Triangular Schlesinger systems and superelliptic curves
aa r X i v : . [ m a t h - ph ] S e p Triangular Schlesinger systems and superelliptic curves
Vladimir Dragovi´c , Renat Gontsov , Vasilisa Shramchenko Abstract
We study the Schlesinger system of partial differential equations in the case when the unknownmatrices of arbitrary size ( p × p ) are triangular and the eigenvalues of each matrix form an arithmeticprogression with a rational difference q , the same for all matrices. We show that such a systempossesses a family of solutions expressed via periods of meromorphic differentials on the Riemannsurfaces of superelliptic curves. We determine the values of the difference q , for which our solutionslead to explicit polynomial or rational solutions of the Schlesinger system. As an application ofthe (2 × Contents a at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Linear independence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 P VI : a torus with three punctures . . . . . . . . . . . . . . . . . . 184.2 Families of rational solutions of P VI : a sphere with three punctures . . . . . . . . . . . 224.3 Further rational solutions of P VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Around Okamoto’s birational transformations and classification of P V I rational solutions 27 Department of Mathematical Sciences, University of Texas at Dallas, 800 West Campbell Road, Richard-son TX 75080, USA. Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia. E-mail:
[email protected] M.S. Pinsker Laboratory no.1, Institute for Information Transmission Problems of the Russian Academy of Sciences,Bolshoy Karetny per. 19, build.1, Moscow 127051 Russia. E-mail: [email protected] Department of mathematics, University of Sherbrooke, 2500, boul. de l’Universit´e, J1K 2R1 Sherbrooke, Quebec,Canada. E-mail:
[email protected] P VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 m punctures . 355.2 Families of algebraic solutions of Garnier systems: a sphere with M + 2 punctures . . 37 We consider the Schlesinger system dB ( i ) = − N X j =1 ,j = i [ B ( i ) , B ( j ) ] a i − a j d ( a i − a j ) , i = 1 , . . . , N, (1)for ( p × p )-matrices B (1) , . . . , B ( N ) depending on the variable a = ( a , . . . , a N ) which belongs to somedisc D of the space C N \ S i = j { a i = a j } . Written in a PDEs form, this becomes ∂B ( i ) ∂a j = [ B ( i ) , B ( j ) ] a i − a j ( i = j ) , ∂B ( i ) ∂a i = − N X j =1 ,j = i [ B ( i ) , B ( j ) ] a i − a j . (2)These equations govern an isomonodromic family of Fuchsian linear differential systems dydz = (cid:18) N X i =1 B ( i ) ( a ) z − a i (cid:19) y, y ( z ) ∈ C p , (3)with varying singular points a , . . . , a N . As follows from the isomonodromic nature of the Schlesingersystem, the eigenvalues β ki of the matrices B ( i ) that solve this system are constant (see proof ofTheorem 3 from [5]). These eigenvalues are called the exponents of the Schlesinger system and of therelated isomonodromic family (3) of Fuchsian systems, at their varying singular points z = a i .As known, due to Bernard Malgrange [37], the Schlesinger system is completely integrable in D ,that is, for any initial data B (1)0 , . . . , B ( N )0 ∈ Mat( p, C ) and any a ∈ D , it has the unique solution B (1) ( a ) , . . . , B ( N ) ( a ) such that B ( i ) ( a ) = B ( i )0 , i = 1 , . . . , N . Moreover, (the pull-backs of) the matrixfunctions B ( i ) are continued meromorphically to the universal cover Z of the space C N \ S i = j { a i = a j } and their polar locus Θ ⊂ Z , called the Malgrange divisor , is described as a zero set of a function τ , holomorphic on the whole space Z . Being locally descended to D , this global τ -function, up to aholomorphic non-vanishing in D factor, coincides with the local one satisfying Miwa’s formula [27] d ln τ ( a ) = 12 N X i =1 N X j =1 ,j = i tr( B ( i ) ( a ) B ( j ) ( a )) a i − a j d ( a i − a j ) . In the present paper we are going to focus on upper triangular matrix solutions B ( i ) = ( b kli ) k,l p ,that is on those with b kli = 0 for k > l , with specific arithmetic restrictions on the exponents . Notethat the exponents in this case coincide with the diagonal entries: β ki = b kki . Triangular solutions of the Schlesinger system are those and only those with triangular initial data, since any set of N triangular matrices evaluating with respect to this system remains triangular, due to the form of the system. p = 2 , N = 3 case andclassical algebraic geometry. It is well known that in such a traceless triangular case, with a = 0 , a =1 , a = x , the off-diagonal matrix element b = b ( x ) of the matrix B (1) satisfies a hypergeometricequation: x (1 − x ) b xx + [ c − ( a + b + 1) x ] b x − ab b = 0 , where a = − P j =1 β j , b = − β , and c = 1 − β + β ). In the special case ( β , β , β ) =(1 / , − / , − / x (1 − x ) b xx + (1 − x ) b x − b = 0 , whose solutions are given by linear combinations of the periods of the differential du/v on the ellipticcurve v = u ( u − u − x )(see for example [38] and [9], formula (2.25), p. 61). Let us note that in this case β i − β i = ± / each tuple { β i , . . . , β pi } forms an arithmetic progression with the same rational difference q = n/m , where n = 0 and m > are coprime. Generalizing the relationship with the Picard–Fuchs equations, we prove that thecorresponding triangular system (2) possesses a family of solutions having algebro-geometric nature,namely they are expressed via periods of meromorphic differentials on the Riemann surfaces X a of a(varying) algebraic plane curve of superelliptic typeˆΓ a = { ( z, w ) ∈ C | w m = ( z − a ) . . . ( z − a N ) } . These expressions for the matrix entries b kli ( a ) are presented in Theorem 1 from Section 2.1.Superelliptic curves are of much interest nowadays as well as some other related classes of curves,like Z m curves or ( m, N )-curves, see [3, 4, 17, 25, 34, 39, 43, 46, 47, 52, 53] and references therein.There are some differences and ambiguity across the literature in definitions of these classes. For us,(following Zarhin, Silverberg, Frey, Shaska and others) superelliptic curves are those which can berepresented by an equation of the form: w m = P N ( z ) , where P N is any polynomial of degree N . Due to the nature of the matter considered in the presentpaper, the zeros of P N are additionally assumed to be distinct, thus the superelliptic curves consideredhere are smooth in the affine part.Note that triangular and, more generally, reducible, Schlesinger systems of arbitrary size p werealready studied by Boris Dubrovin and Marta Mazzocco in [14], where the main question was thefollowing: when are solutions of one Schlesinger system for N ( p × p )-matrices expressed via solutionsof some other “simpler” Schlesinger systems of smaller matrix size or involving less than N matrices?However, there was no restriction imposed on the exponents, and thus there was no discussion of theintegration of such systems in an explicit, in particular algebro-geometric, form. Nevertheless, it wasmentioned that triangular solutions are expressed via Lauricella hypergeometric functions (see alsosome investigations of triangular Schlesinger systems in this context in the case of small dimensions p = 2, p = 3 in [22], [23]). On the other hand, in the papers which provide particular algebro-geometricsolutions to the Schlesinger system ([12], [31], [15] for p = 2, and [17], [33] for an arbitrary p in the case3f quasi-permutation monodromy matrices of the family (3)) the specific character of the triangularcase has not been taken into consideration. The first article on triangular algebro-geometric solutionsof Schlesinger systems (in the case p = 2) is the recent [16], where the hyperelliptic case m = 2 isstudied, and our present work is an improvement and extension of the latter.In the case of n > m and N are coprime, the mentioned meromorphic differentials haveonly one pole, therefore are all of the second kind, i. e. have no residues. Thus their integrationover elements of the homology group H ( X a , Z ) is well defined. The first main result of this paperis Theorem 1 in Section 2.1 which provides families of algebro-geometric solutions of the system (2).Theorem 2 from Section 2.3 answers a delicate question about the dimension of the families of thesolutions obtained in Theorem 1.As observed in Theorem 1 in the case when n is positive and the greatest common divisor of m and N isbigger than 1, denoted ( m, N ) >
