Desingularising Transformations for Complex Differential Equations with Algebraic Singularities
aa r X i v : . [ m a t h . C A ] J a n DESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIALEQUATIONS WITH ALGEBRAIC SINGULARITIES
THOMAS KECKER AND GALINA FILIPUK
Abstract.
In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlev´eequations and their associated Hamiltonian systems, showing that these define regular initial valueproblems at every point of an augmented phase space, a rational surface with certain exceptionaldivisors removed. We show that the construction of the space of initial values remains meaningful forcertain classes of second-order complex differential equations, and more generally, Hamiltonian systems,where all movable singularities of all their solutions are algebraic poles (by some authors denoted thequasi-Painlev´e property), which is a generalisation of the Painlev´e property. The difference here is thatthe initial value problems obtained in the extended phase space become regular only after an additionalchange of dependent and independent variables. Constructing the analogue of space of initial valuesfor these equations in this way also serves as an algorithm to single out, from a given class of equationsor system of equations, those equations which are free from movable logarithmic branch points.
Keywords. Space of initial values, blow-up, movable algebraic singularity, complex differential equation1.
Introduction
Differential equations and their solutions in the complex plane have been studied extensively sincethe 19th century. A main motivation then was that new transcendental functions could be definedand studied as the solutions of certain differential equations. For example, Airy’s equation, Bessel’sequation, Weber-Hermite equation etc., all of which are important in mathematical physics, havesolutions which cannot be expressed in terms of elementary functions. Rather, their solutions canbe given e.g. in terms of power series expansions around a point, convergent in certain domains,defining analytic functions there, or by asymptotic series. Briot and Bouquet [2] noted that the casewhere a differential equation can be integrated directly is extremely rare, and one should thereforestudy the properties of the solutions of a differential equation through the equation itself, as theyhave demonstrated for elliptic functions. All the equations mentioned above are linear differentialequations, with non-constant coefficients. The singularities of their solution are fixed , i.e. they canoccur only at those points where either one of the coefficients of the equation becomes singular or wherethe coefficient multiplying the highest derivative term vanishes. The fixed singularities essentially canbe read off from the equation itself, and the nature of these singularities be determined. The caseis more involved for nonlinear differential equations, for which singularities can develop somewhatspontaneously, depending on the initial data, and a priory the nature of the singularities cannotbe determined by examining the differential equation itself. In particular, the positions of thesesingularities depend on the initial data prescribed for the equation. Roughly speaking, going fromone solution of the equation to a different solution under a small change in the initial values, theposition of the singularities changes in a continuous fashion. Such singularities are thus called movablesingularities . For a detailed discussion and a more exact definition of movable singularities we referto the article [21] by Murata.A main motivation for complex analysts studying differential equations is to find new mathematicalfunctions with properties of interest to solve problems in physics and other areas of mathematics.’Interesting’ or ’good’ mathematical functions for function theorists were considered to have no movablecritical points. In other words, apart from a finite number of fixed singularities, all other (movable)singularities of any solution are poles. An equation of this kind is said to have the
Painlev´e property .For example, S. Kovalevskaya [20] identified all the integrable cases of the equations of motion of aheavy top, by demanding that their complex solutions can be expressed by Laurent series expansions,i.e. solutions with singularities no worse than poles. Apart from the then already known cases ofthe Lagrange and Euler top, she identified one further integrable case, given by certain ratios of theprinciple moments of inertia of the top, which is now known as the Kovalevskaya top.
P. Painlev´e [24] and his pupil B. Gambier [8] took on the challenge of classifying second-orderordinary differential equations of the form(1) y ′′ = R ( z, y, y ′ ) ,R a function rational in y, y ′ with analytic coefficients, with the property now named after Painlev´e.The result of this classification was a list of 50 canonical types of equations, in the sense that anyequation in the class can be obtained from an equation in the list of 50 by applying a M¨obius typetransformation Y ( Z ) = a ( z ) y ( z ) + b ( z ) c ( z ) y ( z ) + d ( z ) , Z = φ ( z ) , where a, b, c, d and φ are analytic functions. Most of the equations in the list were found to beintegrable in terms of formerly (at the time of Painlev´e) known, classical functions, such as Airyfunctions, Hermite functions, Bessel functions, or other special functions (solutions of certain linearsecond-order differential equations with non-constant coefficients), elliptic functions, or by quadrature.Only six equations in the list turned out to produce essentially new analytic functions. These nonlinearequations are now known as the six Painlev´e equations and their non-classical solutions are commonlycalled Painlev´e transcendents. (For particular values of the parameters in the Painlev´e equations thesealso have classical solutions, but not for generic parameters.)One way to detect equations with the Painlev´e property, within a given class, is to first checkwhether they satisfy certain necessary criteria. Of such criteria, although not the one originallypersued by Painlev´e himself (who used the so-called α -method), a very common one is to performthe Painlve´e test , which checks whether the equation admits, at every point in the complex plane,certain formal Laurent series expansions. It is then still a much more difficult task to prove whetheran equation, which passes the Painlev´e test, actually possesses the Painlev´e property. Proofs for thePainlev´e property of all six Painlev´e equations were given in [26], although earlier proofs exist in theliterature, e.g. [12], [23], [30] or [32], see also the book [9]. There also exists a completely differentapproach of proving the Painlev´e property making use of the so-called isomonodromy method [7].In [4, 5, 6], Filipuk and Halburd apply a similar test to certain classes of second-order differentialequations, but with algebraic series expansions in a fractional power of z − z ,(2) y ( z ) = ∞ X j =0 c j ( z − z ) ( j − j ) /n , j , n ∈ N , instead of Laurent series. The test, which again relies on recursively computing the coefficients ofthe series expansion, gives rise to certain resonance conditions, which need to be satisfied in order forthere to be no obstruction to the recurrence. Furthermore, in the papers cited above, Filipuk andHalburd prove that the conditions, within the given classes of equations are sufficient for all movablesingularities to be algebraic poles of the form (2), with the proviso that these are reachable by analyticcontinuation along a path of finite length . The study of this property was continued by Kecker forother classes of second-order equations [16] and certain Hamiltonian systems [18].Thus, by just looking at a nonlinear differential equation, it is far from obvious to see whether ithas the Painlev´e property, or, more generally, what types of movable singularities other than poles itssolutions can develop. In this article we are concerned with an algorithmic method of determining,from a given equation or class of equations, what types of singularities the equation can develop undercertain conditions. Although our point of departure are the Painlev´e equations, we are studyingdifferential equations and Hamiltonian systems which admit different types of movable singularitiesother than poles, such as algebraic poles and logarithmic singularities. We will employ a methodoriginating in algebraic geometry to resolve certain indeterminacies, or base points , which a givenequation acquires in an augmented phase space which includes the points at infinity in the space ofthe dependent variables. We will see that this method essentially gives an algorithmic procedure ofdetermining the possible types of singularities an equation can develop and to give conditions for whichcertain types of singularities, in particular logarithmic singularities, cannot occur. Whereas this canbe seen as an alternative to the Painlev´e test and its generalisation to algebraic series expansions, theresulting space of initial values also allows us to conclude that certain types of singularities are theonly singularities that can occur in the solutions.In a 1979 paper [22], K. Okamoto introduced the space of initial values for each of the Painlev´eequations. These are extended phase spaces, every point of which defines a regular intial value problem ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 3 in some coordinate chart of the space for one of the Painlev´e equations. The space of initial values isobtained by first compactifying the phase space C of ( y, y ′ ) to some rational surface, such as e.g. P or P × P , and then applying a number of blow-ups to certain points of indeterminacy the equationacquires in this augmented space. A blow-up is the simplest form of a bi-rational transformation. Inthis way, we obtain bi-rational coordinate transformations between the original dependent variable y and its derivative and coordinates covering the points at infinity in which the equation is regular. Thespace of initial values is uniformly foliated by the solutions of the Painlev´e equation.Through the space of initial values, every Painlev´e equation is thus assigned a geometric meaning.Sakai [25] classified rational elliptic surfaces by 9-point configurations in P , which correspond to thegeometries of the spaces of initial values of all known discrete and differential Painlev´e equations. Inthis picture, it is more appropriate to divide the Painlev´e differential equations into 8 different types,as the geometry of some of the Painlev´e equations is different for certain choices of parameters. Thiswork has been elaborated on also in the extensive article [15].In the present article, we are mainly concerned with equations and systems of equations that arenot of Painlev´e type, but for which it can be shown that all movable singularities of their solutionsare algebraic poles, such as studied by Shimomura [27, 28], Filipuk and Halburd [4, 5, 6], and Kecker[16, 18]. Although such equations are in general not integrable, the condition on the singularities to bealgebraic, rather than containing e.g. logarithmic branch points, guarantees some degree of regularity.After reviewing the construction of the space of initial values for the second Painlev´e equation