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On Meromorphic Solutions of Some Algebraic Partial Differential Equations onCn
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
214, 1]10 Ž1997.
AY975573
On Meromorphic Solutions of Some Algebraic Partial Differential Equations on C n Zhen-Han Tu* Department of Mathematics, Huazhong Uni¨ ersity of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China; and Department of Mathematics, The Uni¨ ersity of Hong Kong, Pokfulam Road, Hong Kong Submitted by A¨ ner Friedman Received December 7, 1993
Applying the techniques from Nevanlinna theory of value distribution theory of meromorphic functions on C n , we investigate the existence problem of some meromorphic solutions and obtain a satisfactory Malmquist type theorem for a class of algebraic partial differential equations on C n which improves some earlier main results. We give several examples to complement our results. Q 1997 Academic Press
1. INTRODUCTION J. Malmquist w8x proved the following theorem: THEOREM
OF
MALMQUIST.
If the differential equation p
dwrdz s
Ý
q
ai Ž z . w i
is0
Ý bj Ž z . w j , js0
with rational coefficients a i Ž z .4 and bj Ž z .4 , possesses a transcendental meromorphic solution in the complex plane, then the equation is actually degenerated into a Riccati equation, i.e., q s 0 and p F 2. Some other proofs and generalizations of this theorem have been obtained with the aid of Nevanlinna theory of meromorphic functions in the complex plane by various authors Že.g., see w1, 2, 5, 7, 12]14x.. But all *The author’s research was partially supported by the Youth Science Foundation of HUST. 1 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
2
ZHEN-HAN TU
of these studies have been restricted to differential equations in the complex plane. Recently, we w15x established some Malmquist type theorems for a class of algebraic partial differential equations on C n. In this paper, we shall obtain a satisfactory Malmquist type theorem which improves the main results in w15x and complements the Malmquist type result for meromorphic solutions of some algebraic partial differential equations on C n. It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna theory of meromorphic functions in several complex variables Žsee w3, 6, 9, 11, and 15x.. Further, the term ‘‘meromorphic’’ in this paper will always mean meromorphic on C n.
2. STATEMENT OF THE RESULTS Let V Ž z, w . be the first order differential polynomial of meromorphic function w Ž z . on C n with meromorphic coefficients, i.e., V Ž z, w . s
Ý
cŽ i. Ž z . w i 0 Ž wr z1 .
i1
i
??? Ž wr z n . n ,
Ž 1.
Ž i .gI
where z s Ž z1 , . . . , z n . g C n, the coefficients cŽ i.Ž z .4 are meromorphic Žmaybe transcendental . functions, and I is a finite set of multi-indices Ž i . s Ž i 0 , i1 , . . . , i n . for which cŽ i.Ž z . k 0 and i 0 , i1 , . . . , i n are non-negative integers. The degree of a single term of multi-index Ž i . g I in V Ž z, w . is denoted by <Ž i .< s i 0 q i1 q ??? qi n and the total degree of V Ž z, w . is defined by d s max<Ž i .< : Ž i . g I 4 . Let RŽ z, w . be an irreducible rational function in w with meromorphic coefficients, i.e., s
R Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j , is0
Ž 2.
js0
where the coefficients a i Ž z .4 and bj Ž z .4 are meromorphic on C n with a s Ž z . bk Ž z . k 0. We shall consider the algebraic partial differential equation V Ž z, w . s R Ž z, w . , where V Ž z, w . and RŽ z, w . are defined by Ž1. and Ž2., respectively. For Ž3., define SŽ r . s O
ž
s
k
Ý T Ž r , cŽ i. . q Ý T Ž r , ai . q Ý T Ž r , bj . q log r Ž i .gI
is0
js0
to estimate the growth of the coefficients of Eq. Ž3..
Ž 3.
/
Ž 4.
3
ON MEROMORPHIC SOLUTIONS
A meromorphic solution w s w Ž z . of the partial differential equation Ž3. is called admissible Žwith respect to its coefficients cŽ i.4 , a i 4 , and bj 4. if SŽ r . s oŽT Ž r, w .. as r tends q`, possibly outside a set of finite linear measure, where SŽ r . is given by Ž4.. We shall prove THEOREM 1. If Ž3. admits an admissible solution, then we ha¨ e k s 0, i.e., RŽ z, w . in Ž3. is actually degenerated into a polynomial in w. Combining the above result and Theorem 1 in w15x, we immediately obtain: THEOREM 2. ha¨ e
If Ž3. admits an admissible solution w s w Ž z ., then we ks0
and
s F d Ž 2 y u Ž w, ` . . ,
where d is the total degree of V Ž z, w . in w, and u Ž w, `. is the defect of the ¨ alue ` for w Ž z . Ž note 0 F u Ž w, `. F 1 for any meromorphic function w Ž z . on C n .. If all of the coefficients of Ž3. are rational on C n, then any of transcendental meromorphic solutions of Ž3. is admissible. Thus by Theorem 2 the following result is obvious. COROLLARY 3. Suppose that all of the coefficients of Ž3. are rational on C . If Ž3. possesses a transcendental meromorphic solution w s w Ž z ., then we ha¨ e n
ks0
and
s F d Ž 2 y u Ž w, ` . . .