1, or when n is negative, the involved meromorphic differentials haveseveral poles P , . . . , P s and are of the third kind in general, i. e. have non-zero residues, therefore oneshould use elements of H ( X a \ { P , . . . , P s } , Z ) to integrate them correctly. We observe another effectin this case: taking small loops encircling the poles of the differentials, one expresses the matrix entries b kli ( a ) via the residues of the differentials, which turn out to be polynomials or rational functions in thevariables a , . . . , a N . These are the results of Theorem 3 in Section 3.1 for n positive and of Theorem4 from Section 3.2 for n negative.As a consequence of Theorem 3, we calculate explicitly a rational solution of the Painlev´e VI equationwith the parameters α = ( n + 1) , β = − n , γ = n , δ = 9 − n , for each positive integer n not divisible by 3, see Section 4.1, Theorem 5. In the same fashion,Theorem 6 from Section 4.2 gives a one-parameter family of rational solutions of the Painlev´e VIequation with the parameters α = (3 n + 1) , β = − n , γ = n , δ = 1 − n , for each negative integer n . Theorems 7 and 8 from Section 4.3 generalize Theorems 5 and 6,respectively, and provide much larger classes of rational solutions of Painlev´e VI equations. In addition,the corresponding families of Liouvillian solutions of Painlev´e VI equations are presented in Section4.4. The last Section 5 is devoted to the applications to Garnier systems. Some algebraic solutionsof particular Garnier systems are computed explicitly in Section 5.1, Theorem 10, and Section 5.2,Theorem 11. Let us note that the generally non-linear system (1) in the case of triangular ( p × p )-matrices B ( i ) splitsinto a set of p ( p − / N unknowns b kl ( a ) , . . . , b klN ( a )with k, l fixed. Indeed, first for each fixed k = 1 , . . . , p − d b k,k +1 i ( a ) = − N X j =1 ,j = i (cid:0) β k,k +1 i b k,k +1 j ( a ) − β k,k +1 j b k,k +1 i ( a ) (cid:1) d ( a i − a j ) a i − a j , (4)with β k,k +1 i = β ki − β k +1 i , where β ki = b kki , b k,k +11 ( a ) , . . . , b k,k +1 N ( a ). Written in a vector form for the vector b k,k +1 ( a ) = (cid:0) b k,k +11 ( a ) , . . . , b k,k +1 N ( a ) (cid:1) ⊤ ∈ C N , this becomes a Jordan–Pochhammer system d b k,k +1 = Ω b k,k +1 , with the meromorphic (holomorphic in the disc D ) coefficient matrix 1-formΩ = X j
0, with n, m coprime, and are the same for all i = 1 , . . . , N , k = 1 , . . . , p −
1. This choice of β ki − β k +1 i leads to all systems (4) have the same form d b k,k +1 i ( a ) = − nm N X j =1 ,j = i (cid:0) b k,k +1 j ( a ) − b k,k +1 i ( a ) (cid:1) d ( a i − a j ) a i − a j , i = 1 , . . . , N. (7)A similar simplification holds for each inhomogeneous system (5). Note that P Ni =1 d b k,k +1 i ( a ) ≡ ∂b k,k +1 i ∂a j = − nm b k,k +1 i − b k,k +1 j a i − a j , j = i, N X i =1 b k,k +1 i = const . We show that in this particular case of the exponents the triangular Schlesinger system possessesa family of solutions expressed via periods of meromorphic differentials on the compact Riemannsurface of the non-singular algebraic plane curve { ( z, w ) ∈ C | w m = ( z − a ) . . . ( z − a N ) } . Let us denote the corresponding projective curve by Γ a ⊂ C P . There are the following three cases: • if N > m we haveΓ a = { ( z : w : λ ) ∈ C P | w m λ N − m = ( z − λa ) . . . ( z − λa N ) } with one point at infinity ∞ = (0 : 1 : 0) ; • if m > N we haveΓ a = { ( z : w : λ ) ∈ C P | w m = λ m − N ( z − λa ) . . . ( z − λa N ) } with one point at infinity ∞ = (1 : 0 : 0) ; • if m = N we have Γ a = { ( z : w : λ ) ∈ C P | w m = ( z − λa ) . . . ( z − λa N ) } with m points at infinity ∞ = { (1 : 1 : 0), (1 : ε : 0) , . . . , (1 : ε m − : 0) } , where ε = e π i /m . The point at infinity is singular when | m − N | >
1, and non-singular when | m − N | = 1. In the specialcase m = N , the points at infinity are non-singular. In the case of n >
0, this family depends on ( p − − ν )( N −
1) parameters, where ν is the number of integers among1 , , . . . , p − m ; for negative n , the family depends on ( p − N −
1) parameters. (see Remark 4).
6y the well-known theorem on the resolution of singularities (see, for example [30, § X a and a holomorphic mapping π : X a → C P , whose image is Γ a and π : X a \ π − ( {∞} ) → Γ a \ {∞} is a biholomorphism. We introduce differentials Ω ( j )1 ( a ) , . . . , Ω ( j ) N ( a ) given on the affine part ˆΓ a of Γ a by: Ω ( j ) i ( a ) = w jn dz ( z − a i ) , i = 1 , . . . , N, j = 1 , . . . , p − . If n >
0, these differentials are holomorphic on the affine part ˆΓ a of the curve. Their holomorphicityat the points ( a i , ∈ ˆΓ a follows from the parametrization z = a i + t m , w = t O (1) , t → , of ˆΓ a near ( a i , π ∗ Ω ( j ) i of each Ω ( j ) i under the biholomorphic mapping π is a holo-morphic differential on X a \ π − ( {∞} ), with poles at π − ( {∞} ) . In the case n < , differentials Ω ( j ) i ( a ) have poles at the points ( a i ,
0) of ˆΓ a . Their pull-backs π ∗ Ω ( j ) i have poles at π − (( a i , i = 1 , . . . , N and vanish at π − ( {∞} ) as we explain in the next section.For simplicity of notation, we denote the pull-backs π ∗ Ω ( j ) i of the differentials again by Ω ( j ) i , keepingin mind the change of variables R γ π ∗ ω = R π ( γ ) ω in a definite integral, and π − (( a i , a i , Theorem 1
Let the eigenvalues of each matrix B ( i ) , i = 1 , . . . , N , have the same rational difference: β ki − β k +1 i = n/m ( k = 1 , . . . , p − , where n ∈ Z and m are coprime. If n > assume also that m > . Then the following triangular matrices B ( i ) = ( b kli ) satisfy system (2) : b kli ( a ) = I γ l − k Ω ( l − k ) i ( a ) , l > k, where γ , . . . , γ p − are arbitrary elements of (a) H ( X a , Z ) if m , N are coprime and n > , (b) H ( X a \ π − ( {∞} ) , Z ) if m , N are not coprime and n > . (c) H ( X a \ { ( a , , . . . , ( a N , } , Z ) if n < . ( These cycles do not depend on a ∈ D if D is sufficiently small. ) Remark 1
In the case n > , we assume that m > because the case m = 1 is trivial: the differentials Ω ( j ) i are exact in that case and thus the B ( i ) are constant diagonal matrices. Before proving Theorem 1 let us analyze how the local structure of the curve Γ a at its singular pointat infinity depends on the values of m and N , and how the differentials Ω ( j ) i ( a ) behave near theirpoles, the points of the set π − ( {∞} ). 7 .2 The local structure of Γ a at infinity The implicit function theorem cannot give us a local parameter near the singular point of Γ a , forthis purpose one should consider the Puiseux expansions at the point at infinity (using the Newtonpolygon of the curve, see details in [30, §§ π − ( {∞} ). After doing this exercise we arrive to thefollowing two cases, assuming n to be positive.