in thenext section, we will mainly be concerned with equations of the form y ′′ = P ( z, y ) ,P a polynomial in y with analytic coefficients and, more generally, Hamiltonian systems, H = H ( z, x ( z ) , y ( z )) , x ′ ( z ) = ∂H∂y , y ′ ( z ) = − ∂H∂x , where H ( z, x, y ) is polynomial in the last two arguments, with analytic coefficients. Extending thephase space to complex projective space P , certain points at infinity, where the flow of the Hamiltonianvector field becomes indeterminate, are resolved using the method of blowing up these so-called basepoints a number of times, until the indeterminacy disappears, leading to an analogue of the space ofinitial values in which each point defines a regular initial value problem, but possibly only after a changein the dependent and independent variables. For the equations considered in this article, this is a finiteprocedure resulting in differential systems which allow to determine directly, i.e. without having toexplicitly construct solutions, what types of movable singularities the solutions exhibit. In particular itis possible to determine when an equation has logarithmic branch points and to give conditions underwhich these logarithmic singularities are absent. These are the same as the resonance conditions foundby applying a Painlev´e test, testing the system for the existence of formal Laurent series solutions in z − z , or its generalisation multi-valued singularities, testing for formal series solutions in fractionalpowers of z − z . Moreover, in the case of the absence of logarithmic singularities, the procedure allowsus to conclude that the algebraic series obtained are the only possible type movable singularities.2. Okamoto’s space of initial values for the Painlev´e equations
The space of initial values was originally constructed by K. Okamoto [22] for each of the six Painlev´eequations. The idea in the paper is to study the equations in an extended phase space that includes allpoints at infinity to study the behaviour at the singularities. In the case of the Painlev´e equations, theextended phase space (with a certain exceptional divisor removed) covers all possible points, includingpoints at infinity, at which the system defines regular initial values problems. One of the main aims ofthis paper is to show that this construction is also meaningful for a wider class of ordinary differentialequations with singularities other than poles, in particular for equations with algebraic poles. Theother main point we wish to make is that the process of constructing the space of initial values alsoserves as an algorithm to single out, from a given class of equations, those equation for which thesolutions are free from logarithmic singularities. We will first review the process of constructing thespace of initial values here for the case of second Painlev´e equation,(3) P II : y ′′ ( z ) = 2 y + zy + α, α ∈ C . THOMAS KECKER AND GALINA FILIPUK
Note that this is a non-autonomous ( z -dependent) Hamiltonian system by letting x = y ′ and definingthe Hamiltonian to be H = x − y + 2 zy + 2 αy . Okamoto [22] considered a different Hamiltonian, H = x − (cid:0) y + z (cid:1) x − (cid:0) α + (cid:1) y , which, by eliminating x , leads to the same equation.Here, instead of equation (3), we start in fact from the more general class of equations(4) y ′′ ( z ) = 2 y + β ( z ) y + α ( z ) , where α and β are analytic functions. One can easily find necessary conditions for equation (4) tohave the Painlev´e property. This is the so-called Painlev´e test , which is performed by inserting formalLaurent series solution of the form y ( z ) = c − z − z + c + c ( z − z ) + c ( z − z ) + · · · into the equation, with c − = ± c , c , c , . . . recursively leads to certain obstructions for the formal Laurent seriesto exist. The case when these obstructions are absent is equivalent to the resonance conditions β ′′ ( z ) ≡ α ′ ( z ) ≡
0. Thus, β is at most a linear function in z whereas α is a constant. The casewhen β ′ ( z ) = 0 essentially reduces equation (4) to equation (3), up to a rescaling of the independentvariable. When β ′ ( z ) ≡
0, this is an equation with constant coefficients which can be integrated directlyin terms of elliptic functions. We will see below how we can re-discover the resonance conditions usingthe method of blowing up the base points .At a singularity z ∗ of a solution of equation (4), where α ( z ) and β ( z ) are analytic, we havelim z → z ∗ max {| y ( z ) | , | y ′ ( z ) |} = ∞ . This is a consequence of the following lemma by Painlev´e, which in turn follows from Cauchy’s localexistence and uniqueness theorem for analytic solutions of differential equation, see e.g. [11].
Lemma 1.
Given a system of differential equations, y ′ = F ( z, y ) , y = ( y , . . . , y n ) , suppose that F is analytic in a neighbourhood of a point ( z ∗ , η ) , η = ( η , . . . , η n ) ∈ C n . If there existsa sequence ( z i ) i ∈ N , z i → z ∗ as i → ∞ so that y j ( z i ) → η j for all j = 1 , . . . , n , then y is analytic at z ∗ . Therefore, to analyse the behaviour of the solution at a singularity, it suggests itself to include thepoints at infinity of the phase space, i.e. the line at infinity in our case, as we will start constructingthe space of initial values for equation (4) by extending the phase space of the differential equation tothe compact surface P . We introduce coordinates on the three standard charts of P ,(5) [1 : y : x ] = [ u : v : 1] = [ V : 1 : U ] , where y and x = y ′ denote the original phase space variables, and the other two coordinate chartscovering P are given by u = x , v = yx and U = xy , V = y , respectively. In these coordinates, equation(4) is expressed as follows.(6) u ′ ( z ) = − u vβ ( z ) + u α ( z ) + 2 v u , v ′ ( z ) = − u v β ( z ) + u vα ( z ) − u + 2 v u ,U ′ ( z ) = − U V + V α ( z ) + V β ( z ) + 2 V , V ′ ( z ) = − U V.
The line at infinity of P is given by the set I = { u = 0 } ∪ { V = 0 } in these coordinates. On thisline, the vector field defined by (6) is infinite, apart from the point P : ( u, v ) = (0 , . Now suppose a solution y ( z ) develops a singularity at some point z ∗ ∈ C by analytic continuation along some path γ ⊂ C with endpoint z ∗ . The corresponding path tracedout by the solution of the system in P must approach the line I . However, in the vicinity of everypoint of the set I \ {P } , the vector field is tangential to the line I and flowing towards the point P . Therefore, there exists at least a sequence ( z n ) n ∈ N ⊂ γ , z n → z ∗ , such that the correspondingsequence of points in P , with coordinates ( u ( z i ) , v ( z i )), ( U ( z i ) , V ( z i )) in the respective charts, tendsto the point P . A point p ∈ P \ I = C cannot be a limit point of the sequence since by Lemma 1the solution would be analytic at z ∗ after all. Due to the indeterminate form of the vector field at P , one cannot say a priory how the solutions behave at this point, and we investigate the behaviourat this point further by trying to resolve the indeterminacy through blowing up this point. ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 5
Resolution of base points.
A dynamical systems can be interpreted as the flow of a vector field,an arrow at each point of possible initial values for the system. A solution of the system is visualisedby drawing a curve which follows the direction of the arrows in a smooth way. However, there mayexist points in the phase space from which vectors emerge or sink into from all possible directions,such as the point P in the preceding paragraph, at which the vector field is a priori ill-defined. Ingeneral, we start from a rational system of equations, defined in some coordinates ( u i , v i ), u ′ i ( z ) = p i, ( z, u i , v i ) q i, ( z, u i , v i ) , v ′ i ( z ) = p i, ( z, u i , v i ) q i, ( z, u i , v i ) , where we assume that the polynomials p i, , q i, and p i, , q i, are in reduced terms, respectively, i.e.have no common factors that would cancel. The points of indeterminacy of the vector field arethe common zeros ( s, t ) of either pair of polynomials equations p i, ( z, s, t ) = 0 = q i, ( z, s, t ), or p i, ( z, s, t ) = 0 = q i, ( z, s, t ). These base points (which may also depend on z ), at which the behaviourof the system is a priori unknown, can be resolved using the method of blowing up , a process familiarfrom algebraic geometry to resolve singularities of algebraic varieties, see e.g. [10] and the work byHironaka [13]. By a blow-up, the phase space is extended by introducing a new projective line, thepoints on which are in one-to-one correspondence to the different directions emanating from the basepoint. The extended space after the blow-up of a point P i , the centre of the blow-up with coordinates P i : ( u i , v i ) = ( s, t ) ∈ C in the coordinate chart ( u i , v i ), is given by(7) Bl P i ( C ) = (cid:8) (( u i , v i ) , [ w : w ]) ∈ C × P : ( u i − s ) · w = ( v i − t ) · w (cid:9) . To express the differential system in the space obtained by the blow-up, two new coordinate chartsare introduced, covering the portions of the space (7) where w = 0 and w = 0, respectively. Wedenote these coordinates by u i +1 = u i − s, v i +1 = v i − tu i − s ,U i +1 = u i − sv i − t , V i +1 = v i − t. After each blow-up, we thus obtain two new rational systems(8) u ′ i +1 = p i +1 , ( z, u i +1 , v i +1 ) q i +1 , ( z, u i +1 , v i +1 ) v ′ i +1 = p i +1 , ( z, u i +1 , v i +1 ) q i +1 , ( z, u i +1 , v i +1 ) U ′ i +1 = P i +1 , ( z, U i +1 , V i +1 ) Q i +1 , ( z, U i +1 , V i +1 ) V ′ i +1 = P i +1 , ( z, U i +1 , V i +1 ) Q i +1 , ( z, U i +1 , V i +1 )where we assume that the polynomials p i +1 , q i +1 and P i +1 , Q i +1 are already in reduced terms again.Here, the relation U i +1 = v − i +1 holds where either coordinate is non-zero, as [ v i +1 : 1] = [1 : U i +1 ] =[ w : w ] are coordinates on the complex projective line, homeomorphic to P , introduced by the blow-up. This line is also called the exceptional curve, denoted L i , of the blown-up space Bl P i . The pointson L i are said to be infinitely near to the point P i . The canonical projection to the first component π i : Bl P i ( C ) → C , (( u i , v i ) , [ w : w ]) ( u i , v i ) , defines a homeomorphism π i : Bl P i ( C ) \ π − i ( P i ) → C \ {P i } , that is, away from the centre P i and its pre-image, points in C are in one-to one correspondence withpoints in Bl P i . The exceptional curve is the pre-image of P i under the projection, L i = π − i ( P i ) ≃ P . In the coordinates ( u i +1 , v i +1 ), resp. ( U i +1 , V i +1 ), the exceptional curve L i is parametrised by( u i +1 , v i +1 ) = (0 , c ) , c ∈ C ( U i +1 , V i +1 ) = ( C, , C ∈ C , with C = c − for c = 0. After each blow-up, we define the space S i = BL P i ( S i − ), obtained by blowingup S i − at P i , where S = P . Later, the location of the points P i may become z -dependent, and wetherefore denote the blown up spaces by S i ( z ). Furthermore, we define the infinity set I i ( z ) ⊂ S i ( z ) asthe union of the set I i − ( z ) under the blow-up with L i , that is I i ( z ) = I ′ i − ( z ) ∪ L i , where I ′ denotesthe proper transform of the set I under the blow-up. We define I = I \ {P} ⊂ P as the line atinfinity with the initial base point removed. THOMAS KECKER AND GALINA FILIPUK
Sequence of blow-ups.