EXAMPLE 1. The differential equation n
Ý Ž wr z i .
m
m
s Ž y1 . nw m Ž 1 q w .
m
Ž m G 1.
is1
admits an admissible solution w s 1rŽ e z 1 e z 2 ??? e z n y 1.. Here we have s s 2 dŽs 2 m.. Hence the bound of s in Theorem 2 is sharp. Because u Ž w, `. s 1 for any entire function w Ž z . on C n, by Theorem 2 we have: COROLLARY 4.
If Ž3. admits an admissible entire solution, then ks0
and
s F d.
Remark. In this paper, we have restricted our attention to the first order algebraic partial differential equations on C n. In fact, it is very easy
4
ZHEN-HAN TU
to generalize the results in this paper into the general case of higher order algebraic differential equations on C n by the techniques in this paper. We omit these considerations here. From the conclusion in this paper, it might appear that the Malmquist type reasoning is as powerful for algebraic partial differential equations on C n as it has been in the case of algebraic differential equations in the complex plane. But from the following examples, we can see some of their essential differences. EXAMPLE 2. The partial differential equation 2 2 Ž wr z1 . q Ž wr z 2 . q wr z1 q wr z 2 s Ž 1 q 'y 1 . w
has an admissible solution w s e z 1q 'y1 z 2 . By the way, we have that the equation has no admissible solution such a type as w s w Ž z1 . or w s w Ž z 2 . by Theorem 1 in w7x. From the example, we can see: Ž1. The Malmquist type result in higher dimension is never a trivial modification of that in the complex plane. Ž2. It is possible that Ž3. has some admissible solutions, even 1 F s F d y 1 and k s 0. Compared with Theorem 1 in w7x, the result is different from that in the complex plane. EXAMPLE 3. The partial differential equation
2w z12
q
2w z 22
y 2 Ž wr z1 . y 2 Ž wr z 2 . s y2w
has an admissible solution w s e z 1qz 2 hŽ z1 q 'y 1 z 2 . having arbitrarily rapid growth, where hŽ u. is an arbitrary non-constant meromorphic function on the complex plane u. By the way, we have that any solution of the equation such a type as w s w Ž z1 . or w s w Ž z 2 . is entire and finite order by a basic result Žsee w16x as a reference .. From the example, we can see that the growth of admissible solutions of the algebraic partial differential equations on C n seems very different from that of the differential equations on the complex plane.
3. SOME LEMMAS LEMMA 1.
Let s
R Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j is0
js0
5
ON MEROMORPHIC SOLUTIONS
be an irreducible rational function in w with the meromorphic coefficients a i Ž z .4 and bj Ž z .4 . If w s w Ž z . is a meromorphic function on C n, then T Ž r , R Ž z, w Ž z . . . s max s, k 4 T Ž r , w . q O Ž Ý T Ž r , ai . q
Ý T Ž r , bj . . .
For the proof of Lemma 1, see Lemma 4 of w15x. LEMMA 2. If w s w Ž z . is a meromorphic function such that w Ž z . satisfies the equation s
V Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j , is0
Ž 5.
js0
where Ž5. is the same as Ž3., and if k ) s, then n
T Ž r , V Ž z, w Ž z . . . s S Ž r . q O
ž
Ý m Ž r , Ž wr z k . rw . ks1
/
,
where SŽ r . is defined by Ž4.. Lemma 2 extends a result of w4x. Proof. Define s
Ps Ž z, w . s
Ý
k
ai Ž z . w i
and
Q k Ž z, w . s
is0
Ý bj Ž z . w j . js0
We rewrite Q k Ž z, w . as Q k Ž z, w . s bk Ž z . Ž w k q Ž bky1 Ž z . rbk Ž z . . w ky 1 q ??? qb 0 Ž z . rbk Ž z . . s bk Ž z . Ž w k q B1 Ž z . w ky 1 q ??? qBk Ž z . . , where Bi Ž z . s bkyi Ž z .rbk Ž z . Ž i s 0, 1, . . . , k .. Set B Ž z . s max 1, < B1 Ž z . < , < B2 Ž z . < 1r2 , . . . , < Bk Ž z . < 1r k 4 , and let EŽ r . s z s Ž z1 , . . . , z n . g C n : < z1 < 2 q ??? q< z n < 2 s r 2 4 , E1 Ž r . s z g E Ž r . : < w Ž z . < F 2 B Ž z . 4 E2 Ž r . s E Ž r . y E1 Ž r . .
and
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ZHEN-HAN TU
If z g E1Ž r ., then < V Ž z, w Ž z . . < F
Ý < cŽ i. Ž z . < < w < i qi q ? ? ? qi < Ž wr z1 . rw < i 0
F Ž2 BŽ z. .
d
n
1
Ý < cŽ i. < < Ž wr z1 . rw < i
1
??? < Ž wr z n . rw < i n
??? < Ž wr z n . rw < i n .
