Painlevé's theorem extended

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REND. SEM. MAT. UNIV. PADOVA, Vol. 108 (2002)

Painlevé’s Theorem Extended.

CLAUDIO MENEGHINI(*)

ABSTRACT- We extend Painlevé’s determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general

«multiple-valued» Cauchy’s problems. We study C⁰-continuability (near singularities) of solutions.

Foreword and preliminaries.

In this paper we slightly improve Painlevé’s determinateness theorem (see [HIL], th. 3.3.1), investigating the C⁰-continuability of the solutions of finitely «multiple-valued», meromorphic Cauchy’s problems. In particular, we shall be interested in phenomena taking place when the attempt of continuating a solution along an arc leads to singularities of the known terms: we shall see that, under not too restricting hypothe- ses, this process will converge to a limit. Of course we shall formalize

«multiple valuedness» by means of Riemann domains over regions inC² (see also [GRO], p. 43 ff). Branch points will be supposed to lie on algebraic curves. We recall that in the classical statement of the theorem

«multiple valuedness» in the known term is allowed with respect to the independent variable only.

The following theorem extends to the complex domain the so called

«single-sequence criterion» from the theory of real o.d.e.’s (see e.g.

[GIU], th. 3.2); a technical lemma ends the section; the local existence- and-uniqueness theorem is reported in the appendix.

(*) Indirizzo dell’A.: Via A. Panizzi 6, 20146 Milano (MI).

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Claudio Meneghini 94

THEOREM 1. Let W be a C^N-valued holomorphic mapping, solution of the equation W8(z)4F(W(z),z) in V^%C, where F is a C^N-valued holomorphic mapping in a neighbourhood of graph(W). Let z_Q¯V, suppose that there exists a sequence]z_n( Kz_Q,such that, set W(z_n)»4 4W_n, lim

nK QW_n4W_QC^Nand that F is holomorphic at(W_Q,z_Q): then W admits analytical continuation up to z_Q.

PROOF. We deal only with the caseN41: we can findaD0 andbD0 such that the Taylor’s developments

!

k,l40 Q

c_kln(W2W_n)^k(z2z_n)^l at (W_n,z_n) of F are absolutely and uniformly convergent in all closed bidiscsD( (W_n,z_n),a,b). By means of Cauchy’s estimates we can find an upper bound T for

!

k,l40 Q

NcklnNa^kb^l; by theorem 2.5.1 of [HIL] the solutions S_n of W8 4F(W,z), W(z_n)4W_n have radius of convergence at leasta( 12e²^b/2aT)»4s. Thus there existsMsuch thatz_QD(z_M,s); by continuity, SM(z_Q)4W0, and, by uniqueness, SM4S_Q in D(zM,s)O OD(z_Q,s), i.e. W admits analytical continuation up to z_Q. r

LEMMA 2. Let X be a metric space, g: [a,b)KX a continuous arc and suppose that there does not exist lim

tKbg(t): then, for every N-tuple ]x1Rx_N(%X there exists a sequence ]t_i( Kb and neighbourhoods U_k of x_k such that ]g(t_i)(%X0_k

0

41 N

U_k. r

The main theorem.

We recall that aRiemann domainover a regionU^%C^Nis a complex manifold D with an everywhere maximum-rank holomorphic surjective mappingp:DKU^;^D^is^properprovided that so is p(see [GRO] p. 43);

we also recall that A^%^C^N ^is^algebraicif it is the common zero set of K polynomial functions on C^N, with 0EKEN.

Let now8be a curve inC²_(u,_v), (D,p) a proper Riemann domain over C²08,Fa meromorphic function on D, holomorphic outside a curveM^, X0D0M^{, (u}0,v0)4p(X₀) andha local inverse ofp, defined in a bidisc D13D2 around (u0,v0). Let u:D(v0,r)KD1 (with r like in the existence-and-uniqueness theorem in the appendix) be the solution of the Cauchy’s problem: u8(v)4Fih(u(v),v), u(v₀)4u₀ and A₄⁸N Np(M^).

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Painlevé’s theorem extended 95 THEOREM 3 (Painlevé’s determinateness theorem). Suppose that A is algebraic; letg: [ 0 , 1 ]KCbe an embedded C¹arc starting at v₀such that, for each t[ 0 , 1 ], the complex line v4g(t) is not contained inA; suppose that an analytical continuation v of u may be got along gN^{[ 0 , 1 )}: then there exists lim

tK1vig(t), within P¹.