For system (6) we found the initial base point P : ( u , v ) := ( u, v ) =(0 , P i arise ascommon zeros of either of the pair of equations p i +1 , ( z, , v i +1 ) = 0 = q i +1 , ( z, , v i +1 ) ,p i +1 , ( z, , v i +1 ) = 0 = q i +1 , ( z, , v i +1 ) , for the first system, and P i +1 , ( z, U i +1 ,
0) = 0 = Q i +1 , ( z, U i +1 , ,P i +1 , ( z, U i +1 ,
0) = 0 = Q i +1 , ( z, U i +1 , , for the second system. However, any indeterminacy at ( u i , v i ) = (0 , c ), c = 0, of the first system isa base point if and only if this indeterminacy also presents itself at ( U i , V i ) = ( c − ,
0) in the secondsystem, and vice versa. In addition, we can have base points at ( u i , v i ) = (0 ,
0) or ( U i , V i ) = (0 , ± . P : ( u , v ) = (cid:18) x , yx (cid:19) = (0 , ← P : ( U , V ) = (cid:18) y , yx (cid:19) = (0 , ← P ± : ( u , v ) = (cid:18) y , y x (cid:19) = (0 , ± ← P ± : ( u ± , v ± ) = y , y (cid:0) y ∓ x (cid:1) x ! = (0 , ← P ± : ( u ± , v ± ) = y , y (cid:0) y ∓ x (cid:1) x ! = (cid:18) , ∓ β ( z ) (cid:19) ← P ± : ( u ± , v ± ) = y , y (cid:0) y ± (cid:0) xβ ( z ) − xy (cid:1)(cid:1) x ! = (cid:18) , β ′ ( z ) ∓ α ( z ) (cid:19) ← P ± : ( U ± , V ± ) = (cid:18) x/y y − xβ ′ ± (2 xα + xyβ − xy ) , y − xβ ′ ± (2 xα + xyβ − xy )2 x (cid:19) = (0 , . After the blow-up of P ± , the differential system is of the form(9) u ± ′ = − d ± ( z, u ± , v ± ) v ± ′ = 2 α ′ ( z ) ∓ β ′′ ( z ) + p , ( z, u ± , v ± ) u ± · d ± ( z, u ± , v ± ) U ± ′ = U ± ( ± α ′ ( z ) − β ′′ ( z )) + P , ( z, U ± , V ± ) V ± · D ± ( z, U ± , V ± ) V ± ′ = − U ± ( ± α ′ ( z ) − β ′′ ( z )) + P , ( z, U ± , V ± ) V ± · D ± ( z, U ± , V ± )where p , and P ,i , i = 1 ,
2, are polynomials in their second and third arguments. Incidentally, thezero set d ± ( z, u ± , v ± ) = 0 = D ± ( z, U ± , V ± ) is the set I ± ′ ( z ), the proper transform of the exceptionalcurves arising from the cascades of blow-ups P ← P ← · · · ← P +5 resp. P ← P ← · · · ← P − , aswell as the line I , d ± ( z, u ± , v ± ) = ± (2 − u ± ) α ( z ) − ( u ± ) β ( z )) + ( u ± ) β ′ ( z ) + 2( u ± ) v ± ,D ± ( z, U ± , V ± ) = ± (2 − U ± V ± ) α ( z ) − ( U ± V ± ) β ( z )) + ( U ± V ± ) β ′ ( z ) + 2( U ± ) ( V ± ) . ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 7
Remark 1.
After each blow-up we have performed, the resulting vector field is tangent to the ex-ceptional curves in the vicinity of each point on the curve except for the base points, i.e. L \ {P } , L \ {P +3 , P − } and L + i \ {P + i +1 } , i = 3 , , ,
6, respectively L − i \ {P − i +1 } , for i = 3 , , ,
6. Therefore,for a curve γ ⊂ C ending in a movable singularity z ∗ , there exists at least a sequence ( z n ) n ∈ N ⊂ γ , z n → z ∗ , such that the sequence of points ( u i , v i ) or ( U i , V i ) tends towards (one of) the base points.The base point P ± : ( U ± , V ± ) = (0 ,
0) in the second chart of system (9) is only present if thecondition(10) 2 α ′ ( z ) ∓ β ′′ ( z ) ≡ , is not satisfied. This point can be blown up once further, resulting in a system with no further basepoints. However, the solutions of the resulting system give rise to logarithmic singularities. Thisbehaviour is already visible in the systems ( u +6 , v +6 ) and ( u − , v − ), repectively: integrating the firstequation of system (9) with initial data on the exceptional curve after the last blow-up, u ± = 0, andinserting this into the second equations, one obtains u ± = ± ( z − z ) + O (( z − z ) ) , v ± = ( ± α ′ ( z ) − β ′′ ( z )) log( z − z ) + O ( z − z ) . In case of the conditions (10) being satisfied, an additional cancellation of a factor of u ± occursin the second equation of system (9), rendering this system a regular initial value problem on theexceptional curves L ± . Also, in this case the vector field is transversal to these lines. With initialdata ( u ± ( z ) , v ± ( z )) = (0 , h ), one obtains an analytic solution u ± ( z ) = ± ( z − z ) + O (( z − z ) ) , v ± = h + O ( z − z ) , translating into a simple poles for the original variable y . The conditions (10) are exactly the resonanceconditions discussed above, combined giving β ′′ ( z ) = α ′ ( z ) = 0. This is the case in which equation(4) essentially reduces to the second Painlev´e equation, up to re-scaling. We denote by I ( z ) = I +5 ( z ) ∪ I − ( z ) ⊂ S ( z ) the infinity set , that is the proper transforms of the line at infinity I ⊂ P andthe exceptional curves L , L , L +3 , L − , L +4 , L − , L +5 and L − from the first 5 blow-ups of both cascadesof base points. Then, at any point of the set S ( z ) \ I ′ ( z ), the system defines a regular initial valueproblem, which justifies the name ’space of initial values’ for this set. Suppose now a solution y ( z )of the dynamical system, defined in S ( z ), has a movable singularity at some point z ∗ and consider apath γ ⊂ C with endpoint z ∗ . By Remark 1, there exists at least a sequence of points ( z n ) n ∈ N ⊂ γ , z n → z ∗ , such that the solution converges to a point in S ( z ∗ ) \ I ′ ( z ∗ ). Then, by Lemma 1 we canconclude that the solution is analytic in a coordinate chart covering this point, corresponding to eitheran analytic points or a simple pole in the original variable y .In summary, the procedure of blowing up the base points allows us to single out, from the class ofequations (4) with general coefficients, those equations the solutions of which are free from movablelogarithmic singularities. Furthermore, in the absence of logarithmic singularities, the argument in thepreceding paragraph essentially establishes an alternative method of proof of the Painlev´e propertyfor equation (3).We mention that for the alternative (Okamoto) Hamiltonian for equation (3), a different sequenceof base points leads to a related space of initial values. Here, there are originally two base points in P , one at ( u, v ) = (0 , U, V ) = (0 , y ′′ = y + α ( z ), α analytic in z , one finds, after compactifying the equation on P andblowing up a sequence of 9 base points, the condition α ′′ ≡
0. If this condition is satisfied, the systemdefines a regular initial value problem on the exceptional curve from the 9th blow-up, and the equationessentially reduces to the first Painlev´e equation P I . Moreover, in this case there is an analytic solutionaround each point of the space of initial values, which, in the original variable y ( z ) corresponds to apoint where the solution is either analytic or has a double pole. For detailed blow-up calculations seealso the work by Duistermaat and Joshi [3] for the first Painlev´e equation and Howes and Joshi [14]for the second Painlev´e equation, both performed in so-called Boutroux coordinates. THOMAS KECKER AND GALINA FILIPUK Differential equations with movable algebraic singularities
In the papers [27, 28], S. Shimomura studied classes of differential equations with what he calledthe quasi-Painlev´e property . This is a generalisation of the Painlev´e property in the sense that thesolutions of the equations considered may have at most algebraic poles as movable singularities.
Definition 1.