1
Hence we have 1 2p
HE Ž r . ln
q<
V Ž z, w Ž z . . < sn
1
F SŽ r . q
n
1 2p
Ý i kH Ž . lnq < Ž wr z k . rw < sn
ks1
Ž 6.
E1 r
by the definition of B Ž z ., where S Ž r . is defined by Ž4.. If z g E2 Ž r . Žnote < w Ž z .< ) 2 B Ž z . for z g E2 Ž r .., then < Q k Ž z, w . < s < bk Ž z . < < w < k < Ž 1 q B1 Ž z . wy1 q ??? qBk Ž z . wyk . < G < bk Ž z . < < w < k Ž 1 y < B1 Ž z . rw < y ??? y< Bk Ž z . rw k < . G < bk Ž z . < < w < k Ž 1 y B Ž z . r Ž 2 B Ž z . . k
y ??? y Ž B Ž z . . r Ž 2 B Ž z . .
k
.
s 2yk < bk Ž z . < < w Ž z . < k . Hence for z g E2 Ž r . we have < V Ž z, w Ž z . . < s
s
k
Ý a i Ž z . w i Ý bj Ž z . w j is0
F 2 k < bk Ž z .
js0 s
Ý < ai Ž z . < < w Ž z . < iyk . is0
Since k ) s and < w Ž z .< ) 2 B Ž z . ) 1 for z g E2 Ž r ., we have < V Ž z, w Ž z . . < F 2 k < bk Ž z .
s
Ý < ai Ž z . < is0
for z g E2 Ž r .. Therefore we have 1 2p
HE Ž r . ln 2
q<
V Ž z, w Ž z . . < sn s S Ž r . .
Ž 7.
7
ON MEROMORPHIC SOLUTIONS
Combining Ž6. and Ž7. we have m Ž r , V Ž z, w Ž z . . , ` . s
1 2p
žH
E1Ž r .
q
HE Ž r . 2
/
lnq < V Ž z, w Ž z . . < sn
n
s SŽ r . q O
žÝ
m Ž r , Ž wr z i . rw . .
/
is1
Ž 8.
Now we consider the poles of V Ž z, w Ž z ... Let f Ž z . be meromorphic on C n. Then there exist entire functions f 1Ž z . and f 2 Ž z . on C n such that Ž O . l fy1 Ž O .. f 1Ž z . f Ž z . s f 2 Ž z . on C n and the Žcomplex. dimension Ž fy1 1 2 n F n y 2. Define a divisor n Ž f, a. s z g C : af 1Ž z . y f 2 Ž z . s 04 for a g C and n Ž f, `. s z g C n : f 1Ž z . s 04 . We identify n Ž f, a. with its multiplicity function. Hence n Ž f, a.Ž z . G 0 is obvious. We shall verify s
N Ž r , V Ž z, w Ž z . . , ` . F
Ý N Ž r , cŽ i. , `. q Ý N Ž r , ai , `. is0 k
qO
žÝŽ
/
N Ž r , bj , 0 . q N Ž r , bj , ` . . . Ž 9 .
js0
The inequality Ž9. follows from the inequality s
n Ž V Ž z, w Ž z . . , ` . F
Ý n Ž cŽ i. , `. q Ý n Ž ai , `. is0 k
qO
žÝŽ
/
n Ž bj , 0 . q n Ž b j , ` . . .
js0
Ž 10 .
The following inequalities were obvious, s
n Ž Ps , ` . F sn Ž w, ` . q
Ý n Ž ai , `. ,
Ž 11 .
is0
n Ž V , `. F F
½
n
Ý n Ž cŽ i. , `. q max Ý i k n Ž wr z k , `. q i 0 n Ž w, `. Ži. ks1
Ý n Ž cŽ i. , `. q 2 dn Ž w, `. ,
where d is the total degree of V Ž z, w . in w.
5
Ž 12 .
8
ZHEN-HAN TU
By Ž18. in w15x we have k
n Ž Q k , ` . G k n Ž w, ` . y k
Ý Ž n Ž bj , 0 . q n Ž b j , ` . . .
Ž 13 .
js0
If n Ž w, `.Ž z . F Ý kjs0 Ž n Ž bj , 0.Ž z . q n Ž bj , `.Ž z .. for a point z g C n, then by Ž12. we have
n Ž V , `. Ž z . F
Ý n Ž cŽ i. , `. Ž z . k
q 2d
Ý Ž n Ž bj , 0 . Ž z . q n Ž bj , ` . Ž z . . .
