Polynomial Hamiltonian Systems with Movable Algebraic Singularities
aa r X i v : . [ m a t h . C A ] D ec Accepted for publication in the Journal d’Analyse Math´ematique
Polynomial Hamiltonian Systems withMovable Algebraic Singularities
Thomas Kecker
Abstract
The singularity structure of solutions of a class of Hamiltonian systemsof ordinary differential equations in two dependent variables is studied.It is shown that for any solution, all movable singularities, obtained byanalytic continuation along a rectifiable curve, are at most algebraicbranch points.
Singularities of solutions of ordinary differential equations can be classed asbeing either fixed or movable. The set of fixed singularities consists of thepoints in the complex plane where the equation itself becomes singular insome sense. All other singularities of a solution are called movable as theirpositions vary with the initial conditions of the equation.In the 1900’s P. Painlev´e [17] classified all rational second-order equations y ′′ = R ( z, y, y ′ ) , (1.1)for which all solutions are single-valued around their movable singularities,a property known as the Painlev´e property. This classification, with someerrors and gaps which have been filled by R. Fuchs, B. Gambier and others,led to the discovery of six non-linear equations now known as the six Painlev´eequations. The result of the classification is that the solution of any equationof the form (1.1) that has the Painlev´e property can be expressed by thesolutions of classically known function (e.g. elliptic functions, the solutionsof linear differential equations or equations that are solvable by quadrature),1 Thomas Keckerand the solutions of the six non-linear Painlev´e equations. Interestingly, toeach Painlev´e equation is associated an equivalent Hamiltonian system dqdz = ∂H J ∂pdpdz = − ∂H J ∂q , (1.2)with Hamiltonians H J ( z, p, q ), J = 1 , . . . ,
6, polynomial in p and q . Thesewere already noticed by J. Malmquist [10] and later extensively studied byK. Okamoto [12, 13, 14, 15].A classification of systems of equations in two variables that possess thePainlev´e property, y ′ = P ( z, y , y ) y ′ = Q ( z, y , y ) , (1.3)was given by R. Garnier [3] in the autonomous case where P = P ( y , y ) and Q = Q ( y , y ) are homogeneous rational functions of y and y . The solutionsin this case are given by elliptic functions or a combination of rational andexponential functions. J. Goffar-Lombet [4] classified those systems (1.3)with the Painlev´e property where P and Q are certain polynomials of degreeat most 3 in y and y with z -dependent analytic coefficients and showedthat the solutions can be given in terms of classically known functions andPainlev´e transcendents. T. Kimura and T. Matuda [8] extended this result tothe case where the degrees of P and Q are less or equal to 5. They conjecturethat any system (1.3) with the Painlev´e property is equivalent to one of thesystems (1.2).Lifting the restriction of the Painlev´e property, a number of articles havestudied classes of second-order differential equations for which all movablesingularities are at most algebraic branch points. In [19, 20], S. Shimomuraconsidered the equations y ′′ = 2(2 k + 1)(2 k − y k + z, k ∈ N ,y ′′ = k + 1 k y k +1 + zy + α, k ∈ N \ { } , of P I -type and P II -type, respectively, for which he showed that all movablesingularities that can be reached by analytic continuation along a rectifiableolynomial Hamiltonian Systems 3curve are algebraic branch points. More generally, G. Filipuk and R. Halburd[2] have studied equations of the form y ′′ = P ( z, y ) , (1.4)where P is a polynomial in y with analytic coefficients in z that satisfy one ortwo differential relations known as resonance conditions (these are equivalentto the existence of certain formal algebraic series solutions of (1.4)). Equation(1.4) can be seen as a Hamiltonian system with Hamiltonian H ( z, y , y ) = 12 y − ˆ P ( z, y ) , where ˆ P is a polynomial with ˆ P y = P and we have the correspondence y = y , y ′ = y . In this article we consider a much more general class of Hamiltoniansystems in two variables. The way we prove that any movable singularity ofa solution that can be reached by analytic continuation along a finite lengthcurve is an algebraic branch point is based on a method used in certain proofsof the Painlev´e property for the Painlev´e equations. In particular we mentionthe proofs in [7], [16] and [18], see also the book [5]. Consider the Hamiltonian system given by H ( z, y , y ) = α M +1 , ( z ) y M +11 + α ,N +1 ( z ) y N +12 + X ( i,j ) ∈ I α ij ( z ) y i y j , (2.1)where the set of indices I is defined by I = { ( i, j ) ∈ N : i ( N + 1) + j ( M + 1) < ( N + 1)( M + 1) } , (2.2)and α ij ( z ), ( i, j ) ∈ I ∪ { ( M + 1 , , (0 , N + 1) } , are analytic functions in somecommon domain Ω ⊂ C . The Hamiltonian equations are given by y ′ =( N + 1) α ,N +1 ( z ) y N + X ( i,j ) ∈ I jα ij ( z ) y i y j − y ′ = − ( M + 1) α M +1 , ( z ) y M − X ( i,j ) ∈ I iα ij ( z ) y i − y j . (2.3) Thomas KeckerThe set I is chosen so that y N and y M will turn out to be the dominantterms on the right hand sides of the system (2.3) in the vicinity of anysingularity z ∞ for which α M +1 , ( z ∞ ) , α ,N +1 ( z ∞ ) = 0. We define the setΦ = { z ∈ Ω | α M +1 , ( z ) = 0 } ∪ { z ∈ Ω | α ,N +1 ( z ) = 0 } . A singularity z ∞ of some solution ( y ( z ) , y ( z )) of the system (2.3) is called fixed if z ∞ ∈ Φ.A singularity z ∞ / ∈ Φ of a solution of (2.3) is called movable . Intuitively,the position of a movable singularity changes when the initial conditions ofthe system of differential equations are varied, whereas the fixed singularitiesare determined by the equation itself. A more general definition of fixed andmovable singularities for first-order systems of differential equations can befound in [11], see also [9] for the case of second-order differential equations.To determine possible leading order behaviours of a solution ( y , y ) of(2.7) about a movable singularity z ∞ suppose that y ∼ c p ( z − z ∞ ) p , y ∼ c q ( z − z ∞ ) q . Assuming that the leading order terms of the right hand side of (2.3) are y N and y M , respectively, we must have p − N q, q − M p = ⇒ p = − N + 1 M N − , q = − M + 1 M N − . In order for all solutions to have movable algebraic branch points a necessarycondition is the existence of certain formal algebraic series solutions of (2.3), y ( z ) = ∞ X k = − N − c ,k ( z − z ) kMN − , y ( z ) = ∞ X k = − M − c ,k ( z − z ) kMN − , (2.4)about any point z ∈ Ω \ Φ. The main result of this article is that the ex-istence of a certain number of such formal series solutions is also sufficientfor every movable singularity of a solution of (2.3) to be of this form. Thisresult should be compared to the fact that for an ODE, passing the Painlev´etest is not equivalent to having the Painlev´e property. The existence of theseries solutions is equivalent to a number of differential relations betweenthe coefficient functions α ij ( z ), ( i, j ) ∈ I , of the Hamiltonian H , known asresonance conditions, which can be calculated algorithmically. They arisefrom the fact that if one inserts the series (2.4) into the system (2.3) andtries to recursively determine the coefficients c ,k , k = − N − , − N, . . . and c ,k , k = − M − , − M, . . . , the recursion breaks down at certain stages knownas resonances and one is left with identities that need to be satisfied identi-cally, leaving one coefficient at the resonance arbitrary.olynomial Hamiltonian Systems 5
Theorem 1.
