zweiter Ordnung im Komplexen. Dissertation, Aachen. 1986. [7] N. Stcinmetz, Ein Malmquistscher Satz fiir algebraische Differentialgleichungen erster Ordnung, ...
Complex Varruhles. 1989. Vul. 13, pp. 75 83 Reprints available directly from the publ~sher Photocopying permttled hy license only p 1989 (iordon and Breach. Science Publishers. Inc Prtnted in the United States of America
Meromorphic Solutions of Second-Order Algebraic Differential Equations Downloaded by [Universitaetsbibliothek Dortmund] at 00:24 18 January 2015
NORBERT STEINMETZ Mathematisches lnstitut 1 der Universitat Karlsruhe, Englerstr. 2, 0-7500 Karlsruhe, F. R.G.
AMS No. 34A20, 30D35 Dedicated to Alhert kdre~and Wolfgang Fucha Communicated- K F Rarth and R . P. Gilbert (Rewired S @ p r c m h e r 1.2. 19x7)
1. INTRODUCTION A famous theorem of Malmquist states that the first-order differential equation
dw
R rational,
-= R(w, z ) ,
dz
(1)
admits transcendental meromorphic solutions (in the plane) only when it reduces to a Riccati equation
Similarly, in order that the algebraic differential equation
has a transcendental meromorphic solution, a necessary condition is deg P, < 2(n - v ) , 0 < v < n (an excellent reference is Eremenko [I]). The case n = 2 is due to the author
C7l.
For second-order equations dz2
,
R rational,
no analogue of Malmquist's theorem is known. One difficulty arises from the fact that it is possible to derive from (3) an almost arbitrarily complicated equation (4).
76
N . STEINMETZ
The impressive list of 50 (!) equations without moving critical singularities of the solutions (see Ince 141) suggests that equation (4) must reduce to d2w
+ M ( w , Z ) dl? + N ( w . z ) , -
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dz
if there is at least one transcendental solution not satisfying any ccjua:ion ( 3 ) .However, we are far from being able to give a proof. Therefore we will start from equation (5) and prove a theorem of Malmquist type for the reduced equation ( 5 ) . For particular cases the reader is referred to Wittich [9] (L = M = 0 ) . von Rieth [6] (M = N = 0 ) and the author [8] (L = 0 ) .
2. ALGEBRAIC DIFFERENTIAL E O U A T l O N S Ii is assumcd that :hi i i a d i i is f~mi!iai bi:h N~-~an!inna's :hci;rq of rnsromorphi; functions (see Hayman [3]L We consider always algebraic differential equations
where Q(wn, . . . , w , , w , z ) is a polynomial in w,, . . . , w , , w with rational coefficients. We remark that in most theorems on algebraic differential equations the phrases "rational coefficients" and "transcendental solutions" may be replaced by L:- - - - E L --a ....a-:--:Ll--I..&:--,.'' --.:&L-..& I I I W U I I I U I ~ I I I LLUGIIILIGIILS ~ I I U ~ U I I I I S S I U I C SULULIUIIS W I L I I U U L addiiig a iiew idea to the proof (an admissible solution grows faster than the coefficients of (6)). In certain cases, as will be seen, it is even easier to work with admissible solutions. On the other hand, it is sometimes convenient to know that theorems like Malmquist's are also true in this general setting. The main tool in our investigations of the Painlev6 equation ( 5 )will be the following theorem which probably will be useful also for other classes of algebraic differential equations. --a-
??
THEOREM 1 Let f, fo, . . . , fm be meromorphic in the plane, f nonrurional, and let Po,. . . , Pm be polynomials such that
Assume further that there are nonnegatiue integers k,, k , , . . . , km such that f
Jj -
f and
kJ
. regular except possibly at the zeros o f f zs
DIFFERENTIAL EQUATIONS
77
Then there exist
+
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distinct complex numbers T,, . . . , T,, I,, 1 polynomials Qo, . . . .Qm ( Q j # 0 if' Pj # 0 ) and a meromorphic Jimction lz such that
nnd Reindr-/i I! turns out that ( 1 1 ) interpol;rtes equation (71 at the zeros of f - T,. In applications ,fi will be a polynomial R j ( ~'"I,/ . . . j". j; 3 ) , and if J' is entire we may choose k, = deg Q j , in general the choice k, = weight Q j suffices. For example, if jj = f" we have k j = 4. in particular cases, however, kj may be smaller. Thus, if jj = f f " - 2f '' we may choose kj = 3 (to satisfy (0) kj = 2 is enough), since at a pole of f'of order p, j; has a pole of order at most 3 if p = 1 and at most 2p + 2 d 3 p if p b 2.
.
3.
PROOF OF THEOREM 1
Let
TE
C satisfy the following conditions:
(ii) f - 7 has no multiple zeros, (iii) Pj(r, z ) f 0 (whenever Pj # 0) and (iv) the coefficients of Pj are regular at zeros of f - .r (0