Ž 14 .
js0
If n Ž w, `.Ž z . ) Ý kjs0 Ž n Ž bj , 0.Ž z . q n Ž bj , `.Ž z .. for a point z g C n, then z is a pole of Q k Ž z, w Ž z ... By Ž11., Ž13., and s - k we have
n Ž V , ` . Ž z . s n Ž PsrQk , ` . Ž z . s n Ž Ps , ` . Ž z . y n Ž Q k , ` . Ž z . s
F sn Ž w, ` . Ž z . q
Ý n Ž ai , `. Ž z . is0 k
ž
y k n Ž w, ` . Ž z . y k
js0
s
F
Ý Ž n Ž bj , 0 . Ž z . q n Ž b j , ` . Ž z . .
/
k
Ý n Ž a i , ` . Ž z . q k Ý Ž n Ž bj , 0 . Ž z . q n Ž b j , ` . Ž z . . . is0
js0
Ž 15 . Combining Ž14. and Ž15. we have
n Ž V , `. Ž z . F
Ý n Ž ai , `. Ž z . q Ý n Ž cŽ i. , `. Ž z . k
q max 2 d, k 4
Ý Ž n Ž bj , 0 . Ž z . q n Ž bj , ` . Ž z . . js0
for any z g C n, and Ž10. follows. Combining Ž8. and Ž9., we get Lemma 2.
9
ON MEROMORPHIC SOLUTIONS
4. PROOF OF THEOREM 1 Let w s w Ž z . be an admissible solution of Ž3.. Suppose k G 1. We rewrite Eq. Ž3. as V Ž z, w Ž z . . y P1 Ž z, w Ž z . . s P2 Ž z, w Ž z . . rQk Ž z, w Ž z . . ,
Ž 16 .
where P1Ž z, w . and P2 Ž z, w . are polynomials in w, deg w Ž P2 Ž z, w .. - k, deg w Ž P1Ž z, w .. s max s y k, 04 , and the coefficients of P1Ž z, w . and P2 Ž z, w . in w are rational of a i Ž z .4 and bj Ž z .4 . Since Ps Ž z, w .rQk Ž z, w . is irreducible in w, we have P2 Ž z, w Ž z .. k 0. By Lemma 1 we have T Ž r, P2 Ž z, w Ž z ..rQk Ž z, w Ž z ... s kT Ž r, w . q SŽ r .. By Lemma 2 we have T Ž r, V y P1 . s SŽ r . q O ŽÝ nks1 mŽ r, Ž wr z k .rw ... Hence by Ž16. we have n
kT Ž r , w . s S Ž r . q O
ž
Ý m Ž r , Ž wr z i . rw . is1
/
.
Ž 17 .
By the Lemma of the logarithmic derivative in several complex variables Žsee w3x. and the condition that w s w Ž z . is an admissible solution, we have that Ž17. is a contradiction. Hence k s 0. Theorem 1 follows.
REFERENCES 1. S. Bank, On the growth of solutions of algebraic differential equations, Trans. Amer. Math. Soc. 240 Ž1978., 195]212. 2. S. Bank and R. Kaufman, On the growth of meromorphic solutions of the differential equation Ž y9. m s RŽ z, y ., Acta Math. 144 Ž1980., 233]248. 3. A. Biancofiore and W. Stoll, Another proof of the lemma of the logarithmic derivative in several complex variables, in ‘‘Recent Developments in Several Complex Variables,’’ pp. 29]45, Annals of Mathematics Studies, Vol. 100, Princeton Univ. Press, Princeton, 1981. 4. J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 Ž1962., 17]27. 5. F. Gackstatter and I. Laine, Zur Theorie der gewohnlichen Differentialgleichungen im ¨ Komplexen, Ann. Polon. Math. 38 Ž1980., 259]287. 6. P. Griffiths, Entire holomorphic mappings in one and several complex variables,’’ in ‘‘Annals of Mathematics Studies,’’ Vol. 85, Princeton Univ. Press, Princeton, 1976. 7. K. Ishizaki, On a conjecture of Gackstatter and Laine on some differential equations, Proc. Japan Acad. Ser. A 67 Ž1991., 270]273. 8. J. Malmquist, Sur les fonctions ` a une nombre finis des branches definies par les ´ du premier ordre, Acta Math. 36 Ž1913., 297]343. ´equations differentielles ´
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ZHEN-HAN TU
9. J. Noguchi and T. Ochiai, Geometric function theory in several complex variables, in ‘‘Translations of Math. Monographs,’’ Vol. 80, Amer. Math. Soc., Providence, Rhode Island, 1990. 10. B. Shiffman, A general second main theorem for meromorphic functions on C n , Amer. J. Math. 106 Ž1984., 509]531. 11. W. Stoll, ‘‘Holomorphic Functions of Finite Order in Several Complex Variables,’’ Amer. Math. Soc., Providence, Rhode Island, 1974. 12. Sh. Strelitz, On meromorphic solutions of algebraic differential equations, Trans. Amer. Math. Soc. 258 Ž1980., 431]440. 13. N. Toda, On the growth of meromorphic solutions of some higher order differential equations, J. Math. Soc. Japan 38 Ž1986., 439]451. 14. Z.-H. Tu, A Malmquist type theorem of some generalized higher order algebraic differential equations, Complex Variables Theory Appl. 17 Ž1992., 265]275. 15. Z.-H. Tu, Some Malmquist type theorems of partial differential equations on C n , J. Math. Anal. Appl. 179 Ž1993., 41]60. ´ 16. G. Valiron, ‘‘Lectures on the General Theory of Integral Functions,’’ Edouard Privat, Toulouse, 1923.