Suppose that at every point z ∈ Ω \ Φ the Hamiltonian system(2.3) admits formal series solutions of the form (2.4) for every pair of values ( c , − N − , c , − M − ) satisfying c MN − , − N − = − (cid:0) α ,N +1 ( z ) α M +1 , ( z ) N ( M N − N +1 (cid:1) − ,c , − M − = ( M N − α M +1 , ( z ) c M , − N − . Let γ ⊂ Ω be a finite length curve with endpoint z ∞ ∈ Ω \ Φ such that asolution ( y , y ) can be analytically continued along γ up to, but not including z ∞ . Then the solution is represented by series (2.4) at z = z ∞ , y ( z ) = ∞ X k = − N +1 d C ,k ( z − z ∞ ) kdMN − ,y ( z ) = ∞ X k = − M +1 d C ,k ( z − z ∞ ) kdMN − , (2.5) where d = gcd { M + 1 , N + 1 , M N − } , convergent in some punctured,branched, neighbourhood of z ∞ . Remark 1.
As mentioned above, the existence of the formal series solutions(2.4) is an assumption on the form of the equations, each formal series beingequivalent to a differential relation between the coefficients α ij ( z ) . Whereasthe existence of the formal series is clearly necessary for the solution to berepresented by (2.5) near a movable singularity, the theorem states that everymovable singularity is of this form. If ( N + 1) ∤ ( M N − , i.e. d = 1 , thenthere is really only one leading order behaviour for the solution (2.5) as thechoice of branch for c , − N − can be absorbed into the choice of branch for ( z − z ∞ ) / ( MN − . In general there will be d possible leading order behaviours. We assume in the following and for the rest of the article that N ≥ M .In the neighbourhood of any movable singularity one can let˜ y ( z ) = (cid:0) α M +1 , ( z ) N α ,N +1 ( z ) (cid:1) MN − (cid:18) y ( z ) + α M, z ) α M +1 , ( z ) (cid:19) , ˜ y ( z ) = (cid:0) α M +1 , ( z ) α ,N +1 ( z ) M (cid:1) MN − (cid:18) y ( z ) + α ,N ( z ) α ,N +1 ( z ) (cid:19) , to achieve that the transformed Hamiltonian ˜ H is of the same form as in (2.1)but with ˜ α M +1 ≡ ≡ ˜ α ,N +1 and ˜ α N ≡ α M ≡ N = M ). Thomas KeckerIn the following we will assume that the Hamiltonian is already given in thisnormalised form and readily omit the tildes again, H ( z, y , y ) = y M +11 + y N +12 + X ( i,j ) ∈ I ′ α ij ( z ) y i y j , (2.6)where I ′ = I \ { (0 , N ) } , the Hamiltonian equations being y ′ =( N + 1) y N + X ( i,j ) ∈ I ′ jα ij ( z ) y i y j − ,y ′ = − ( M + 1) y M − X ( i,j ) ∈ I ′ iα ij ( z ) y i − y j . (2.7)For N ≥ M , condition (2.2) in fact implies that j ≤ N − i, j ) ∈ I ′ . We will make repeated use of the following lemma by Painlev´e, see e.g. [6].
Lemma 1.
Let F k ( z, y , . . . , y m ) , k = 1 , . . . , m , be analytic functions in aneighbourhood of a point ( z ∞ , η , . . . , η m ) ∈ C m +1 . Let γ be a curve with endpoint z ∞ and suppose that ( y , . . . , y m ) are analytic on γ \ { z ∞ } and satisfy y ′ k = F k ( z, y , . . . , y m ) , k = 1 , . . . , m. Suppose there is a sequence ( z n ) n ∈ N ⊂ γ such that z n → z ∞ and y k ( z n ) → η k ∈ C as n → ∞ for all k = 1 , . . . , n .Then the solution can be analytically continued to include the point z ∞ .Proof. We can choose some r such that all the functions F k , k = 1 , . . . , m are analytic in the set D = {| z − z ∞ | ≤ r, | y k − η k | ≤ r, k = 1 , . . . , m } andlet M = max {| F k ( z, y , . . . , y m ) | : ( z, y , . . . , y m ) ∈ D, k = 1 , . . . , m } . Fromsome n onwards, {| z − z n | < r/ , | y k − y k ( z n ) | < r/ , k = 1 , . . . , m } ⊂ D .By Cauchy’s local existence and uniqueness theorem, a solution around z n isdefined at least in the disc of radius ρ = r (cid:16) − e − m +1) M (cid:17) . For some n wehave z ∞ ∈ B ( z n , ρ ).The next lemma is needed to show that an auxiliary function W , which willbe constructed from the Hamiltonian H in section 5, is bounded along γ . W will be shown to satisfy a first-order linear differential equation of the form(3.1) below. The lemma converts this into an integral representation for W .olynomial Hamiltonian Systems 7 Lemma 2.