(a) Let N and m be coprime. In this case the set π − ( {∞} ) consists of one point P , hence thedifferentials Ω ( j ) i ( a ) have the only pole and are all of the second kind. That is why the integrationis correctly defined along the elements of H ( X a , Z ) in this case.In a local parameter t in a neighbourhood of the point P ∈ X a , t ( P ) = 0, the mapping π : X a → Γ a (the parametrization of Γ a ) can be chosen to have the form z = 1 /t m , w = 1 t N (1 − a t m ) /m . . . (1 − a N t m ) /m . The genus g ( X a ) of the Riemann surface X a equals g ( X a ) = 12 ( m − N − N and m be not coprime, that is let there be an integer s > N = sN , m = sm , with coprime N and m . In this case the set π − ( {∞} ) consists of s points P , . . . , P s , and thedifferential Ω ( j ) i ( a ) has s poles, one at each of the points P , . . . , P s at infinity, being of the thirdkind in general. Thus, for the integration of Ω ( j ) i ( a ) to be well-defined, one uses the elements of H ( X a \ { P , . . . , P s } , Z ) as integration contours.In a local parameter t at each point P k ∈ π − ( {∞} ), t ( P k ) = 0, the mapping π : X a → Γ a (theparametrization of Γ a ) can be chosen to have the form z = 1 /t m , w = ε k − t N (1 − a t m ) /m . . . (1 − a N t m ) /m , ε = e π i /s , which implies the coordinate representation of the differentials near the poles P k , k = 1 , . . . , s :Ω ( j ) i = w jn dzz − a i = ν k (1 − a t m ) jn/m . . . (1 − a N t m ) jn/m t jnN +1 (1 − a i t m ) dt, ν k = − m ε jn ( k − . (8)The genus g ( X a ) of the Riemann surface X a equals g ( X a ) = 12 (cid:0) ( m − N − − s + 1 (cid:1) in this case.Assuming n to be negative, we see from (8) that the differential Ω ( j ) i ( a ) vanishes at the points P , . . . , P s at infinity. In this case it has N poles ( a , , . . . , ( a N , ∈ X a and for the integration of Ω ( j ) i ( a ) to bewell-defined, one uses the elements of H ( X a \ { ( a , , . . . , ( a N , } , Z ) as integration contours. Remark 2
A non-singular case N = m can be regarded as a particular case of ( b ) , with X a = Γ a , {∞} = { P , . . . , P m } , and s = N = m , N = m = 1 . .3 Proof of Theorem 1 Note that for each fixed i = 1 , . . . , N , the functions b kli with the same l − k , defined in Theorem 1,coincide. As for every s such that k < s < l , there exists t such that k < t < l and l − s = t − k (andhence s − k = l − t ), the inhomogeneity (6) of system (5) vanishes. Therefore suffices to prove thatthe functions b kl , . . . , b klN satisfy (5) with F kli = 0: d b kli ( a ) = − ( l − k ) nm N X j =1 ,j = i (cid:0) b klj ( a ) − b kli ( a ) (cid:1) d ( a i − a j ) a i − a j , or, written in an equivalent PDEs form, ∂b kli ∂a j = − ( l − k ) nm b kli − b klj a i − a j , j = i, N X i =1 b kli = const . Differentiating the equality w m = P ( z, a ) := ( z − a ) . . . ( z − a N ) with respect to a j , we obtain mw m − ∂w∂a j = − P ( z, a ) z − a j or, equivalently, ∂w∂a j = − m wz − a j . Thus for j = i one has ∂b kli ∂a j = I γ l − k ∂ Ω ( l − k ) i ( a ) ∂a j = − nm ( l − k ) I γ l − k w ( l − k ) n dz ( z − a i )( z − a j ) == − n ( l − k ) m ( a i − a j ) I γ l − k (cid:16) z − a i − z − a j (cid:17) w ( l − k ) n dz = − ( l − k ) nm b kli − b klj a i − a j . The proof of P Ni =1 b kli = const is also a straightforward computation: for every fixed a there holds mw m − dw = N X i =1 P ( z, a ) dzz − a i and thus m dw = N X i =1 w dzz − a i . Using this we obtain N X i =1 b kli = I γ l − k N X i =1 w ( l − k ) n dz ( z − a i ) = m ( l − k ) n I γ l − k dw ( l − k ) n , which is zero as an integral of an exact differential over a cycle. This proves Theorem 1. (cid:3) emark 3 As explained in Section 2.2, the number of independent contours in the homology groups H ( X a , Z ) or H ( X a \ π − ( {∞} ) , Z ) is L = ( m − N −