214, 1]10 Ž1997.
AY975573
On Meromorphic Solutions of Some Algebraic Partial Differential Equations on C n Zhen-Han Tu* Department of Mathematics, Huazhong Uni¨ ersity of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China; and Department of Mathematics, The Uni¨ ersity of Hong Kong, Pokfulam Road, Hong Kong Submitted by A¨ ner Friedman Received December 7, 1993
Applying the techniques from Nevanlinna theory of value distribution theory of meromorphic functions on C n , we investigate the existence problem of some meromorphic solutions and obtain a satisfactory Malmquist type theorem for a class of algebraic partial differential equations on C n which improves some earlier main results. We give several examples to complement our results. Q 1997 Academic Press
1. INTRODUCTION J. Malmquist w8x proved the following theorem: THEOREM
OF
MALMQUIST.
If the differential equation p
dwrdz s
Ý
q
ai Ž z . w i
is0
Ý bj Ž z . w j , js0
with rational coefficients a i Ž z .4 and bj Ž z .4 , possesses a transcendental meromorphic solution in the complex plane, then the equation is actually degenerated into a Riccati equation, i.e., q s 0 and p F 2. Some other proofs and generalizations of this theorem have been obtained with the aid of Nevanlinna theory of meromorphic functions in the complex plane by various authors Že.g., see w1, 2, 5, 7, 12]14x.. But all *The author’s research was partially supported by the Youth Science Foundation of HUST. 1 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
2
ZHEN-HAN TU
of these studies have been restricted to differential equations in the complex plane. Recently, we w15x established some Malmquist type theorems for a class of algebraic partial differential equations on C n. In this paper, we shall obtain a satisfactory Malmquist type theorem which improves the main results in w15x and complements the Malmquist type result for meromorphic solutions of some algebraic partial differential equations on C n. It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna theory of meromorphic functions in several complex variables Žsee w3, 6, 9, 11, and 15x.. Further, the term ‘‘meromorphic’’ in this paper will always mean meromorphic on C n.
2. STATEMENT OF THE RESULTS Let V Ž z, w . be the first order differential polynomial of meromorphic function w Ž z . on C n with meromorphic coefficients, i.e., V Ž z, w . s
Ý
cŽ i. Ž z . w i 0 Ž wr z1 .
i1
i
??? Ž wr z n . n ,
Ž 1.
Ž i .gI
where z s Ž z1 , . . . , z n . g C n, the coefficients cŽ i.Ž z .4 are meromorphic Žmaybe transcendental . functions, and I is a finite set of multi-indices Ž i . s Ž i 0 , i1 , . . . , i n . for which cŽ i.Ž z . k 0 and i 0 , i1 , . . . , i n are non-negative integers. The degree of a single term of multi-index Ž i . g I in V Ž z, w . is denoted by <Ž i .< s i 0 q i1 q ??? qi n and the total degree of V Ž z, w . is defined by d s max<Ž i .< : Ž i . g I 4 . Let RŽ z, w . be an irreducible rational function in w with meromorphic coefficients, i.e., s
R Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j , is0
Ž 2.
js0
where the coefficients a i Ž z .4 and bj Ž z .4 are meromorphic on C n with a s Ž z . bk Ž z . k 0. We shall consider the algebraic partial differential equation V Ž z, w . s R Ž z, w . , where V Ž z, w . and RŽ z, w . are defined by Ž1. and Ž2., respectively. For Ž3., define SŽ r . s O
ž
s
k
Ý T Ž r , cŽ i. . q Ý T Ž r , ai . q Ý T Ž r , bj . q log r Ž i .gI
is0
js0
to estimate the growth of the coefficients of Eq. Ž3..
Ž 3.
/
Ž 4.