Let γ be a finite length curve in the complex plane and let P ( z ) , Q ( z ) and R ( z ) be bounded functions on γ . Then any solution of the equation W ′ = P W + Q + R ′ , (3.1) is also bounded on γ .Proof. Choosing a point z ∈ γ the solution can be written as W ( z ) = R ( z ) + I ( z ) (cid:18) C + Z zz ( Q ( ζ ) + P ( ζ ) R ( ζ )) I ( ζ ) − dζ (cid:19) , where C = W ( z ) − R ( z ) is an integration constant and I is the integratingfactor I ( z ) = exp (cid:18)Z zz P ( ζ ) dζ (cid:19) . Since P , Q and R are bounded on γ and γ has finite length, I ( z ) and I ( z ) − are bounded and hence W ( z ) itself is bounded on γ . In this section we will show that the curve γ leading up to a singularitycan be modified to a curve ˜ γ , still of finite length, such that it avoids thezeros of a solution ( y , y ) of (2.3). This is a technical necessity to showthat the auxiliary function W , to be constructed in section 5, is boundedon γ . The proof runs along the lines of a lemma by S. Shimomura [18] inwhich he showed that for a solution y ( z ) of a second order ODE of the form y ′′ = E ( z, y )( y ′ ) + F ( z, y ) y ′ + G ( z, y ), by modifying a curve γ ending in asingularity one can achieve that y is bounded away on ˜ γ from some fixedvalue c for which the equation is non-singular.Consider a differential system of two equations in y and y of the form y ′ = F ( z, y , y ) y ′ = F ( z, y , y ) (4.1)where F , F ∈ O D [ y , y ] are polynomials in y , y with coefficients analyticin some domain D which we take to be a disc D = { z ∈ C : | z − a | ≤ R } . Thomas KeckerWe assume that F , F are of the form F ( z, y , y ) = α N y N + M X j =0 N − X k =0 α jk ( z ) y j y kk ,F ( z, y , y ) = α M y M + M − X j =0 N X k =0 α jk ( z ) y j y kk , (4.2)where N ≥ N , M ≥ M and α N , α M are constants with | α N | ≥ | α M | ≥
1. Let
K > | α ijk ( z ) | < K for all i, j, k and z ∈ D . Also, let N := N, M := M and C := 2 N +1 ( M + 1)( N + 1) K . Lemma 3.
Let < ∆ < and θ := min { ∆ C , R } . Let ( y , y ) be a solutionof (4.1) analytic at a point c for which | c − a | < R . Suppose that | y ( c ) | < θ and | y ( c ) | > C . Then ( y ( z ) , y ( z )) is analytic on the disc | z − c | < θ | y ( c ) | and satisfies | y ( z ) | ≥ θ and | y ( z ) | ≥ on the circle | z − c | = θ | y ( c ) | .Proof. Let ρ = y ( c ) N , ζ = ρ ( z − c ) and define η i ( ζ ) := y i ( z ), i = 1 , ζ by a dot we have ˙ η i ( ζ ) = ρ − y ′ i ( z )and η i ( ζ ) = η i (0) + Z ζ ˙ η i ( ˜ ζ ) d ˜ ζ, where η i (0) = y i ( c ). Define the functions M i ( r ) = max | ζ |≤ r | η i ( ζ ) | , i = 1 , r = sup { r : M ( r ) < ∆ , M ( r ) < | ρ | /N } . Clearly we have r > | ζ | < min { r , R } we have, since | z − a | ≤ | z − c | + | c − a | < R | ρ | + R ≤ R , | η i ( ζ ) | ≤ | y i ( c ) | + | ρ | − | ζ | M i X j =0 N i X k =0 K ∆ j k | ρ | kN ≤ | y i ( c ) | + | ζ | N K ( N +1)( M +1) . (4.3)Now suppose that r < θ . Then, for | ζ | < r < R we have the estimates | η ( ζ ) | <θ (1 / N ( N + 1)( M + 1) K ) < ∆ , | η ( ζ ) | < | y ( c ) | + θ N ( M + 1)( N + 1) K < | y ( c ) | , in contradiction to the definition of r . Therefore we must have r ≥ θ ,showing that (4.3), i = 1 ,
2, is valid for | ζ | < θ and therefore that η andolynomial Hamiltonian Systems 9 η are analytic for | ζ | < θ . We now obtain estimates for η and η in theopposite direction on the circle | ζ | = θ : | η ( ζ ) | ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z ζ ρ − α N η ( ˜ ζ ) N d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ ρ − M X i =0 N − X j =0 α ij ( z ) η i η j d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | η (0) |≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ η ( ˜ ζ ) − η (0) η (0) ! N d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ | ρ | − N N − ( M + 1) N K − θ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ N X n =1 (cid:18) Nn (cid:19) η ( ˜ ζ ) − η (0) η (0) ! n ! d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ ≥ θ − θ N X n =1 (cid:18) Nn (cid:19) (cid:18) ∆ C (cid:19) n − θ ≥ θ , | η ( ζ ) | ≥| y ( c ) | − θ N ( M + 1)( N + 1) K ≥ . Remark 2.
In Lemma 3 the role of y and y can be interchanged if in everyexpression one simultaneously replaces M ↔ N . Using Lemma 3 and Remark 2 we can now show that a curve ending in amovable singularity of a solution ( y , y ) of the system (4.1) can be modifiedby arcs of circles in such a way that both y and y are bounded away from0 on the modified curve. The argument is very similar to the one in [18]. Lemma 4 (1st curve modification) . Suppose ( y , y ) is a solution of (4.1),analytic on a finite length curve γ ⊂ D up to, but not including its endpoint z ∞ ∈ D . Then we can deform γ , if necessary, in the region where ( y , y ) isanalytic, to a curve ˜ γ , still of finite length, such that y and y are boundedaway from on ˜ γ in a neighbourhood of z ∞ .Proof. Let γ be parametrised by arclength such that γ (0) = z , γ ( l ) = z ∞ where l is the length of γ . Define the two sets S i := { s : 0 < s < l and | y i ( γ ( s )) | ≤ θ/ } , i = 1 , . s → l − min {| y | , | y |} = 0, otherwise there is nothingto show. Therefore the union S ∪ S contains values arbitrarily close to l .There now exists some number 0 < s < l with the following two properties:(i) S ∩ S ∩ [ s , l ) = ∅ , (ii) whenever s ∈ S i , s > s , we have | y − i ( γ ( s )) | > C .Namely, if this was not the case we could find a sequence z i = γ ( s i ), s i → l ,such that ( y ( z i ) , y ( z i )) is bounded and hence, by Lemma 1, the solutioncould be analytically continued to z ∞ in contradiction to the assumption.Denote S = ( S ∪ S ) ∩ [ s , l ) and let s = inf { s ∈ S : s > s } . Suppose that s ∈ S i