PROOF. Suppose, on the contrary, that such limit does not exist: for everynCv, setW_n4pr1(AO(C3 ]n() ): by hypothesisWv₁is finite or empty. The former case is trivial; as to the latter, say,W_v₁4 ]l_k(^k41Rq. For eachk41Rqand everyeD0 , setD_ke4D(l_k,e),T_ke4¯(D_ke); then there exists r_eD0 such that vD(v1,r_e) ¨ Wv%_k

0

41 q

Dke: set now

. / ´

M_e4 max

Xp²¹(0^qk41T_k3D(v₁,re()NF(X)N

M_Re4 max

Xp²¹(¯(D( 0 ,R) )3D(v₁,r_e()NF(X)N for each RD0 ) ; Introduce the compact set URe4D(O,R)0_k41

0

q

D_ke: for every v D(v1,r_e), p²¹(C_u3 ]v() is a Riemann surface, hence, by maximum principle, and by the arbitrariness of v in D(v₁,r_e),

Xp²¹(U_Re3D(v₁,re)) ¨ NF(X)N Gmax (M_e,M_Re) .

Now we have assumed thatvig(t) does not admit limit astK1 , hence, by lemma 2 (withX4P¹, ]x_k( 4 ]lk(N]Q(), there exist: a sequence ]t_i( K1 , e small enough and R large enough such that ]v(]g(t_i)()(%

%URe. Without loss of generality, we may suppose that]g(t_i)(%D(v₁,r_e).

Sincepis proper,p²¹(Uu3D(v1,r)) is compact, hence we can extract a convergent subsequence V_k from p²¹]v(g(t_i) ), g(t_i)(, whose limit we shall call V. By hypothesis there exists a holomorphic function element (V^,vA) such thatV^&g( [ 0 , 1 ) ) andvA(v₀)4u(v₀); moreover,Fihcould be analytically continuated across .vig3gN^{[ 0 , 1 )}, since vA8is finite at each point ofg( [ 0 , 1 ) ); by constrution,Fis holomorphic atVandrk(p

*(V) )4 42 , henceFihadmits analytical continuation up toV. Therefore, by theorem 1, vA admits analytical continuation up to v₁, hence there exists lim

tK1vig(t)4lim

tK1vA_ig(t), which is a contradiction. r

A simple example.

Consider w8 4 2

4

k8

k

3z1w²O4

4

k

(z1w²)³, w( 1 )41 , where the branches of the roots are those ones which take positive values on the

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Claudio Meneghini 96

positive real axis (this is in fact the choice ofh); the problem is solved by w(z)4kz, which admits analytical continuation, for example, on the backwards oriented semi-closed interval [ 1 , 0 ): note that nor 3z1w²4 40 norz1w²40 contains any complex linez4const; hence, as expected,

lim

tK0¹,tR

kt exists, being in fact 0.

REMARK. The algebraicity assumption aboutA in the main theorem could be weakened (we thank the referee for pointing out this argu- ment): indeed it would be enough to suppose that, for every nC_v, pr₁(AO(C3 ]n() ) is finite.

Appendix: the existence-and-uniqueness theorem.

Let W0 be a complexN-tuple, z0C; letF be aC^N-valued holomorphic mapping in

»

j41 N

D(W₀^j,b)3D(z₀,a), (a,bR) withC⁰-norm M and C⁰-norm of each ¯F/¯w^j (j41RN) not exceeding KR.

THEOREM. If rEmin (a,b/M, 1 /K), there exists a unique holomorphic mappingW:D(z₀,r)K

»

j41 N

D(W₀^j,b) such thatW8 4F(W(z),z) and W(z₀)4W0. (See e.g. [HIL], th 2.2.2, [INC] p. 281-284).

R E F E R E N C E S

[GIU] E. GIUSTI, Analisi Matematica, vol. 2, Bollati Boringhieri, 1989.

[GRO] ROBERTC. GUNNING - HUGOROSSI, Analytic functions of several com- plex variables, Prentice Hall, 1965.

[HIL] EINAR HILLE, Ordinary differential equations in the complex domain, John Wiley & sons, 1976.

[INC] E. L. INCE,Ordinary differential equations, Dover, 1956 (originally pub- lished in 1926).

Manoscritto pervenuto in redazione il 16 maggio 2002.