3
ON MEROMORPHIC SOLUTIONS
A meromorphic solution w s w Ž z . of the partial differential equation Ž3. is called admissible Žwith respect to its coefficients cŽ i.4 , a i 4 , and bj 4. if SŽ r . s oŽT Ž r, w .. as r tends q`, possibly outside a set of finite linear measure, where SŽ r . is given by Ž4.. We shall prove THEOREM 1. If Ž3. admits an admissible solution, then we ha¨ e k s 0, i.e., RŽ z, w . in Ž3. is actually degenerated into a polynomial in w. Combining the above result and Theorem 1 in w15x, we immediately obtain: THEOREM 2. ha¨ e
If Ž3. admits an admissible solution w s w Ž z ., then we ks0
and
s F d Ž 2 y u Ž w, ` . . ,
where d is the total degree of V Ž z, w . in w, and u Ž w, `. is the defect of the ¨ alue ` for w Ž z . Ž note 0 F u Ž w, `. F 1 for any meromorphic function w Ž z . on C n .. If all of the coefficients of Ž3. are rational on C n, then any of transcendental meromorphic solutions of Ž3. is admissible. Thus by Theorem 2 the following result is obvious. COROLLARY 3. Suppose that all of the coefficients of Ž3. are rational on C . If Ž3. possesses a transcendental meromorphic solution w s w Ž z ., then we ha¨ e n
ks0
and
s F d Ž 2 y u Ž w, ` . . .
EXAMPLE 1. The differential equation n
Ý Ž wr z i .
m
m
s Ž y1 . nw m Ž 1 q w .
m
Ž m G 1.
is1
admits an admissible solution w s 1rŽ e z 1 e z 2 ??? e z n y 1.. Here we have s s 2 dŽs 2 m.. Hence the bound of s in Theorem 2 is sharp. Because u Ž w, `. s 1 for any entire function w Ž z . on C n, by Theorem 2 we have: COROLLARY 4.
If Ž3. admits an admissible entire solution, then ks0
and
s F d.
Remark. In this paper, we have restricted our attention to the first order algebraic partial differential equations on C n. In fact, it is very easy
4
ZHEN-HAN TU
to generalize the results in this paper into the general case of higher order algebraic differential equations on C n by the techniques in this paper. We omit these considerations here. From the conclusion in this paper, it might appear that the Malmquist type reasoning is as powerful for algebraic partial differential equations on C n as it has been in the case of algebraic differential equations in the complex plane. But from the following examples, we can see some of their essential differences. EXAMPLE 2. The partial differential equation 2 2 Ž wr z1 . q Ž wr z 2 . q wr z1 q wr z 2 s Ž 1 q 'y 1 . w
has an admissible solution w s e z 1q 'y1 z 2 . By the way, we have that the equation has no admissible solution such a type as w s w Ž z1 . or w s w Ž z 2 . by Theorem 1 in w7x. From the example, we can see: Ž1. The Malmquist type result in higher dimension is never a trivial modification of that in the complex plane. Ž2. It is possible that Ž3. has some admissible solutions, even 1 F s F d y 1 and k s 0. Compared with Theorem 1 in w7x, the result is different from that in the complex plane. EXAMPLE 3. The partial differential equation
2w z12
q
2w z 22
y 2 Ž wr z1 . y 2 Ž wr z 2 . s y2w
has an admissible solution w s e z 1qz 2 hŽ z1 q 'y 1 z 2 . having arbitrarily rapid growth, where hŽ u. is an arbitrary non-constant meromorphic function on the complex plane u. By the way, we have that any solution of the equation such a type as w s w Ž z1 . or w s w Ž z 2 . is entire and finite order by a basic result Žsee w16x as a reference .. From the example, we can see that the growth of admissible solutions of the algebraic partial differential equations on C n seems very different from that of the differential equations on the complex plane.
3. SOME LEMMAS LEMMA 1.
Let s
R Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j is0
js0
5
ON MEROMORPHIC SOLUTIONS
be an irreducible rational function in w with the meromorphic coefficients a i Ž z .4 and bj Ž z .4 . If w s w Ž z . is a meromorphic function on C n, then T Ž r , R Ž z, w Ž z . . . s max s, k 4 T Ž r , w . q O Ž Ý T Ž r , ai . q
Ý T Ž r , bj . . .
For the proof of Lemma 1, see Lemma 4 of w15x. LEMMA 2. If w s w Ž z . is a meromorphic function such that w Ž z . satisfies the equation s
V Ž z, w . s
k
Ý a i Ž z . w i Ý bj Ž z . w j , is0
Ž 5.
js0
where Ž5. is the same as Ž3., and if k ) s, then n
T Ž r , V Ž z, w Ž z . . . s S Ž r . q O
ž
Ý m Ž r , Ž wr z k . rw . ks1
/
,
where SŽ r . is defined by Ž4.. Lemma 2 extends a result of w4x. Proof. Define s
Ps Ž z, w . s
Ý
k
ai Ž z . w i
and
Q k Ž z, w . s
is0
Ý bj Ž z . w j . js0
We rewrite Q k Ž z, w . as Q k Ž z, w . s bk Ž z . Ž w k q Ž bky1 Ž z . rbk Ž z . . w ky 1 q ??? qb 0 Ž z . rbk Ž z . . s bk Ž z . Ž w k q B1 Ž z . w ky 1 q ??? qBk Ž z . . , where Bi Ž z . s bkyi Ž z .rbk Ž z . Ž i s 0, 1, . . . , k .. Set B Ž z . s max 1, < B1 Ž z . < , < B2 Ž z . < 1r2 , . . . , < Bk Ž z . < 1r k 4 , and let EŽ r . s z s Ž z1 , . . . , z n . g C n : < z1 < 2 q ??? q< z n < 2 s r 2 4 , E1 Ž r . s z g E Ž r . : < w Ž z . < F 2 B Ž z . 4 E2 Ž r . s E Ž r . y E1 Ž r . .