and let r = θ | y − i ( γ ( s )) | . Lemma 3 now shows that that y and y are analytic for | z − γ ( s ) | < r and that | y i ( z ) | ≥ θ/ | y − i ( z ) | ≥ C = { z : | z − γ ( s ) | = r } . We now recursively define a sequenceof points s n and circles C n with radii r n as follows: Let s n +1 = inf { s ∈ S : s > s n + r n } . If s n +1 ∈ S i ( i = 1 or 2), then let r n +1 = θ | y − i ( γ ( s n )) | .By Lemma 3, for every circle C n , n = 1 , , . . . , we have | y ( z ) | , | y ( z ) | ≥ θ for all z ∈ C n . Also, P ∞ n =1 r n ≤ P ∞ n =1 | s n +1 − s n | ≤ l which implies r n → n → ∞ . The centres s n of the circles accumulate at z ∞ : If this was notthe case we would have s n → s ∞ for some s ∞ < l , but thenlim n →∞ max {| y ( γ ( s n ) | , | y ( γ ( s n ) |} ≥ lim n →∞ θ r n = ∞ , in contradiction to the fact that ( y ( z ) , y ( z )) is analytic on γ \ { z ∞ } . Wenow define ˜ γ in the following way. Suppose for convenience that γ has no self-intersections (otherwise we could shorten γ by omitting pieces between self-intersections). Let γ ext be an infinite non-intersecting extension of γ such that γ ext ( s ) → ∞ for s → ±∞ which divides the complex plane into parts C + and C − such that C + , γ ext and C − are pairwise disjoint and C + ∪ γ ext ∪ C − = C .Now let D = γ ∪ S ∞ n =1 D n where D n = { z : | z − γ ( s n ) | ≤ r n } and define˜ γ = ∂D ∩ ( C + ∪ γ ext ). Then ( y , y ) is analytic on ˜ γ and | y ( z ) | , | y ( z ) | ≥ θ for all z ∈ ˜ γ . Furthermore, ˜ γ has length less than (1 + 2 π ) l .We will now specialise the results obtained so far in this section to theHamiltonian system (2.7) which is of the form (4.1) with N = N , M = M .Lemma 4 is not quite enough to show that the auxiliary function W in section5, rational in y and y , is bounded. We need to show that certain terms ofthe form y k y l are bounded. To do so we will apply a second curve modificationwhere we can now make use of the fact that y and y are already boundedaway from 0 on γ . We rewrite the system of equations (2.7) in the variables u = y · y − N +1 M +1 and u = y for some branch of y M +1 .olynomial Hamiltonian Systems 11The system of equations in the variables u , u becomes u ′ =( N + 1) u N − N +1 M +1 (cid:0) u M +11 (cid:1) + X ( i,j ) ∈ I ′ (cid:18) j + i N + 1 M + 1 (cid:19) α ij u i u ( i − N +1 M +1 + j − u ′ = − ( M + 1) u M u M N +1 M +1 − X ( i,j ) ∈ I ′ iα ij u i − u ( i − N +1 M +1 + j . (4.4)Let K > | iα ij ( z ) | < K and (cid:12)(cid:12)(cid:0) j + i N +1 M +1 (cid:1) α ij ( z ) (cid:12)(cid:12) < K for all ( i, j ) ∈ ˜ I = I ′ ∪ { ( M + 1 , , (0 , N + 1) } , z ∈ D . As before let C = 2 N +1 K ( M + 1)( N + 1). Suppose ( u ( z ) , u ( z )) is a solution of (4.4),corresponding to a solution ( y ( z ) , y ( z )) of (2.7) on a curve γ , which byLemma 4 we assume to be such that y and y = u are bounded away from0 on γ . The following Lemma is somewhat similar to Lemma 3, the proof,however, requires some modifications. Lemma 5.
Let < ∆ < − N − ( N + 1) − < and θ := min { ∆ C , R } . Let ( u , u ) be a solution of (4.4) analytic at c with | c − a | ≤ R and supposethat | u ( c ) | < θ and | u ( c ) | > (4 C ) M +1 . Then ( u ( z ) , u ( z )) is analytic inthe disc | z − c | < θ | u ( c ) | and on the circle | z − c | = θ | u ( c ) | we have | u ( c ) | ≥ θ and | u ( c ) | ≥ .Proof. Let ρ = u ( c ) L , where L = N − N +1 M +1 ≤ N −
1. For i = 1 , η i ( ζ ) := u i ( z ), where ζ = ρ ( z − c ), and define M i ( r ) = max | ζ |≤ r | η i ( ζ ) | , m i ( r ) = min | ζ |≤ r | η i ( ζ ) | . Let r = sup (cid:26) r : M ( r ) < ∆ , M ( r ) < | ρ | /L , m ( r ) > | ρ | /L (cid:27) , (4.5)which is positive as | η (0) | < ∆ and | η (0) | = | ρ | /L . We have η i ( ζ ) = η i (0) + Z ζ ˙ η i ( ζ ) dζ , where η i (0) = u i ( c ) and ˙ η i ( ζ ) = ρ − u ′ i ( z ). For | ζ | < min { r , R } we have,since | z − a | ≤ | z − c | + | c − a | < R | ρ | + R < R , | η ( ζ ) | ≤| u ( c ) | + | ρ | − | ζ | X ( i,j ) ∈ ˜ I \{ (0 , } K ∆ i | ( i − N +1 M +1 + j − | | ρ | (( i − N +1 M +1 + j − /L ≤| u ( c ) | + | ζ | N K ( M + 1)( N + 1) , (4.6)2 Thomas Kecker | η ( ζ ) | ≤| u ( c ) | + | ρ | − | ζ | X ( i,j ) ∈ ˜ I i =0 K ∆ i − | ( i − N +1 M +1 + j | | ρ | (( i − N +1 M +1 + j ) /L ≤| u ( c ) | (cid:0) | ζ | N K ( M + 1)( N + 1) (cid:1) , | η ( ζ ) | ≥| u ( c ) | (cid:0) − | ζ | N K ( M + 1)( N + 1) (cid:1) , (4.7)where we have used condition (2.2) which implies ( i − N +1 M +1 + j − ≤ L for( i, j ) ∈ ˜ I \ { (0 , } and therefore (cid:12)(cid:12) ( i − N +1 M +1 + j − (cid:12)(cid:12) ≤ N . Now supposingthat r < θ one would obtain the estimates | η ( ζ ) | ≤ θ (1 / N K ( M + 1)( N + 1)) < ∆ , | η ( ζ ) | ≤| u ( c ) | (cid:0) θ N K ( M + 1)( N + 1) (cid:1) < | ρ | /L , | η ( ζ ) | ≥| u ( c ) | (cid:0) − θ N K ( M + 1)( N + 1) (cid:1) > | ρ | /L , in contradiction to the definition (4.5) of r . Therefore we must have r ≥ θ ,implying that the estimates (4.6), (4.7) are valid for | ζ | < θ and that u , u are analytic for | ζ | < θ . On the circle | ζ | = θ we now have | η ( ζ ) | ≥ ( N + 1) (cid:12)(cid:12)(cid:12)(cid:12)Z ζ ρ − η ( ˜ ζ ) L d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)Z ζ ρ − ( N + 1) η M +11 η N − N +1 M +1 d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ ρ − X ( i,j ) ∈ I ′ (cid:18) j + i N + 1 M + 1 (cid:19) α ij η i η ( i − N +1 M +1 + j − d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | η (0) |≥ ( N + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ η ( ˜ ζ ) − η (0) η (0) ! L d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ N + 1)∆ M +1 L − θ | ρ | − L ( M +1) N K ( M + 1)( N + 1) − θ ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z ζ d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ζ η ( ˜ ζ ) − η (0) η (0) ! L − d ˜ ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ ≥ θ − θ N X n =1 (cid:18) Nn (cid:19) (cid:18) ∆4 C (cid:19) n ≥ θ , | η ( ζ ) | ≥ | ρ | /L > . olynomial Hamiltonian Systems 13 Lemma 6 (2nd curve modification) . Let ( y , y ) be a solution of the system(2.7), analytic on the finite length curve γ ending in a movable singularity z ∞ , such that y and y are bounded on γ . Then, after a possible deformationof γ in the region where y , y are analytic, one can achieve that y k y l is boundedon ˜ γ for all k, l ≥ for which l ( N + 1) − k ( M + 1) ≥ .Proof. Define the set S = { s : 0 < s < l and | u ( γ ( s )) | ≤ θ/ } . There existssome s , 0 < s < l , such that on S ∩ [ s , l ] one has | u ( z ) | > (4 C ) M +1 . For,if this was not the case, one would have a sequence of points ( z n ) on γ with z n → z ∞ as n → ∞ such that u ( z n ) is bounded and u ( z n ) is bounded andbounded away from zero. Lemma 1 applied to the system (4.4) would thenimply that u , u are analytic at z ∞ in contradiction to the assumption. Bythe same method as in the proof of Lemma 4 one can now deform the curve γ by arcs of circles such that u and u are bounded away from 0 on themodified curve ˜ γ , that is, u − ( M +1)1 = y N +12 y M +11 and u − = y are bounded on ˜ γ .By writing y k y l = (cid:18) y N +12 y M +11 (cid:19) l · y l ( N +1) − k ( M +1)2 ! / ( M +1) , one can conclude that y k y l is bounded on ˜ γ if l ( N + 1) − k ( M + 1) ≥ In this section we will show the existence of a function W that remainsbounded whenever a solution ( y ( z ) , y ( z )) develops a movable singularityby analytic continuation along a finite length curve. Formally inserting theseries expansions (2.4) for y and y into H ′ = dHdz = ∂H∂z = X ( i,j ) ∈ I ′ α ′ ij ( z ) y ( z ) i y ( z ) j , (5.1)yields a formal series expansion for H ′ in ( z − z ) MN − . Heuristically, W isconstructed from H by adding certain terms, rational in y and y , whichwould cancel all terms of H ′ with negative powers of ( z − z ) MN − . Note,however, that terms of power ( z − z ) − cannot be cancelled in this way, since4 Thomas Keckerthese would correspond terms of H that are logarithmic in z − z and cannotbe obtained by rational expressions in y and y . We define W ( z, y , y ) = y M +11 + y N +12 + X ( i,j ) ∈ I ′ α ij ( z ) y i y j + X ( k,l ) ∈ J β kl ( z ) y k y l , (5.2)where the β kl ( z ) are certain analytic functions to be determined in terms ofthe α ij ( z ) and their derivatives, and the index set J is given by J = { ( k, l ) ∈ N : 1 ≤ k ≤ N +1 , − M N < k ( M +1) − l ( N +1) < M + N +2 } . Note that the pairs of indices in the set J are in one-to-one correspondencewith the elements of the set I \ { (0 , } , which can easily be seen by setting k = j + 1 and l = M − i . Thus for every unbounded term α ′ ij ( z ) y i y j in (5.1)there is one function β kl to compensate for. However, we will see that not allthe functions β kl can be used. The other essential ingredient is the existenceof the formal series solutions (2.4), which will ensure that the terms of power( z − z ) − vanish identically. We will now show formally that W is bounded. Lemma 7.
The coefficients β kl ( z ) , ( k, l ) ∈ J , in (5.2) can be chosen suchthat the function W is bounded on the curve ˜ γ .Proof. Taking the total z -derivative of (5.2) one obtains W ′ = X ( i,j ) ∈ I ′ α ′ ij y i y j + X ( k,l ) ∈ J (cid:18) β ′ kl y k y l + kβ kl y k − y ′ y l − lβ kl y k y ′ y l +11 (cid:19) = X ( i,j ) ∈ I ′ α ′ ij y i y j − X ( i,j ) ∈ I ′ X ( k,l ) ∈ J ( ik + jl ) α ij β kl y i − l − y k + j − + X ( k,l ) ∈ J (cid:18) β ′ kl y k y l − k ( M + 1) β kl y M − l y k − − l ( N + 1) β kl y N + k y l +11 (cid:19) = X ( i,j ) ∈ I ′ α ′ ij y i y j + X ( k,l ) ∈ J ( l ( N + 1) − k ( M + 1)) β kl y M − l y k − + X ( k,l ) ∈ J (cid:18) β ′ kl y k y l − l ( N + 1) β kl y k − y l +11 W (cid:19) + X ( i,j ) ∈ I ′ X ( k,l ) ∈ J ( l ( N − j + 1) − ik ) α ij β kl y i − l − y k + j − + X ( k,l ) ∈ J X ( k ′ ,l ′ ) ∈ J l ( N + 1) β kl β k ′ l ′ y k + k ′ − y l + l ′ +11 , (5.3)olynomial Hamiltonian Systems 15where we have used (5.2). All terms in (5.3) are now either of the form y i y j with ( i , j ) ∈ I , or of the form y j y i with i ≥ j ( M + 1) − i ( N + 1) < ( M + 1)( N + 1). Note also that for the coefficients y k − y l +11 of W , ( k, l ) ∈ J , wehave ( l + 1)( N + 1) − ( k − M + 1) ≥
0, i.e. by Lemma 6 these are boundedon ˜ γ . By repeating the process of replacing powers y N +12 using (5.2) one canachieve in a finite number of steps that the terms of the form y j y i either have j ≥ N + 1 with i ( N + 1) − j ( M + 1) ≥ j ≤ N and j ( M + 1) − i ( N + 1) ≤ M N −
1, equalityholding if and only if ( i , j ) = (1 , N ). We now manipulate the terms of theform y j y i , j ≤ N , in the following way( M +1)( j + 1) y j y i = − ( j + 1) y ′ y j y M + i − X ( i,j ) ∈ I ′ i ( j + 1) α ij y j + j y M − i + i +11 = − y j +12 y M + i ! ′ − ( M + i ) y j +12 y ′ y M + i +11 − X ( i,j ) ∈ I ′ i ( j + 1) α ij y j + j y M − i + i +11 = − y j +12 y M + i ! ′ − ( N + 1)( M + i ) y N + j +12 y M + i +11 − X ( i,j ) ∈ I ′ ( i ( j + 1) + j ( M + i )) α ij y j + j y M − i + i +11 = − y j +12 y M + i ! ′ − ( N + 1)( M + i ) y j y M + i +11 W + X ( i,j ) ∈ I ′ (( N + 1)( M + i ) − j ( M + i ) − i ( j + 1)) α ij y j + j y M − i + i +11 + X ( k,l ) ∈ J ( N + 1)( M + i ) β kl y k + j y M + l + i +11 + ( N + 1)( M + i ) y j y i . (5.4)6 Thomas KeckerThus, unless j ( M + 1) − i ( N + 1) = M N −