By an algebraic pole we denote a singularity z ∗ of y ( z ), which, in a cut neighbourhoodof z ∗ , can be represented by a convergent Puiseux series,(11) y ( z ∗ ) = ∞ X j =0 c j ( z − z ∗ ) ( j − j ) /n , j , n ∈ N . For n = 1 this includes the notion of an ordinary pole. If the number n is chosen minimal and n > y has an n th-root type algebraic pole at z ∗ .Shimomura proved that, for the classes of equations(12) P ( k ) I : y ′′ = 2(2 k + 1)(2 k − y k + z ( k ∈ N ) ,P ( k ) II : y ′′ = k + 1 k y k +1 + zy + α ( k ∈ N \ { } , α ∈ C ) , the only types of movable singularities that can occur, by analytic continuation of a local solutionalong finite-length paths , are of the algebraic form(13) y ( z ) = ( z − z ∗ ) − k − − (2 k − k − z ∗ ( z − z ∗ ) + h ( z − z ∗ ) k k − + ∞ X j c j ( z − z ∗ ) j k − for P ( k ) I , where h ∈ C is an integration constant, and(14) y ( z ) = ω k ( z − z ∗ ) − k − kω k z ∗ z − z ∗ ) − k − k α k + 1 ( z − z ∗ ) + h ( z − z ∗ ) k + ∞ X j c j ( z − z ∗ ) jk for P ( k ) II , where h is an integration constant and ω k ∈ { , e iπ/k } , so here there are two essentiallydifferent types of leading-order behaviours of singularities. He called this property the quasi-Painlev´eproperty . The proofs in [27, 28] of these facts rely on similar methods as the proofs of the Painlev´eproperty for the Painlev´e property in [26]. In fact, for k = 1, the equations P ( k ) I and P ( k ) II reduce tothe first and second Painlev´e equations, respectively.Already in an earlier (1953) paper, R.A. Smith considered the class of equations(15) y ′′ ( z ) + f ( y ) y ′ ( z ) + g ( y ) = h ( z ) , where f and g are polynomials in y . He showed that, under the condition deg( g ) < deg( f ), the onlytypes of movable singularities that can occur by analytic continuation along finite length paths arealgebraic poles of the form(16) y ( z ) = ∞ X j =0 c j ( z − z ) ( j − /n , n = deg( f ) . Here, as in the cases of equations P ( k ) I and P ( k ) II , it is easy to verify that there exist formal seriessolutions of the form (16), (13) or (14), respectively. Namely, inserting a formal series into therespective equation, one can determine the coefficients recursively without obstruction. A harderproblem is to show that all movable singularities are of this form. As mentioned above, this is similarto the difference in difficulty of showing that an equation passes the Painlev´e test and showing thatthe equation has the Painlev´e property (if it has). The problem thus posed is, for a given differentialequation, to determine a list of possible types of movable singularities that can occur in solutions ofthe equation and show that these are the only ones. In the cases of the equations by Smith (15) andShimomura (12), this was shown under the proviso that paths along which we obtain a singularitythrough analytically continuation, are of finite length. In [29], Smith gave an example of a solutionwith a singularity not of the form (16), which can be obtained only by analytic continuation of acertain solution along a path of infinite length. This singularity, at which the solution behaves verydifferently, is an accumulation point of algebraic singularities of the form (16). We make the following ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 9
Definition 2.
We say that a differential equation or system of equations has the quasi-Painlev´eproperty in the strict sense if all movable singularities of all solutions are algebraic poles.Departing from the works by Smith and Shimomura, Filipuk and Halburd [4, 5, 6] studied moregeneral classes of differential equations with movable algebraic poles. In [4], a class of second-orderequations of the form(17) y ′′ ( z ) = N X n =0 a n ( z ) y n , is studied. After a simple transformation, this equation can be brought into the normalised form(18) y ′′ ( z ) = ˜ a N y N + N − X n =0 ˜ a n ( z ) y n , with a conveniently chosen constant ˜ a N , where the y N − term is now missing. By inserting intoequation (18) a formal series expansion of the form(19) y ( x ) = ∞ X j =0 c j ( z − z ) ( j − /N , and recursively computing the coefficients c j , one finds a necessary condition for the singularities ofthe solution to be algebraic. Namely, the recurrence relation is of the form(20) ( j + N − j − N − c j = P j ( c , c , . . . , c j − ) , j = 1 , , . . . , where P j is a polynomial in the coefficients c , . . . , c j − . The coefficient c N +2 cannot be determinedin this way and the recurrence relation (20) is satisfied if and only if P N +2 is identically zero, in whichcase c N +2 is a free parameter. This resonance condition , P N +2 ≡
0, is necessary for the existenceof the formal algebraic series solutions (19). By expanding the coefficient ˜ a n ( z ) in Taylor series, onecan show that the resonance conditions are equivalent to ˜ a ′′ N − ( z ) = 0 for even N , plus an additionaldifferential relation between the coefficient functions when N is odd. Note that each formal seriessolution (19), with distinct leading-order behaviour, gives rise to one resonance condition. The mainresult in [4] is that all resonance conditions being satisfied is also sufficient for all movable singularitiesof any solution of the equation, reachable by analytic continuation along finite length curves, to bealgebraic poles of the form (19). This was achieved by constructing certain auxiliary functions, orapproximate first integrals, showing that certain quantities are bounded in the vicinity of any movablesingularity, from which a regular initial value problem can be defined.In this article, we will construct the analogue of the space of initial values for some of these equations( N = 4 and N = 5), showing that there exists a regular initial value problem at each point of thiscompact space, away from the exceptional curves introduced by a number of blow-ups. The advantageof this approach is that, due to the compactness of the augmented space, we do not need constructany auxiliary functions as mentioned above. To obtain the regular initial value problem, an additionalchange of the dependent and independent variables is necessary after the blow-ups. Furthermore, withthe approach in this article we can show that, for these equations and also the Hamiltonian systemsconsidered in Section 6, all movable singularities are algebraic poles, i.e. these equations in fact havethe strict quasi-Painlev´e property in the sense of Definition 2. This is due to the fact that for theseequations, blowing up the base points is a finite procedure, i.e. the sequence of base points terminatesand the indeterminacies can be resolved completely. We will see that, in the resulting compact space,a solution approaching the singularity has a limit point somewhere on the exceptional curve after thelast blow-up, where the system defines a regular initial value problem, after a change in dependentand independent variable. By Lemma 1 we can conclude that there exists an analytic solution nearthis point.Smith [29] showed that the equation y ′′ + 4 y y ′ + y = 0, which is contained in the class (15), does notpossess the quasi-Painlev´e property in the strict sense. The sequence of base points neede to resolvethe indeterminacy in the vector field does not terminate, which is a hint that this and other equationsin this class may have a more complicated singularity structure. The class of equations (17) is contained in the wider class of polynomial Hamiltonian systemsstudied by one of the authors in [18], H ( z, x, y ) = x M + y N + X i,j iN + jM 0, i.e. α is either a linearfunction in z or constant. We will recover this condition using an appropriate cascade of 14 blow-ups.The equation is first extended to complex projective space P by introducing homogeneous coordi-nates as in (5) above. The system of equations is presented as follows: u ′ ( z ) = − u v α ( z ) + 2 u vβ ( z ) + 2 u γ ( z ) + 5 v u ,v ′ ( z ) = − u v α ( z ) + 2 u v β ( z ) + 2 u vγ ( z ) − u + 5 v u ,U ′ ( z ) = − U V − V α ( z ) − V β ( z ) − V γ ( z ) − V ,V ′ ( z ) = − U V. We see that there is an initial base point in the first chart at ( u, v ) = (0 , P : ( u, v ) = (cid:18) x , yx (cid:19) = (0 , ← P : ( U , V ) = (cid:18) y , yx (cid:19) = (0 , ← P : ( u , v ) = (cid:18) y , y x (cid:19) = (0 , ← P : ( U , V ) = (cid:18) xy , y x (cid:19) = (0 , ← P : ( u , v ) = (cid:18) xy , y x (cid:19) = (0 , ← P : ( u , v ) = xy , y (cid:0) y − x (cid:1) x ! = (0 , ← P : ( u , v ) = xy , y (cid:0) y − x (cid:1) x ! = (0 , ← P : ( u , v ) = xy , y (cid:0) y − x (cid:1) x ! = (0 , ← P : ( u , v ) = xy , y (cid:0) y − x (cid:1) x ! = (cid:18) , − α ( z ) (cid:19) ← P : ( u , v ) = xy , y (cid:0) x α ( z ) − x y + 3 y (cid:1) x ! = (0 , ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 11 ← P : ( u , v ) = xy , y (cid:0) x α ( z ) − x y + 3 y (cid:1) x ! = (0 , − β ( z )) ← P : ( u , v ) = xy , y (cid:0) x β ( z ) + 2 x y α ( z ) − x y + 3 y (cid:1) x ! = (cid:18) , α ′ ( z ) (cid:19) ← P : ( u , v ) = (cid:18) , α ( z ) − γ ( z ) (cid:19) = xy , y (cid:0) x y α ( z ) + 9 x y β ( z ) − x α ′ ( z ) − x y + 9 y (cid:1) x ! ← P : ( u , v ) = (0 , β ′ ( z ))= xy , − y (cid:0) x α ( z ) − x γ ( z ) + 4 x y α ′ ( z ) − x y α ( z ) − x y β ( z ) + 9 x y − y (cid:1) x ! . After blowing up P , the differential system is of the form(21) u ′ = − 81 + p , ( z, u , v )2 u · d ( z, u , v ) ,v ′ = − α ′′ ( z ) + p , ( z, u , v ) u · d ( z, u , v ) ,U ′ = 36 α ′′ ( z ) + P , ( z, U , V ) U V · D ( z, U , V ) ,V ′ = − 81 + P , ( z, U , V )2 U V · D ( z, U , V ) , where p ,i and P ,i , i = 1 , u , v and U , V , respectively, so that on theexceptional curve L : { u = 0 } ∪ { V = 0 } introduced by the last blow-up, p ,i ( z, , v ) = 0 = P ,i ( z, U , d ( z, u , v ) and D ( z, U , V ) in the denominators of (21), also called the ex-ceptional divisor , represents the set I ′ in these coordinates, that is the proper transforms of theexceptional curves L , . . . , L introduced by the first 13 blow-ups together with the line at infinity I ⊂ P , d = 9 + 9 u v − u α + 12 u α − u β − u γ + 4 u α ′ + 18 u β ′ ,D = 9 + 9 U V − U V α + 12 U V α − U V β − U V γ + 4 U V α ′ + 18 U V β ′ . Remark 2. It is important to note that, after the blow-up of each point P i , i = 1 , . . . , 13, the resultingvector field is infinite on the exceptional curve L i and tangent to this curve in the vicinity of everypoint on the curve apart from the base point P i +1 , and flowing towards this point. Thus, for a curve γ ⊂ C which ends in a movable singularity z ∗ , there exists at least a sequence of points ( z n ) n ∈ N ⊂ γ , z n → z ∗ , such that the corresponding points with coordinates ( u i ( z n ) , v i ( z n )), resp. ( U i ( z n ) , V i ( z n )),converge to the base point P i +1 as z n → z ∗ .Since the blow-ups are bi-rational transformations, one can always solve for the original coordinates.E.g. we can give the dependence of y on u , v , as follows:(22) y = u − (cid:18) − u α + u (cid:18) − β + u (cid:18) α ′ u (cid:0) α − γ + 3 u (cid:0) u v + 2 β ′ (cid:1)(cid:1)(cid:19)(cid:19)(cid:19) − . Integrating the system (21) when α ′′ ( z ) = 0 would result in logarithmic behaviour for v , since, toleading order, u = r − 32 ( z − z ) / + O (cid:16) ( z − z ) / (cid:17) . Inserting this into the second equation of would result in v ′ = 827 α ′′ ( z ) z − z + O (cid:16) ( z − z ) − / (cid:17) , from which the logarithmic behaviour v = α ′′ ( z ) log( z − z ) + O ( z − z ) follows. As discussedabove, α ′′ ( z ) ≡ u resp. V occurs in the second and third equation ofsystem (21), which becomes(23) u ′ = − 81 + p , ( z, u , v )2 u · d ( z, u , v ) ,v ′ = 72 α ( z ) α ′ ( z ) + 162 γ ′ ( z ) + ˜ p , ( z, u , v ) u · d ( z, u , v ) ,U ′ = 72 α ( z ) α ′ ( z ) + 162 γ ′ ( z ) + ˜ P , ( z, U , V ) U V · D ( z, U , V ) ,V ′ = − 81 + P , ( z, U , V )2 U V · D ( z, U , V ) , where ˜ p , and ˜ P , are polynomials in u , v resp. U , V with ˜ p , ( z, , v ) = 0 = ˜ P , ( z, U , L : { u = 0 } ∪ { V = 0 } ,and the system can be integrated, to leading order, e.g. in the coordinates u , v as u = r − 32 ( z − z ) / + O (cid:16) ( z − z ) / (cid:17) ,v = h + √ (cid:18) α ( z ) α ′ ( z ) + 2 γ ′ ( z ) (cid:19) ( z − z ) / + O (( z − z ) / ) , where h is the second integration constant (besides z ). In this way, every point on the line L introduced by the last blow-up, parameterised by ( u , v ) = (0 , h ), gives rise to an algebraic seriessolution. Denoting by S ( z ) the space obtained by blowing up the sequence of 14 base points, whichare themselves z -dependent, and the set I ( z ) as above, the analogue of the space of initial valuescan be defined as S ( z ) \ I ′ ( z ). Thus, away from the set I ′ ( z ), every point in the space we haveconstructed gives rise to an initial value problem with either analytic solutions or power series solutionsin ( z − z ) / . The latter solutions are transversal to the exceptional curve L from the last blow-up.Moreover, we can conclude that these singularities are the only possible types a solution can have. Proposition 1. The class of equations y ′′ = y + ( az + b ) y + β ( z ) y + γ ( z ) , where β and γ are analytic functions and a, b ∈ C , has the quasi-Painlev´e property in the strict sense,with cubic-root type algebraic poles.Proof. Making a change in dependent and independent variables, the system (23) becomes(24) dzdu = 2 u · d ( z, u , v ) − 81 + p , ( z, u , v ) ,dv du = 2 · α ( z ) α ′ ( z ) + 162 γ ′ ( z ) + ˜ p , ( z, u , v ) − 81 + p , ( z, u , v ) ,dzdV = 2 U V · D ( z, U , V ) − 81 + P , ( z, U , V ) ,dU dV = 2 U · α ( z ) α ′ ( z ) + 162 γ ′ ( z ) + ˜ P , ( z, U , V ) − 81 + P , ( z, U , V ) , which, for initial data ( z, u , v ) = ( z , , h ) resp. ( z, U , V ) = ( z , H, 0) on the exceptional curve L , defines a regular initial value problem. By Remark 2, for a any curve γ ending in a movablesingularity z ∗ , there exists at least a sequence of points ( z n ) ⊂ γ , z n → z ∗ , on which the solution( u ( z n ) , v ( z n )), resp. ( U ( z n ) , V ( z n )) converges to the line L ≃ P , a compact set. Restrictingif necessary to a subsequence, by Lemma 1 system (24) has an analytic solution in ( z, v ) of the form z ( u ) = z ∗ − u + O ( u ) , v ( u ) = h + 2 (cid:18) α ( z ∗ ) α ′ ( z ∗ ) + 2 γ ′ ( z ∗ ) (cid:19) u + O ( u ) . Inverting these power series one finds an algebraic series expansion for ( u , v ) in terms of ( z − z ∗ ) / ,which by (22) corresponds to cubic-root type algebraic pole in the original variable y ( z ). (cid:3) ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 13 Second-order equation with polynomial right-hand side of degree y ′′ = a N y N + P N − n =0 a n ( z ) y n with odd N > N = 5,(25) y ′′ ( z ) = 3 y + α ( z ) y + β ( z ) y + γ ( z ) y + δ ( z ) , where the coefficient a = 3 is chosen for computational convenience. As was shown in [4], in the odd N case, two resonance conditions are necessary and sufficient for the solutions of the equation to havealgebraic poles as movable singularities. These can be found by inserting the formal series expansion(26) y ( z ) = ∞ X j =0 c j ( z − z ) ( j − / into equation (25) and computing, for each possible leading coefficient c , the obstruction in therecurrence relation (20) to determine the coefficients c j , j = 1 , , . . . . In the odd N case, there aretwo essentially different leading-order behaviours corresponding to the initial coefficients c ∈ { , − } ,yielding two distinct resonances. In this case, these conditions are equivalent to α ′′ ( z ) ≡ γ ( z ) + 4 α ( z )) ′ ≡ 0. We will now show that we can recover these conditions through the constructionof the analogue of the space of initial values for equation (25), and moreover, that the singularities ofthe form (26) are the only type of movable singularity for equation (25).Extending the phase space of the equation in the variables ( y, x ) = ( y, y ′ ) to P via the relations[1 : y : x ] = [ u : v : 1] = [ V : 1 : U ], one finds the following systems of equations: u ′ ( z ) = − u v α ( z ) + u v β ( z ) + u vγ ( z ) + u δ ( z ) + 3 v u ,v ′ ( z ) = − u v α ( z ) + u v β ( z ) + u v γ ( z ) + u vδ ( z ) − u + 3 v u ,U ′ ( z ) = 3 − U V + V α ( z ) + V β ( z ) + V γ ( z ) + V δ ( z ) V ,V ′ ( z ) = − U V. There is a single base point in the chart u, v at ( u, v ) = (0 , ± . P : ( u, v ) = (cid:18) x , yx (cid:19) = (0 , ← P : ( U , V ) = (cid:18) y , yx (cid:19) = (0 , ← P : ( u , v ) = (cid:18) y , y x (cid:19) = (0 , ← P ± : ( u , v ) = (cid:18) y , y x (cid:19) = (0 , ± ← P ± : ( u ± , v ± ) = y , y (cid:0) y ∓ x (cid:1) x ! = (0 , ← P ± : ( u ± , v ± ) = y , y (cid:0) y ∓ x (cid:1) x ! = (cid:16) , ∓ α (cid:17) ← P ± : ( u ± , v ± ) = (cid:18) y , y ± y ( αx − y x )4 x (cid:19) = (cid:18) , ∓ β (cid:19) ← P ± : ( u ± , v ± ) = y , y ± y (cid:0) xyα + 4 xβ − xy (cid:1) x ! = (cid:18) , (cid:0) α ′ ± (cid:0) α − γ (cid:1)(cid:1)(cid:19) ← P ± : ( u ± , v ± ) = y , y (cid:0) y − xα ′ ∓ (cid:0) xy − xy α + 9 xα − xyβ − xγ (cid:1)(cid:1) x ! = (cid:18) , β ′ ± (cid:18) αβ − δ (cid:19)(cid:19) . Due to the bi-rational nature of the blow-ups, the collected coordinate transformations in each of thecascades of 9 blow-ups can be inverted,(27) y = 1 u , x = y ′ = u − (cid:18) u (cid:18) − α u (cid:18) − β u (cid:18) (cid:0) α − γ + 4 α ′ (cid:1) + u (cid:18) u v + 112 (cid:0) αβ − δ + 4 β ′ (cid:1)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19) − . In the coordinates after blowing up P ± the system is of the form(28) u ± ′ = − u ± · d ± ( z, u ± , v ± ) ,v ± ′ = ∓ α ′′ ( z ) − α ( z ) α ′ ( z ) + 48 γ ( z ) + p ± , ( z, u ± , v ± )( u ± ) · d ± ( z, u ± , v ± ) ,U ± ′ = ± α ′′ ( z ) + 6 α ( z ) α ′ ( z ) − γ ′ ( z ) + P ± , ( z, U ± , V ± )( V ± ) · D ± ( z, U ± , V ± ) ,V ± ′ = − − U ( α ( z ) α ′ ( z ) + 8 γ ′ ( z ) − α ′′ ( z )) + P ± , ( z, U ± , V ± )( U ± ) V ± · D ± ( z, U ± , V ± ) , where p , and P ,i , i = 1 , p ± , ( z, , c ) = 0 and P ± ,i ( z, C, 0) = 0. The zero set of d ± , resp. D ± is the exceptionaldivisor, i.e. the proper transform of the line at infinity I ∈ P and the exceptional curves L , . . . , L ± from the blow-ups of the two cascades P ← · · · ← P +8 and P ← · · · ← P − , respectively: d ± = ± (cid:0) − u ± ) α + 9( u ± ) α − u ± ) β + 24( u ± ) αβ − u ± ) γ − u δ (cid:1) + 12( u ± ) α ′ + 32( u ± ) β ′ + 96( u ± ) v ± ,D ± = ± (cid:0) − U ± V ± ) α + 9( U ± V ± ) α − U ± V ± ) β + 24( U ± V ± ) αβ − U ± V ± ) γ − U ± V ± ) δ (cid:1) + 12( U ± V ± ) α ′ + 32( U ± V ± ) β ′ + 96( U ± ) ( V ± ) . Integrating the first equation in (28) yields u ± = i √ z − z ) / + O (( z − z )) , where the sign of the square root can be absorbed into the choice of branch for ( z − z ) . Insertingthis intermediate result into the second equation, we see that v ± has a logarithmic singularity, v ± ( z ) = 196 (cid:0) ± α ′′ ( z ) + 3 α ( z ) α ′ ( z ) − γ ′ ( z ) (cid:1) log( z − z ) + O (cid:16) ( z − z ) / (cid:17) , unless the condition(29) ± α ′′ ( z ) + α ( z ) α ′ ( z ) − γ ( z ) = 0is satisfied. In the latter case, a cancellation of one factor of u ± resp. V ± occurs in the second andthird equation of system (28). Then, by changing the role of dependent and independent variables,the system is of the following form:(30) dzdu ± = − u ± · d ± ( z, u ± , v ± )96 ,dv ± du ± = − ˜ p ± , ( z, u ± , v ± )96 ,dzdV ± = ( U ± ) V ± · D ± ( z, U ± , V ± ) − 96 + P ± , ( z, U ± , V ± ) ,dU ± dV ± = ˜ P ± , ( z, U ± , V ± ) − 96 + P ± , ( z, U ± , V ± ) , where ˜ p ± , = u ± p ± , and ˜ P ± , = V ± P , are polynomials in u ± , v ± and U ± , V ± , respectively. Forinitial values ( z, u ± , v ± ) = ( z , , h ), respectively ( z, U ± , V ± ) = ( z , H, ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 15 L ± : { u ± = 0 } ∪ { V ± = 0 } , this defines a regular initial value problem with analytic solutions, e.g. z ( u ± ) = z − 12 ( u ± ) + O (cid:0) ( u ± ) (cid:1) ,v ± ( u ± ) = h + O ( u ± ) . Inverting these expansions we find the algebraic series solutions(31) u ± ( z ) = i √ z − z ) / + O ( z − z ) , v ± ( z ) = h + O (( z − z ) / ) , which by (27) correspond to square-root type algebraic poles in the variable y . In the case where bothconditions (29) are satisfied, amounting to α ′′ ( z ) ≡ , (cid:0) α ( z ) − γ ( z ) (cid:1) ′ ≡ , we can thus prove, by similar arguments as in Proposition 1, that the algebraic series (31) are theonly possible type of movable singularities that can occur. Proposition 2. The class of equations (32) y ′′ = y + ( az + b ) y + β ( z ) y + (cid:18) 116 ( az + b ) + c (cid:19) y + δ ( z ) , where β ( z ) and δ ( z ) are analytic in z and a, b, c ∈ C constants, has the quasi-Painlev´e property in thestrict sense, with square-root type algebraic poles. Denoting by S ( z ) the space obtained by blowing up P along the cascades P ← · · · ← P +9 and P ←· · · ← P − , and by I ′ ( z ) the proper transform of the set I ( z ) = I ∪ L ∪ L ∪ L ∪ S i =4 L + i ∪ S i =4 L − i in S ( z ), we obtain S ( z ) \ I ′ ( z ) as the analogue of the space of initial values for equation (32).6. Hamiltonian systems with algebraic singularities In the previous section we have seen how we can resolve the base points of the second-order equationsof the form y ′′ = P ( z, y ), extending the phase space of ( y, y ′ ). These equations are in fact Hamiltoniansystems by letting(33) H ( z, x, y ) = 12 x − ˜ P ( z, y ) , ∂ ˜ P∂y = P ( z, y ) , where x = y ′ and we let N = deg y P . In the previous sections we considered the cases N = 4 and N = 5, whereas the case N = 3 was discussed in section 2 which led to the second Painlev´e equation.The case N = 2 leads to the first Painleve equation. In fact, all six Painlev´e equations can be writtenas Hamiltonian systems H ( z, x, y ) with rational coefficients. The blow-ups leading to the space ofinitial values for all these Painlev´e Hamiltonian systems where performed by Okamoto [22].In [18], one of the authors studied the class of Hamiltonian systems, H ( z, x ( z ) , y ( z )) = M X i =0 N X j =0 α ij ( z ) x i y j ,x ′ ( z ) = ∂H∂z , y ′ ( z ) = − ∂H∂x , which, similar to the equations y ′′ = P ( z, y ), under certain resonance conditions, have the propertythat all their movable singularities are algebraic poles. We consider here the case where the coefficientsof the dominant terms are constant, which amounts to saying there are no fixed singularities,(34) H = 1 N y N − M x M + X i,j α ij ( z ) x i y j , so that the leading order behaviour of solutions would be of the form(35) x ( z ) = c ( z − z ) − MMN − M − N + · · · , y ( z ) = d ( z − z ) − NMN − M − N + · · · . Using the method of compactifying the phase space and blowing up the base points, we will see howto obtain the conditions by which the expansions (35) yield algebraic poles, i.e. when they are freefrom logarithmic singularities. The case min { M, N } = 2 can be reduced essentially to the case (33),representing second-order equations. We will thus look at some examples with M, N ≥ 3. The case M = N = 3, discussed in the next paragraph, is interesting as it leads to a system of equations related to the fourth Painlev´e equation, i.e. in this case the singularities are simple (ordinary) poles. The spaceof initial values for this system was already computed in [19] and is reproduced here for completeness.We then consider the cases M = 3 , N = 4 and M = N = 4 in the following two sections, which have5th-root type and square-root type algebraic poles, respectively. Constructing the analogue of thespace of initial values for these systems allows us to conclude that these are the only possible typesof movable singularities, and in particular, that these systems have the quasi-Painlev´e property in thestrict sense.6.1. Case M = N = 3 : a system with the Painlev´e property. We consider the Hamiltonian system(36) H ( z, x ( z ) , y ( z )) = 13 (cid:0) y − x (cid:1) + γ ( z ) xy + β ( z ) x + α ( z ) y, which was introduced in [17]. If α and β are constants and γ ( z ) a function at most linear in z , itwas shown that the system of equations derived from (36) has the Painlev´e property. Below we willsee that, by applying the procedure of compactifying the system (36) with general analytic functions α ( z ), β ( z ), γ ( z ), after blowing up and resolving the base points of the system, these conditions canbe recovered.Extending the system to P we obtain, in the three standard coordinate charts [1 : y : x ] = [ u : v :1] = [ V : 1 : U ], x ′ ( x ) = y + γ ( z ) x + α ( z ) , y ′ ( z ) = x − γ ( z ) y − β ( z ) ,u ′ ( z ) = − v − u α ( z ) − uγ ( z ) , v ′ ( z ) = − − v + u vα ( z ) + u β ( z ) + 2 uvγ ( z ) u ,V ′ ( z ) = − U + V β ( z ) + V γ ( z ) , U ′ ( z ) = − − U − V α ( z ) − U V β ( z ) − U V γ ( z ) V . We can see that initially there are three base points on the line at infinity of P , given by P ρ : ( u, v ) = (0 , ρ ) ↔ ( U, V ) = ( ρ − , , ρ ∈ { , ω, ¯ ω } , where ω = − ± i √ is a third root of unity. Keeping ρ as a symbol representing either of the threeroots of unity, each base point is resolved by a sequence of three blow-ups. We denote the coordinatesof the three respective sequences of blow-ups with superscripts ρ ∈ { , ω, ¯ ω } : P ρ : ( u ρ , v ρ ) = (cid:18) x , yx (cid:19) = (0 , ρ ) ← P ρ : ( u ρ , v ρ ) = (cid:18) x , y − ρx (cid:19) = (0 , − ¯ ργ ( z )) ← P ρ : ( u ρ , v ρ ) = (cid:18) x , x ( y − ρx + ¯ ργ ( z )) (cid:19) = (cid:0) , γ ′ ( z ) − ρβ ( z ) − ¯ ρα ( z ) (cid:1) , where ¯ ρ denotes the complex conjugate of ρ . After