and
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ZHEN-HAN TU
If z g E1Ž r ., then < V Ž z, w Ž z . . < F
Ý < cŽ i. Ž z . < < w < i qi q ? ? ? qi < Ž wr z1 . rw < i 0
F Ž2 BŽ z. .
d
n
1
Ý < cŽ i. < < Ž wr z1 . rw < i
1
??? < Ž wr z n . rw < i n
??? < Ž wr z n . rw < i n .
1
Hence we have 1 2p
HE Ž r . ln
q<
V Ž z, w Ž z . . < sn
1
F SŽ r . q
n
1 2p
Ý i kH Ž . lnq < Ž wr z k . rw < sn
ks1
Ž 6.
E1 r
by the definition of B Ž z ., where S Ž r . is defined by Ž4.. If z g E2 Ž r . Žnote < w Ž z .< ) 2 B Ž z . for z g E2 Ž r .., then < Q k Ž z, w . < s < bk Ž z . < < w < k < Ž 1 q B1 Ž z . wy1 q ??? qBk Ž z . wyk . < G < bk Ž z . < < w < k Ž 1 y < B1 Ž z . rw < y ??? y< Bk Ž z . rw k < . G < bk Ž z . < < w < k Ž 1 y B Ž z . r Ž 2 B Ž z . . k
y ??? y Ž B Ž z . . r Ž 2 B Ž z . .
k
.
s 2yk < bk Ž z . < < w Ž z . < k . Hence for z g E2 Ž r . we have < V Ž z, w Ž z . . < s
s
k
Ý a i Ž z . w i Ý bj Ž z . w j is0
F 2 k < bk Ž z .
js0 s
Ý < ai Ž z . < < w Ž z . < iyk . is0
Since k ) s and < w Ž z .< ) 2 B Ž z . ) 1 for z g E2 Ž r ., we have < V Ž z, w Ž z . . < F 2 k < bk Ž z .
s
Ý < ai Ž z . < is0
for z g E2 Ž r .. Therefore we have 1 2p
HE Ž r . ln 2
q<
V Ž z, w Ž z . . < sn s S Ž r . .
Ž 7.
7
ON MEROMORPHIC SOLUTIONS
Combining Ž6. and Ž7. we have m Ž r , V Ž z, w Ž z . . , ` . s
1 2p
žH
E1Ž r .
q
HE Ž r . 2
/
lnq < V Ž z, w Ž z . . < sn
n
s SŽ r . q O
žÝ
m Ž r , Ž wr z i . rw . .
/
is1
Ž 8.
Now we consider the poles of V Ž z, w Ž z ... Let f Ž z . be meromorphic on C n. Then there exist entire functions f 1Ž z . and f 2 Ž z . on C n such that Ž O . l fy1 Ž O .. f 1Ž z . f Ž z . s f 2 Ž z . on C n and the Žcomplex. dimension Ž fy1 1 2 n F n y 2. Define a divisor n Ž f, a. s z g C : af 1Ž z . y f 2 Ž z . s 04 for a g C and n Ž f, `. s z g C n : f 1Ž z . s 04 . We identify n Ž f, a. with its multiplicity function. Hence n Ž f, a.Ž z . G 0 is obvious. We shall verify s
N Ž r , V Ž z, w Ž z . . , ` . F
Ý N Ž r , cŽ i. , `. q Ý N Ž r , ai , `. is0 k
qO
žÝŽ
/
N Ž r , bj , 0 . q N Ž r , bj , ` . . . Ž 9 .
js0
The inequality Ž9. follows from the inequality s
n Ž V Ž z, w Ž z . . , ` . F
Ý n Ž cŽ i. , `. q Ý n Ž ai , `. is0 k
qO
žÝŽ
/
n Ž bj , 0 . q n Ž b j , ` . . .
js0
Ž 10 .
The following inequalities were obvious, s
n Ž Ps , ` . F sn Ž w, ` . q
Ý n Ž ai , `. ,
Ž 11 .
is0
n Ž V , `. F F
½
n
Ý n Ž cŽ i. , `. q max Ý i k n Ž wr z k , `. q i 0 n Ž w, `. Ži. ks1
Ý n Ž cŽ i. , `. q 2 dn Ž w, `. ,
where d is the total degree of V Ž z, w . in w.