1, one can solve (5.4) for y j y i : y j y i = 1 M N − i ( N + 1) − j ( M + 1) ( N + 1)( M + i ) y j y M + i +11 W + X ( i,j ) ∈ I ′ ( i ( j + 1) + j ( M + i ) − ( N + 1)( M + i )) α ij y j + j y M − i + i +11 − X ( k,l ) ∈ J ( N + 1)( M + i ) β kl y k + j y M + l + i +11 + y j +12 y M + i ! ′ . (5.5)Again, in (5.5) the coefficient y j y M + i of W is bounded by Lemma 6 since wehave ( M + i + 1)( N + 1) − j ( M + 1) >
0. Also, the term y j y M + i is boundedby Lemma 6 since ( M + i )( N + 1) − ( j + 1)( M + 1) >
0. Therefore, theterm (cid:16) y j y M + i (cid:17) ′ is bounded when integrated over the finite length curve ˜ γ . Forthe terms of type y k + j y M + l + i , ( k, l ) ∈ J , we find( M + l + i + 1)( N + 1) − ( k + j )( M + 1) ≥ , which are therefore all bounded, and for the terms y j + j y M − i + i , ( i, j ) ∈ I ′ ,( j + j )( M + 1) − ( M − i + i + 1)( N + 1) < j ( M + 1) − i ( N + 1) . We can thus replace y j y i by terms which are bounded or proportional to W with bounded factor, and a sum of terms of the form y j y i with j = j + j , i = M − i + i + 1, such that the quantity j ( M + 1) − i ( N + 1) is strictlydecreasing. Performing this process iteratively a finite number of times weeventually end up only with terms y jn y in for which j n ( M + 1) − i n ( N + 1) ≤ γ .We thus arrive at a first-order differential equation for W of the form W ′ = P ( z, y − , y ) W + X ( i,j ) ∈ I γ ij ( z ) y i y j + γ − N ( z ) y N y + Q ( z, y − , y ) + ddz R ( z, y − , y ) , olynomial Hamiltonian Systems 17where P , Q and R are polynomial in their last two arguments and for eachmonomial y k y l we have l ( N + 1) − k ( M + 1) ≥
0, i.e. they are bounded on ˜ γ .We will now show that, by a suitable choice of the β kl and the existence ofthe formal series solutions (2.4), all the coefficients γ ij , ( i, j ) ∈ I , as well as γ − N , are identically 0.We determine the functions β kl = β j +1 ,M − i recursively starting with thepairs ( i, j ) ∈ I for which the quantity i ( N + 1) + j ( M + 1) is maximal. From(5.3) we see that γ ij ( z ) = α ′ ij ( z ) + ( M N − − i ( N + 1) − j ( M + 1)) β j +1 ,M − i ( z ) + · · · , (5.6)where the dots stand for expressions involving only terms β k ′ l ′ = β j ′ +1 ,M − i ′ for which i ′ ( N + 1) + j ′ ( M + 1) is strictly greater than i ( N + 1) + j ( M + 1).We can thus determine β kl = β j +1 ,M − i for all pairs ( i, j ) ∈ I for which i ( N +1)+ j ( M +1) > M N −
1. However, when i ( N +1)+ j ( M +1) = M N − β j +1 ,M − i in (5.6) vanishes. We now make use of the existenceof the formal series solutions (2.4) to show that also γ ij ≡ n = N +1 d and m = M +1 d where d = gcd { M + 1 , N + 1 } . Consider the d terms γ − ,N ( z ) y N y , γ m − ,N − n ( z ) y m − y N − n , . . . , γ M − m,n − ( z ) y M − m y n − . Whenone inserts the formal series solutions (2.4) into these expressions they haveleading order ( z − z ) − . But, as explained in Remark 1, there are essentially d formal series solutions corresponding to the different choices of the leadingcoefficients c , − N − , c , − M − such that c MN − , − N − = − MN − N +1 . Inserting anyof the series into (5.2) shows that W has a Laurent series expansion in powersof ( z − z ) / ( MN − . Therefore, the coefficient of ( z − z ) − in W ′ vanishes sinceotherwise W would have logarithmic terms in its expansion. The coefficientsof ( z − z ) − in W ′ , for the different choices of ( c , − N − , c , − M − ), are − M N − (cid:0) γ − ,N ( z ) + ω γ m − ,N − n ( z ) + · · · + ω d − γ M − m,n − ( z ) (cid:1) =0 − M N − (cid:0) γ − ,N ( z ) + ω γ m − ,N − n ( z ) + · · · + ω d − γ M − m,n − ( z ) (cid:1) =0... − M N − (cid:0) γ − ,N ( z ) + ω d γ m − ,N − n ( z ) + · · · + ω d − d γ M − m,n − ( z ) (cid:1) =0 , where ω i , i = 1 , . . . , d , are the d distinct roots of ω d = −
1. This system of d equations shows γ − ,N ( z ) = γ m − ,N − n ( z ) = · · · = γ M − m,n − ( z ) = 0 . z in a neighbourhood of z . Therefore we have shown in fact that γ − ,N = γ m − ,N − n = · · · = γ M − m,n − ≡ . The functions β j +1 ,M − i with i ( N + 1) + j ( M + 1) = M N − β j +1 ,M − i with i ( N + 1) + j ( M + 1) < M N − γ ij ≡ i, j ) ∈ I ∪ { ( − , N ) } . We have thus arrived at afirst-order linear differential equation for W of the form W ′ = P ( z, y − , y ) W + Q ( z, y − , y ) + R ′ ( z, y − , y ) , (5.7)where P , Q and R are bounded on ˜ γ near a movable singularity z of asolution ( y ( z ) , y ( z )). Lemma 2 now shows that W is bounded on ˜ γ . To show that a movable singularity is an algebraic branch point we will nowintroduce coordinates u and v for which there exists a regular initial valueproblem. The coordinate u is defined by y = u − N +1 d , (6.1)where a choice of branch is made. We also define w = y u M +1 d . (6.2)From (5.2) one obtains an algebraic equation for w ,0 = w N +1 + X ( i,j ) ∈ I ′ α ij ( z ) u ( M +1)( N +1) − i ( N +1) − j ( M +1) d w j + X ( k,l ) ∈ J β kl ( z ) u ( M +1)( N +1)+ l ( N +1) − k ( M +1) d w k + 1 − W u ( M +1)( N +1) d , (6.3)all the exponents of u being positive integers. The solutions of this equationfor w will be denoted by w , . . . , w N +1 . They are analytic functions of u , z and W in some neighbourhood of u = 0, z = z ∞ and W = W for anyolynomial Hamiltonian Systems 19 W ∈ C . We express the w n as power series in u and W with analyticcoefficients in z , w n = F n ( z, u, W ) = ω n ∞ X j,k =0 a jkn ( z ) u j W k , where ω n , n = 1 , . . . , N + 1, are the distinct roots of ω N +1 = − a n ≡ W is of the form − N +1 u ( M +1)( N +1) d W . Wedenote ¯ F n ( z, u ) = P ( M +1)( N +1) d j =0 a j n ( z ) u j and define functions v n by w n = ω n (cid:18) ¯ F n ( z, u ) − N + 1 u ( M +1)( N +1) d v n (cid:19) , (6.4)so that in the limit u → v n agrees to leading order with W . From thedefiniton (6.2) of w we see that the choice of branch for ω n can partiallybe absorbed into the original choice of branch for u if 1 < d < M + 1,and completely be absorbed if d = 1, so that there are essentially only d inequivalent choices for ( u, v n ). From (6.1) and (2.7) we obtain the differentialequation satisfied by u : u ′ = − dN + 1 u N +1 d +1 " ( N + 1) ω Nn (cid:18) u − M +1 d ¯ F n ( z, u ) − N + 1 u ( M +1) Nd v n (cid:19) N + X ( i,j ) ∈ I ′ jα ij ( z ) u − i N +1 d ω j − n (cid:18) u − M +1 d ¯ F n ( z, u ) − N + 1 u ( M +1) Nd v n (cid:19) j − . (6.5)Taking the reciprocal of (6.5) and changing the role of the dependent andindependent variables u and z one obtains, extracting the highest power of u on the right hand side, an equation of the form, dzdu = u MN − d − A ( u, z, v ) , (6.6)where A ( u, z, v ) is analytic in ( u, z, v ) at (0 , z ∞ , v ) for any v ∈ C , and A (0 , z ∞ , v ) = ω n d . We drop the index n from now on. Reinserting (6.4) into(6.3) yields an expression for W in terms of u and v of the form W = v + G ( z, u, v ) , (6.7)0 Thomas Keckerwhere G is a polynomial in v of degree N + 1 and analyic in z and u near u = 0, satisfying G ( z, , v ) = 0. We differentiate (6.7) with respect to z , W ′ = v ′ + G z + G u u ′ + G v v ′ , (6.8)and compare this with equation (5.7), which can be written in the form W ′ = ˜ P ( z, u, v ) W + ˜ Q ( z, u, v ) + ddz ˜ R ( z, u, v )= ˜ P ( v + G ) + ˜ Q + ˜ R z + ˜ R u u ′ + ˜ R v v ′ , (6.9)where ˜ P , ˜ Q and ˜ R are polynomial in u and v . One can solve (6.8) and (6.9)for v ′ to obtain an equation of the form v ′ = B ( z, u, v ) u ′ + C ( z, u, v ) , (6.10)where B and C are analytic in their arguments. Multiplying (6.10) by (6.6)one obtains and equation for v as function of u : dvdu = dvdz dzdu = B ( z, u, v ) + u MN − d − A ( z, u, v ) C ( z, u, v ) . (6.11)Equations (6.6) and (6.11) together form a regular initial value problem for z and v as functions of u near u = 0 with z (0) = z ∞ and v (0) = v . We can now complete the proof of Theorem 1.
Proof.
By Lemma 7, after a possible modification of γ , the auxiliary function W is bounded along γ . Consider a sequence ( z n ) ⊂ γ such that z n → z ∞ as n → ∞ . Suppose that the sequence ( y ( z n )) is bounded. Then the func-tional form of W ( z, y , y ) implies that the sequence ( y ( z n )) is also bounded.However, Lemma 1 now implies that the solution ( y , y ) can be analyticallycontinued to z ∞ , in contradiction to the assumption in the theorem. There-fore, the sequence ( y ( z n )) must tend to infinity since otherwise it would havea bounded subsequence. In the coordinates u, v introduced in the previoussection we therefore have that u ( z n ) → v ( z n ) is bounded. Hence thereexists some subsequence ( z n k ) such that v ( z n k ) → v for some v ∈ C . Equa-tions (6.6) and (6.11) now form a regular initial value problem for z and v asolynomial Hamiltonian Systems 21functions of u with initial values z ∞ and v at u = 0. Lemma 1 then showsthat z and v are analytic at u = 0. Since A (0 , z ∞ , v ) = 0 in (6.6), z has aconvergent power series expansion of the form z = z ∞ + ∞ X k =0 ξ k u k + MN − d , in a neighbourhood of u = 0. Taking the MN − d -th root,( z − z ∞ ) dMN − = ∞ X k =1 η k u k , and inverting the power series, one shows that u has a convergent seriesexpansion u = ∞ X k =1 ζ k ( z − z ∞ ) kdMN − . By the definition (6.1) of u , one obtains a series expansion for y , y ( z ) = ∞ X k = − N +1 d C ,k ( z − z ∞ ) kdMN − , convergent in a branched, punctured neighbourhood of z ∞ . Also, from thedefinition (6.2) we find, since w = 0 at z = z ∞ , y ( z ) = ∞ X k = − M +1 d C ,k ( z − z ∞ ) kdMN − . If M = 1 the Hamiltonian (2.6) can essentially be reduced to the form H ( z, y , y ) = 12 y + P ( z, y ) . The Hamiltonian system thus corresponds to the second-order differentialequation y ′′ = P y ( z, y ), the case of which was treated in [2]. This caseincludes the Painlev´e equations P I (for N = 2) and P II (for N = 3). For N ≥ M, N ≥ M = N = 2 The Hamiltonian here is of the form H ( z, y , y ) = 13 y + 13 y + α ( z ) y y + β ( z ) y + γ ( z ) y , where we have chosen a slightly different normalisation than the one in (2.6).The resonance conditions in this case are α ′′ ≡ β ′ ≡ γ ′ ≡
0. One istherefore essentially left with H ( z, y , y ) = 13 y + 13 y + zy y + βy + γy , the corresponding system of differential equations being y ′ = y + zy + γy ′ = − y − zy − β. (8.1)About any movable singularity z ∞ a solution is represented by y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , with C , − = − C , − = C , − , i.e. there are three possible leadingorder behaviours about any movable singularity which in this case are simplepoles. Theorem 1 in this case states that every local solution ( y , y ) extendsto meromorphic functions in the whole complex plane, i.e. the system (8.1)has the Painlev´e property. It is therefore of interest how its solutions canbe expressed in terms of the six Painlev´e transcendents. Therefore we let y = y and eliminate y from (8.1). This yields the following scalar differentialequation of second order and second degree in y , (cid:0) y ′′ + zy ′ − (1 − z ) y − γz (cid:1) = 4 (cid:0) y + β (cid:1) ( y ′ − zy − γ ) . (8.2)A Painlev´e type classification for equations of second order and second degreehas been done by C. Cosgrove and G. Scoufis in [1]. They found six inequiv-alent types of equations in the class ( y ′′ ) = F ( z, y, y ′ ) which they denotedby SD-I – SD-VI. All of these can be solved in terms of the Painlev´e tran-scendents P I – P V I . In fact, equation (8.2) is of the modified form denotedby SD-IV’.A (equation 5 .