blowing up P ρ , the system of equations takes thefollowing form:(37) u ρ ′ = p ρ , ( z, u ρ , v ρ ) ,v ρ ′ = ¯ ρα ′ ( z ) + ρβ ′ ( z ) − γ ′′ ( z ) + p ρ , ( z, u ρ , v ρ ) u ρ ,U ρ ′ = U ρ ( γ ′′ ( z ) − ¯ ρα ′ ( z ) − ρβ ′ ( z )) + P , ( z, U ρ , V ρ ) V ρ ,V ρ ′ = − ¯ ρ + P , ( z, U ρ , V ρ ) U ρ . We see that there is an additional base point at P ρ : U ρ = V ρ = 0. This point can be blown uponce more, rendering a system free from base points. The point P ρ , however, is only present if thecondition(38) ¯ ρα ′ ( z ) + ρβ ′ ( z ) − γ ′′ ( z ) ≡ u ρ and V ρ cancel in the second resp. third equation of system (37), which thendefines a regular initial value problem at every point of the exceptional curve L ρ : ( u ρ , v ρ ) = (0 , h ).Denoting by S ( z ) the space obtained by blowing up P along the three cascades of base points ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 17 P ρ ← P ρ ← P ρ , ρ ∈ { , ω, ¯ ω } , the space of initial values is S ( z ) \ (cid:16)S ρ ∈{ ,ω, ¯ ω } I ρ ′ ( z ) (cid:17) , where I ρ ( z ) isthe union of the proper transforms of the exceptional curves L ρi , i = 1 , 2, from the first two blow-upsin the cascade of base points, and the line at infinity I ⊂ P .Together, the three conditions (38) for ρ ∈ { , ω, ¯ ω } are required for the absence of logarithms,which result in γ ′′ = β ′ = α ′ ≡ 0. Thus, α and β are constant and γ ( z ) = az + b is at most linearin z . In case a = 0, the Hamiltonian system is autonomous and can be integrated directly using theHamiltonian as first integral. When a = 0, by a re-scaling of z , x and y the system can be normalisedto the form(39) H = 13 ( y − x ) + zxy + αy + βx, x ′ = y + zx + α, y ′ = x − zy − β. This system is in fact closely related to the Hamiltonian system defining the fourth Painlev´e equation,and was introduced in [17] and investigated further in [31]. By similar arguments as in section 2,constructing the space of initial values gives an alternative method of proof for the Painlev´e propertyof system (39).6.2. Case M = 3 , N = 4 . With a slightly different normalisation as given in (34) we consider theHamiltonian of the form(40) H ( z, x ( z ) , y ( z )) = y − x + X i,j =0 i + j ≤ α ij ( z ) x ( z ) i y ( z ) j , with analytic coefficients α ij ( z ). Extending the system to P yields the three systems of equations x ′ ( z ) = 4 y + a x + 2 a xy + a x + 2 a y + a ,y ′ ( z ) = 3 x − a xy − a y − a x − a y − a ,u ′ ( z ) = − u vα + u α + u α + 2 uvα + uα + 4 v u ,v ′ ( z ) = − u v α + u vα + 2 u vα + u α + 2 u α + 3 uv α + 3 uvα − u + 4 v u ,U ′ ( z ) = − − U V α − U V α + 3 U V − U V α − U V α − U V α − V α − V α − V ,V ′ ( z ) = − U + 2 U V α + 2 U α + V α + V α + α , which have a single base point at ( u, v ) = (0 , P : ( u, v ) = (0 , ← P : ( U , V ) = (0 , ← P : ( U , V ) = (0 , ← P : ( U , V ) = (0 , ← P : ( U , V ) = (1 , ← P : ( U , V ) = ( α , ← P : ( U , V ) = (cid:0) α + α , (cid:1) ← P : ( U , V ) = (cid:0) α α + α , (cid:1) ← P : ( U , V ) = (cid:0) α + 3 α α + α + α , (cid:1) ← P : ( U , V ) = (cid:0) α + 4 α α + 3 α α + 3 α α + α , (cid:1) ← P : ( U , V ) = (cid:18) − α ′ + α + 5 α α + 6 α α + α + 6 α α + 3 α α +3 α α + α , ← P : ( U , V ) = (cid:18) − α α ′ − α ′ + α + 6 α α + 10 α α + 10 α α +4 α α + 3 α α + 12 α α α + 3 α + 6 α α + 3 α α , (cid:1) ← P : ( U , V ) = (cid:18) − α α ′ − α α ′ − α α ′ + α + 7 α α + 15 α α + 15 α α + 10 α α + 10 α α + 30 α α α + 12 α α α + α +2 α + α + 6 α α + 3 α (cid:0) α + α (cid:1) , (cid:1) ← P : ( U , V ) = (cid:18) − α α ′ − α α ′ − α α α ′ − α α ′ − α ′ + α + 8 α α + 21 α α + 21 α α + 20 α α + 10 α α + 60 α α α + 5 α α + 10 α α + 4 α α + 12 α α α +30 α α α + α (cid:0) α + 30 α α + 6 α + 4 α (cid:1) + α , (cid:1) ← P : ( U , V ) = (cid:18) − α α ′ − α α ′ − α α α ′ − α α α ′ − α α ′ − α ′ − α α ′ − α α ′ + α + 9 α α + 28 α α + 28 α α + 35 α α + 15 α α + 105 α α α + 15 α α + 30 α α + 30 α α α + 90 α α α + 4 α α + α + α (cid:0) α + 60 α α + 30 α + 20 α (cid:1) α + 2 α + 6 α α +10 α α + 10 α α + 4 α α + 2 α (cid:0) α + 2 α (cid:1) , (cid:1) ← P : ( U , V ) = (cid:18) − α α ′ − α α ′ − α α α ′ − α α α ′ − α α ′ − α α ′ − α α α ′ − α α α ′ − α ′ − α α ′ − α α ′ − α α ′ − α α ′ + 16 α ′′ + α + 10 α α + 36 α α + 36 α α + 28 α α + 56 α α + 168 α α α + 105 α α α + 35 α α + 70 α α + 20 α α + 210 α α α + 90 α α α + 60 α α α + 6 α α + 10 α α + 60 α α α + 20 α α α + 60 α α α + 10 α α + 20 α α α +2 α (cid:0) α + 2 α (cid:1) + α (cid:0) α + α (cid:0) α + 60 α α + 30 α + 20 α (cid:1)(cid:1) , (cid:1) . After the 16th blow-up, the system of equations take the following form:(41) u ′ = − p , ( z, u , v ) u v · d ( z, u , v ) ,v ′ = − α ′ ( z ) + α ( z ) α ′′ ( z )) + p , ( z, u , v ) u v · d ( z, u , v ) ,U ′ = 240(2 α ′ ( z ) + 2 α ( z ) α ′′ ( z ) + 3 α ′′ ( z )) + P , ( z, U , V ) V · D ( z, U , V ) ,V ′ = − P , ( z, U , V ) V · D ( z, U , V ) , where p ,i and P ,i , i = 1 , 2, are polynomials in u , v and U , V , respectively, such that p ,i ( z, , v ) = 0 = P ,i ( z, U , 0) on the exceptional curve L : { u = 0 } ∪ { V = 0 } . Thepolynomial expressions d ( z, u , v ) and D ( z, U , V ), whose zero sets are the proper transforms inthese coordinates of the exceptional curves L i , i = 1 , . . . , 15, of all previous blow-ups, and satisfy d ( z, , v ) = 60 = D ( z, U , 0) on the curve L . Due to their length we omit writing down the fullexpressions. Thus, unless the condition(42) 2 α ′ ( z ) + 2 α ( z ) α ′′ ( z ) + 3 α ′′ ( z ) = (2 α + 3 α ) ′′ ≡ V ∼ ( z − z ) / .Inserting this into the third equation, U has a logarithmic branch point. Thus, for the absenceof logarithmic singularities we require condition (42) to be satisfied, which amounts to the function ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 19 α ( z ) + 3 α ( z ) being at most linear in z . In this case, one factor of u and V cancel in thesecond resp. third equation of system (41), and by interchanging the role of dependent and independentvariables, we can write the equations in the form dzdV = V · D ( z, U , V ) − P , ( z, U , V ) ,dU dV = ˜ P , ( z, U , V ) − P , ( z, U , V ) , which, for initial data ( z, U , V ) = ( z , H, 0) on the exceptional curve L from the last blow-upbecomes a regular initial value problem with analytic solutions z ( V ) = z − V + O ( V ) , U ( V ) = H + O ( V ) . Inverting the power series for z − z leads to series expansions for U and V in ( z − z ) / , whichtranslate to 5th-root type algebraic poles in the original variables x, y . Using similar arguments as inProposition 1, we have thus shown: Proposition 3. Under the condition α ( z ) + 3 α ( z ) = az + b , a, b ∈ C , the Hamiltonian systemderived from (40) has the quasi-Painlev´e property in the strict sense, with th-root type algebraic poles. Denoting by S ( z ) the extended space obtained from P by blowing up the cascade of base points P ← · · · ← P , we define the analogue of the space of initial values for the system by S ( z ) \ I ′ ( z ),where I ( z ) = I ∪ S i =1 L i .6.3. Case M = N = 4 . The differential system studied in this section arises from the Hamiltonian H ( z, x, y ) = 14 (cid:0) y − x (cid:1) + X i,j =0 i + j ≤ α i,j ( z ) x i y j , where the α i,j ( z ) are analytic functions. Similar to the case M = N = 3, the geometry of the analogueof the space of initial values of this system is much more symmetric than the M = 3 , N = 4 case.Although the same number, 16, of blow-ups is required to regularise the system, the calculations area lot less involved here. As before, we write down the extended system of equations in the threestandard charts of P : x ′ = y + α , + 2 yα , + xα , + 2 xyα , + x α , ,y ′ = x − α , − yα , − y α , − xα , − xyα , ,u ′ = − v + u α , + 2 u vα , + u α , + 2 uvα , + uα , u ,v ′ = − − v + u vα , + 2 u v α , + u α , + 2 u vα , + 3 uv α , + 2 u α , + 3 uvα , u ,U ′ = − − U − V α , − V α , − U V α , − U V α , − U V α , − U V α , − U V α , V ,V ′ = − U − V α , − V α , − V α , − U V α , − U V α , V . We observe that there are six base points (denoted with superscripts), namely P : ( u, v ) = (0 , P : ( U, V ) = (0 , P : ( u, v ) = (0 , ↔ ( U, V ) = (1 , P i : ( u, v ) = (0 , i ) ↔ ( U, V ) = ( − i, P − : ( u, v ) = (0 , − ↔ ( U, V ) = ( − , P − i : ( u, v ) = (0 , − i ) ↔ ( U, V ) = ( i, . We note that the points P and ˜ P can be resolved by one blow-up each, only the transforms of thepoints P ρ , ρ ∈ { , i, − , − i } are still visible in the systems obtained after the blow-ups of P and ˜ P . We now resolve the four base points