5
Ž 12 .
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ZHEN-HAN TU
By Ž18. in w15x we have k
n Ž Q k , ` . G k n Ž w, ` . y k
Ý Ž n Ž bj , 0 . q n Ž b j , ` . . .
Ž 13 .
js0
If n Ž w, `.Ž z . F Ý kjs0 Ž n Ž bj , 0.Ž z . q n Ž bj , `.Ž z .. for a point z g C n, then by Ž12. we have
n Ž V , `. Ž z . F
Ý n Ž cŽ i. , `. Ž z . k
q 2d
Ý Ž n Ž bj , 0 . Ž z . q n Ž bj , ` . Ž z . . .
Ž 14 .
js0
If n Ž w, `.Ž z . ) Ý kjs0 Ž n Ž bj , 0.Ž z . q n Ž bj , `.Ž z .. for a point z g C n, then z is a pole of Q k Ž z, w Ž z ... By Ž11., Ž13., and s - k we have
n Ž V , ` . Ž z . s n Ž PsrQk , ` . Ž z . s n Ž Ps , ` . Ž z . y n Ž Q k , ` . Ž z . s
F sn Ž w, ` . Ž z . q
Ý n Ž ai , `. Ž z . is0 k
ž
y k n Ž w, ` . Ž z . y k
js0
s
F
Ý Ž n Ž bj , 0 . Ž z . q n Ž b j , ` . Ž z . .
/
k
Ý n Ž a i , ` . Ž z . q k Ý Ž n Ž bj , 0 . Ž z . q n Ž b j , ` . Ž z . . . is0
js0
Ž 15 . Combining Ž14. and Ž15. we have
n Ž V , `. Ž z . F
Ý n Ž ai , `. Ž z . q Ý n Ž cŽ i. , `. Ž z . k
q max 2 d, k 4
Ý Ž n Ž bj , 0 . Ž z . q n Ž bj , ` . Ž z . . js0
for any z g C n, and Ž10. follows. Combining Ž8. and Ž9., we get Lemma 2.
9
ON MEROMORPHIC SOLUTIONS
4. PROOF OF THEOREM 1 Let w s w Ž z . be an admissible solution of Ž3.. Suppose k G 1. We rewrite Eq. Ž3. as V Ž z, w Ž z . . y P1 Ž z, w Ž z . . s P2 Ž z, w Ž z . . rQk Ž z, w Ž z . . ,
Ž 16 .
where P1Ž z, w . and P2 Ž z, w . are polynomials in w, deg w Ž P2 Ž z, w .. - k, deg w Ž P1Ž z, w .. s max s y k, 04 , and the coefficients of P1Ž z, w . and P2 Ž z, w . in w are rational of a i Ž z .4 and bj Ž z .4 . Since Ps Ž z, w .rQk Ž z, w . is irreducible in w, we have P2 Ž z, w Ž z .. k 0. By Lemma 1 we have T Ž r, P2 Ž z, w Ž z ..rQk Ž z, w Ž z ... s kT Ž r, w . q SŽ r .. By Lemma 2 we have T Ž r, V y P1 . s SŽ r . q O ŽÝ nks1 mŽ r, Ž wr z k .rw ... Hence by Ž16. we have n
kT Ž r , w . s S Ž r . q O
ž
Ý m Ž r , Ž wr z i . rw . is1
/
.
Ž 17 .
By the Lemma of the logarithmic derivative in several complex variables Žsee w3x. and the condition that w s w Ž z . is an admissible solution, we have that Ž17. is a contradiction. Hence k s 0. Theorem 1 follows.
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9. J. Noguchi and T. Ochiai, Geometric function theory in several complex variables, in ‘‘Translations of Math. Monographs,’’ Vol. 80, Amer. Math. Soc., Providence, Rhode Island, 1990. 10. B. Shiffman, A general second main theorem for meromorphic functions on C n , Amer. J. Math. 106 Ž1984., 509]531. 11. W. Stoll, ‘‘Holomorphic Functions of Finite Order in Several Complex Variables,’’ Amer. Math. Soc., Providence, Rhode Island, 1974. 12. Sh. Strelitz, On meromorphic solutions of algebraic differential equations, Trans. Amer. Math. Soc. 258 Ž1980., 431]440. 13. N. Toda, On the growth of meromorphic solutions of some higher order differential equations, J. Math. Soc. Japan 38 Ž1986., 439]451. 14. Z.-H. Tu, A Malmquist type theorem of some generalized higher order algebraic differential equations, Complex Variables Theory Appl. 17 Ž1992., 265]275. 15. Z.-H. Tu, Some Malmquist type theorems of partial differential equations on C n , J. Math. Anal. Appl. 179 Ž1993., 41]60. ´ 16. G. Valiron, ‘‘Lectures on the General Theory of Integral Functions,’’ Edouard Privat, Toulouse, 1923.