87 in [1]), which is solved in terms of P IV .olynomial Hamiltonian Systems 23 M = 2 , N = 3 In this case the normalised Hamiltonian (2.6) is H = y + y + α y y + α y y + α y y + α y + α y + α y + α y . The only resonance condition is (cid:0) α − α (cid:1) ′′ = 0 , and if it satisfied the solutions near a movable singularity z ∞ are given by y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , with C , − = − − , C , − = 5 C , − , where the choice for C , − can completelybe absorbed into the choice of branch for ( z − z ∞ ) . M = N = 3 The normalised Hamiltonian is given by H = y + y + α y y + α y y + α y + α y y + α y + α y + α y . In order for the solutions to have only movable algebraic singularities thefollowing conditions need to be satisfied, (cid:0) α − α (cid:1) ′ = 0 , α ′ = 0 , (cid:0) α − α (cid:1) ′ = 0 . The solutions are given by y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , y ( z ) = ∞ X k = − C ,k ( z − z ∞ ) k , about any movable singularity z ∞ , where C , − = − , C , − = 2 C , − , thechoice for C , − however can only partially be absorbed into the choice ofbranch for ( z − z ∞ ) , i.e. there are 4 possible leading order behaviours ofthe solution near any movable singularity.4 Thomas Kecker For a class of Hamiltonian systems of ordinary differential equations we havefound that the only movable singularities obtained by analytic continuationalong finite length curves are algebraic branched points, in particular thesesingularities are isolated and the solutions are locally finitely branched. Thepossibility of movable singularities obtained by analytic continuation alongan infinite length curve is discussed by R. Smith in [21] for certain second-order differential equations. There it is shown that a singularity of this type isnon-isolated, more specifically it is an accumulation point of algebraic singu-larities, and they cannot be ruled out at this stage for the systems presentedhere. It remains an interesting task to classify the structure of movable sin-gularities for wider classes of differential equations. The author would like toexpress his sincere gratitude to Prof. R. Halburd for his invaluable supportand many interesting discussions.
References [1] C. M. Cosgrove and G. Scoufis,
Painlev´e classification of a class of dif-ferential equations of the second order and second degree , Stud. Appl.Math. (1993), 25–87[2] G. Filipuk and R. G. Halburd, Movable algebraic singularities of second-order ordinary differential equations , J. Math. Phys. (2009), 023509[3] R. Garnier, Sur des syst´emes diff´erentiels du second ordre dont l’int´e-grale g´en´erale est uniforme , Ann. Sci. ´Ecole Norm. Sup. (1960), 123–144[4] J. Goffar-Lombet, Sur des syst´emes polynomiaux d’´equations diff´eren-tielles dont l’int´egrale g´en´erale est ´a points critiques fixes , Acad. Roy.Belg. Cl. Sci. Mem. (1974), 1–76[5] Gromak, V. I. and Laine, I. and Shimomura, S. Painlev´e differentialequations in the complex plane , De Gruyter Studies in Mathematics,Berlin, 2002[6] E. Hille,
Ordinary Differential Equations in the Complex Domain ,Wiley-Interscience, New-York, 1976olynomial Hamiltonian Systems 25[7] A. Hinkkanen and I. Laine,
Solutions of the first and second Painlev´eequations are meromorphic , J. Anal. Math. (1999), 345–77[8] T. Kimura and T. Matuda, On systems of differential equations of ordertwo with fixed branch points , Proc. Japan Acad. Ser. A Math. Sci. (1980), 445–449[9] T. Kimura, Sur les points singuliers essentiels mobiles des ´equationsdiff´erentielles du second ordre , Comment. Math. Univ. St. Paul. (1956), 81–94[10] J. Malmquist, Sur les ´equations diff´erentielles du second ordre, dontl’int´egrale g´en´erale a ses points critiques fixes , Ark. f¨or Mat., Astron.och Fys. (1923), 1–89[11] Y. Murata, On fixed and movable singularities of systems of rationaldifferential equations of order n , J. Fac. Sci. Univ. Tokyo Sect. IA Math. (1988), 439–506[12] K. Okamoto, Studies on the Painlev´e equations. I. Sixth Painlev´e equa-tion P VI , Ann. Mat. Pura Appl. (1987), 337–381[13] K. Okamoto, Studies on the Painlev´e equations. II. Fifth Painlev´e equa-tion P V , Japan. J. Math. (1987), 47–76[14] K. Okamoto, Studies on the Painlev´e equations. III. Second and fourthPainlev´e equations, P II and P IV , Math. Ann. (1986), 221–255[15] K. Okamoto, Studies on the Painlev´e equations. IV. Third Painlev´eequation P III , Funkcial. Ekvac. (1987), 305–332[16] K. Okamoto and K. Takano, The proof of the Painlev´e property by MasuoHukuhara , Funkcial. Ekvac. (2001), 201–217[17] P. Painlev´e, M´emoire sur les ´equations diff´erentielles dont l’int´egraleg´en´erale est uniforme , Bull. Soc. Math. France (1900), 201–261[18] S. Shimomura, Proofs of the Painlev´e property for all Painlev´e equations ,Japan. J. Math. (2003), 159–180[19] S. Shimomura, A class of differential equations of PI-type with the quasi-Painlev´e property , Ann. Mat. Pura Appl. (2007), 267–2806 Thomas Kecker[20] S. Shimomura,
Nonlinear differential equations of second Painlev´e typewith the quasi-Painlev´e property along a rectifiable curve , Tohoku Math.J. (2008), 581–595[21] R. A. Smith, On the singularities in the complex plane of solutions of y ′′ + y ′ f ( y ) + g ( y ) = P ( x ), Proc. London Math. Soc.3