
is regular at ρ
1
if one of the following conditions is satisfied
(r
5
) m
1
≥ n
1
and m
2
≥ 3, or
(r
6
) n
1
>m
1
and (m
1
+ 1)(m
2
− 3) ≥ m
1
(n

− 1)(n
2
− m
2
).
Functional Equation P(f)=Q(g) in Complex Numbers Field 327
Thus, ω
1
is regular on C if one of the following conditions is satisfied
(a) m
1
≥ n
1
,m
1
≥ 3,m
2
≥ n
2
and m
2
≥ 2,
(b) m
1
≥ n
1
,n
2
>m
2

1
),
(d) n
1
>m
1
,n
2
>m
2
, (m
1
+ 1)(m
2
− 2) ≥ (m
1
− 1)(n
1
− m
1
)
and (m
2
+ 1)(m
1
− 3) ≥ m
2
(n
2
− m

≥ 3 and (m
2
+ 1)(m
1
− 2) ≥ (m
2
− 1)(n
2
− m
2
),
(c

) n
1
>m
1
≥ m
2
≥ n
2
,m
1
≥ 2 and (m
1
+ 1)(m
2
− 3) ≥ m
1
(n

2
− 1)(n
2
− m
2
).
Hence, ω
1
and ω
2
are regular on C if one of the following conditions holds
(i) m
1
≥ m
2
≥ 3,m
1
≥ n
1
,m
2
≥ n
2
,
(ii) m
1
≥ n
1
,n
2

1
− m
1
),
(iv) n
1
>m
1
≥ m
2
,n
2
>m
2
, (m
1
+ 1)(m
2
− 3) ≥ m
1
(n
1
− m
1
)
and (m
2
+ 1)(m
1
− 3) ≥ m

1
,ω
2
are linearly independent. Therefore, the curve C is
Brody hyperbolic if one of conditions of the proposition is satisfied.

Remark that if m
1
≥ m
2
≥ 3 and m
1
+ m
2
− 4 ≥ max{n
i
1
,n
i
2
}, then
ord
ρ
1
,F
L
m
1
+m
2

−
(m
1
–1)(n
1
+1)
m
1
+1
}ord
ρ
1
,F
(X − α
1
Z)ifm
1
<n
1
,
ord
ρ
2
,F
L
m
1
+m
2
–3

2
+1
}ord
ρ
2
,F
(X–α
2
Z)ifm
2
<n
2
,
ord
ρ
1
,F
L
m
1
+m
2
–3
(Y –β
1
Z)
m
1
≥


1
Z)ifm
1
<n
1
,
ord
ρ
2
,F
L
m
1
+m
2
–3
(Y –β
2
Z)
m
2
–1
≥





(m
1

Z)ifm
2
<n
2
.
328 Nguyen Trong Hoa
We obtain
ord
ρ
1
,F
L
m
1
+m
2
−3
(Y − β
1
Z)
m
1
−1
≥ 0, ord
ρ
2
,F
L
m
1

1
+m
2
−3
(Y − β
2
Z)
m
2
−1
≥ 0.
Thus, we have ω
1
and ω
2
are regular on C. Therefore, we obtain the following
corollary.
Corollary 3.7. If the hypotheses of Proposition 3.5 are satisfied, then the curve
C is Brody hyperbolic if m
1
≥ m
2
≥ 3 and m
1
+ m
2
− 4 ≥ max{n
i
1
,n

(C)
has its image contained in C defined by (3). If C is Brody hyperbolic, then
f = g. From this it follows that P = Q, contrary to the fact that P(x) − Q(y)
has no linear factors of the form ax + by + c. Hence, we prove that under the
assumptions of the theorems, the curve C is Brody hyperbolic.

Proof of Theorem 2.1. Theorem 2.1 immediately follows from Lemmas 3.3, 3.4,
Proposition 3.5 and Remark 3.9.

Proof of Corollary 2.2. From Theorem 2.1, if

j|β
j
/∈Λ
m
j
≥ 3, then the func-
tional equation P (f)=Q(g) has no solution in the set of non-constant mero-
morphic functions. Since m
j
≥ 1, we conclude that if l − J ≥ 3, then p =

j|β
j
/∈Λ
m
j
≥ 3. If l − J =2, then there only exist two zeros β
1
,β

j
/∈∆
n
j
≥ k − I, therefore, if k − I ≥ n − m + 3 then the curve C is
Brody hyperbolic. If k − I =2andn
1
+ n
2
≥ n − m + 3 then

j|α
j
/∈∆
n
j
= n
1
+
n
2
≥ n−m+3. If k−I = 1 and n
1
≥ n−m+3 then

j|α
j
/∈∆
n
j

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