P ρ , ρ ∈ { , i, − , − i } , in the coordinates ( u, v ), where we usethe superscript ρ to denote the coordinates following the blow-ups. For each point P ρ , we find thefollowing sequence of four blow-ups: P ρ : ( u, v ) = (cid:18) x , yx (cid:19) = (0 , ρ ) ← P ρ : ( u ρ , v ρ ) = (cid:18) x , y − ρx (cid:19) = (0 , α , + ρα , ) ← P ρ : ( u ρ , v ρ ) = (cid:18) x , x (cid:0) y − ρx + ¯ ρα , + ρ α , (cid:1)(cid:19) = (cid:16) , ρ α , − ¯ ρ α , − ρ α , − ¯ ρα , − ρα , (cid:17) ← P ρ : ( u ρ , v ρ ) = (cid:18) x , x (cid:16) − ρx + xy + ¯ ρα , + ρ α , + ¯ ρxα , − ρ α , + ρα , + ρ α , + ¯ ρ α , (cid:17)(cid:19) = (cid:18) , − i (cid:0) iα , + 2 α , − α , α , + α , − α , α , − α , α , − iα , α , + 4 iα , α , − α , α , + 2 iα ′ , + 2 α ′ , (cid:1)(cid:1) . After blowing up P ρ , the system of equations takes the following form:(43) u ρ ′ = − ¯ ρ + p , ( z, u ρ , v ρ ) u ρ ,v ρ ′ = ρ α ′ , ( z ) + ρ (cid:0) α ′ , ( z ) − α , ( z ) α ′ , ( z ) (cid:1) + ¯ ρ (cid:0) α ′ , ( z ) + α , ( z ) α ′ , ( z ) (cid:1) + p , ( z, u ρ , v ρ )( u ρ ) ,U ρ ′ = − ρ α ′ , ( z ) − ρ (cid:0) α ′ , ( z ) − α , ( z ) α ′ , ( z ) (cid:1) − ¯ ρ (cid:0) α ′ , ( z ) + α , ( z ) α ′ , ( z ) (cid:1) + P , ( z, U ρ , V ρ )( V ρ ) ,V ρ ′ = − ¯ ρ + P , ( z, U ρ , V ρ )( U ρ ) V ρ . Thus, unless the condition(44) ρ α ′ , ( z ) + ρ (cid:0) α ′ , ( z ) − α , ( z ) α ′ , ( z ) (cid:1) + ¯ ρ (cid:0) α ′ , ( z ) + α , ( z ) α ′ , ( z ) (cid:1) = 0is satisfied, the system admits logarithmic singularities, u ρ =( − ρ ) / ( z − z ) / + O ( z − z ) ,v ρ = (cid:0) ρ α ′ , ( z ) + ρ (cid:0) α ′ , ( z ) − α , ( z ) α ′ , ( z ) (cid:1) + ¯ ρ (cid:0) α ′ , ( z ) + α , ( z ) α ′ , ( z ) (cid:1)(cid:1) log( z − z )+ O (( z − z ) / ) . For the solutions of the system to be free from logarithmic branch points, condition (44) must besatisfied for all ρ ∈ { , i, − , − i } . Then, one factor of u ρ and V ρ cancel in the second resp. thirdequation of system (43).To see that the singularities z ∗ of the solution are all of this form, we can write the system in theform dzdu ρ = u ρ − ¯ ρ + p , ( z, u ρ , v ρ ) ,dv ρ du ρ = p , ( z, u ρ , v ρ ) − ¯ ρ + p , ( z, u ρ , v ρ ) , which has analytic solutions for initial values ( z, u ρ , v ρ ) = ( z ∗ , , h ) on the exceptional curve L ρ , z = z ∗ − ρ ( u ρ ) + O (( u ρ ) ) , v ρ = h + O ( u ρ ) , which can be inverted to find square-root type algebraic series expansions for ( u ρ , v ρ ): u ρ = ( z − z ∗ ) / + O ( z − z ∗ ) , v ρ = h + ( z − z ∗ ) / + O ( z − z ∗ ) . The conditions (44), for ρ ∈ { , i, − , − i } , decouple into three linear independent conditions amongthe α i,j ( z ) and their derivatives, namely(2 α , ( z ) − α , ( z ) ) ′ = α ′ , ( z ) = (2 α , ( z ) + α , ( z ) ) ′ ≡ , that is, the functions 2 α , ( z ) − α , ( z ) , α , ( z ) and 2 α , ( z ) + α , ( z ) have to be equal to a constanteach. This is in agreement with the resonance conditions found in [18] for this Hamiltonian system. ESINGULARISING TRANSFORMATIONS FOR COMPLEX DIFFERENTIAL EQUATIONS 21 Proposition 4. Given the Hamiltonian (45) H = 14 (cid:0) y − x (cid:1) + α , x y + α , xy + ( α , + a ) x + ( α , + b ) y + cxy + α , x + α , y, where α , ( z ) , α , ( z ) , α , ( z ) , α , ( z ) are analytic functions and a, b, c ∈ C are constants, the systemderived from this Hamiltonian has the quasi-Painlev´e property in the strict sense, with square-root typealgebraic poles. Let S ( z ) denote the space obtained by blowing up P along the four cascades of base points, P ρ ← P ρ ← P ρ ← P ρ , ρ ∈ { , i, − , − i } , and I ( z ) = I ∪ S i =1 L i ∪ S i =1 L ii ∪ S i =1 L − i ∪ S i =1 L − ii .The space of initial values for the Hamiltonian system (45) is S ( z ) \ I ′ ( z ), at each point of which thesystem either defines an analytic solution or a solution with square-root type algebraic branch point.7. Discussion For the examples of second-order equations in Sections 4 and 5, as well as the Hamiltonian systems inSection 6 we have constructed, under the conditions by which these systems do not admit logarithmicsingularities, the analogue of the space of initial values in the sense of Okamoto’s space for the Painlev´eequations. In this case, the solutions are transversal to the exceptional curve introduced by the lastblow-up for any cascade of base points. The difference to the Painlev´e case is that, in order to obtainregular initial value problems in the coordinates of the extended phase space, an additional changeof dependent and independent variable is needed. The existence of these regular systems allows usto conclude, together with a compactness argument and Painlev´e’s lemma (Lemma 1), that the onlymovable singularities that can occur in these equations are algebraic poles.This procedure thus firstly serves as an algorithm to determine, for a given second-order equationor system of two equations, what types of singularities their solutions can develop and give conditionsunder which there are no logarithmic singularities. In the latter case, the construction of the analogueof the space of initial values allows us to show that the algebraic poles are the only movable singularities,i.e. these equations have the quasi-Painlev´e property in the strict sense.In the examples considered in this article, it is crucial that the cascades of blow-ups required toresolve the base points terminate. By a powerful theorem by Hironaka [13] for singular algebraicvarieties, the singularities of an arbitrary algebraic variety can always be resolved by a finite numberof blow-ups. This is not the case, however, for flows of vector fields, where one can reduce thesingularities, but not necessarily complete resolve them (see Chapter 6 in [1]). An example where thesequence of blow-ups does not terminate is given by Smith’s equation y ′′ + 4 y y ′ + y = 0, which isnot of Hamiltonian form. This is a hint that for this equation more complicated movable singularitiesexist than considered in this article. In fact, as noted earlier, Smith himself showed that there do existsingularities besides the algebraic poles (16), which are are known to be accumulation points of suchalgebraic poles. It would be an important step to find conditions (necessary and / or sufficient) for adifferential equation to decide whether such behaviour is possible or not. We believe that, at least forthe second-order equation in [4] and the Hamiltonian systems in [18] such behaviour is not possible,and that the equations have the strict quasi-Painlev´e property in the sense of Definition 2, althoughwe cannot show this in general without performing the blow-up calculations and checking that theseterminate in each individual case.In the Hamiltonian setting, the level sets H ( z, x, y ) = c define, for generic z and analytic functions α i,j ( z ) in the Hamiltonian H ( z, x, y ; α ij ), algebraic curves in P . For the Painlev´e Hamiltonians, andalso the system with cubic Hamiltonian in section 6.1, these level sets have genus g = 1, i.e. representelliptic curves. This also expresses itself in the fact that the Painlev´e transcendents, in general, areasymptotic to elliptic functions in certain sectors of the complex plane. For Hamiltonians of higherdegree, the level sets H ( z, x, y ) = c are algebraic curves of hyper-elliptic type, which, for all the otherexamples considered in this article have genus g = 2. The number of blow-ups required here, rangingfrom 14 to 16, is substantially larger than in the Painleve´e case. We would like to propose severalquestions which will require further investigation.Can one predict, from the form of the Hamiltonian H ( z, x, y ), how many blow-ups will be requiredto completely resolve all base points?Can a classification, similar to Sakai’s classification [25] for the Painlev´e equations, be given forHamiltonians defining algebraic curve of genus g ≥ 2, and, do there exist discrete (difference) equationswith a similar meaning as the discrete Painlev´e equations? Acknowledgements GF acknowledges the support of the National Science Center (Poland) through the grant OPUS2017/25/B/BST1/00931. TK acknowledges the support of the London Mathematical Society (LMS)and the Faculty of Mathematics, Informatics and Mechanics of the University of Warsaw (MIMUW)for grants to visit Warsaw in the years 2014, 2015 and 2016; these visits, were this research wasinitiated, were essential for the success of the project. References [1] J. M. Aroca, H. Hironaka and J. L. 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