1) = 2 g + s − , where s = ( N, m ) is the greatestcommon divisor of m and N and g is the genus of the Riemann surface X a . If N and m are coprime,then there are L = 2 g basic cycles in H ( X a , Z ) . If ( N, m ) = s > then there are s points P , . . . , P s inthe set π − ( {∞} ) and thus L = 2 g + s − basis cycles in H ( X a \{ P , . . . , P s } , Z ) . In the homology group H ( X a \{ ( a , , . . . , ( a N , } , Z ) , the number of generators is L = ( m − N − N − s = 2 g + N − . Denoting A , . . . , A L generators of H ( X a \ π − ( {∞} ) , Z ) , in the case n > , and generators of H ( X a \ { ( a , , . . . , ( a N , } , Z ) in the case n < , we see that Theorem 1 gives us the followingfamily of solutions for b kli for each pair of indices l > k : taking γ l − k = P Lj =1 c ( l − k ) j A j with c ( l − k ) j ∈ C we have b kli ( a ) = L X j =1 c ( l − k ) j I A j Ω ( l − k ) i ( a ) , c ( l − k ) j ∈ C . The number of independent parameters describing this family will be discussed in Section 2.4.
Note that for each fixed pair ( k, l ), 1 k < l p , the vector (cid:0) b kl ( a ) , . . . , b klN ( a ) (cid:1) ⊤ = (cid:16)I γ Ω ( l − k )1 ( a ) , . . . , I γ Ω ( l − k ) N ( a ) (cid:17) ⊤ is a solution of the Jordan–Pochhammer linear differential system of size N , where the cycle γ belongsto H ( X a , Z ) or H ( X a \{ P , . . . , P s } , Z ) in the case of positive n and to H ( X a \{ ( a , , . . . , ( a N , } , Z )in the case of negative n . As P Ni =1 b kli = 0, the complete integrability of the latter system implies thatthis vector belongs to an ( N − N -dimensional solution space of thesystem. Thus it is natural to ask whether among the columns of the matrix B ( a ) = H A Ω ( l − k )1 ( a ) . . . H A L Ω ( l − k )1 ( a )... ... H A Ω ( l − k ) N ( a ) . . . H A L Ω ( l − k ) N ( a ) , there are N − C . In case the answer is positive, we have an ( N − n is positive, then L = ( m − N −
1) = 2 g + s − s = ( N, m ) , see Remark 3, and the contours of integration A , . . . , A L are generators of H ( X a \ { P , . . . , P s } , Z ) . In the case of negative n , we have L = ( m − N −
1) + N − s = 2 g + N − A , . . . , A L are some generators of the group H ( X a \{ ( a , , . . . , ( a N , } , Z ) . Theorem 2
Let n and m be comprime. If n is positive and l − k is not divisible by m or if n isnegative, then among the columns of the matrix B there are N − linearly independent over C . emark 4 Let n > and ν be the number of integers among , , . . . , p − that are divisible by m . Then Theorem 2 implies that the algebro-geometric expressions of Theorem 1 generate a ( p − − ν )( N − -parameter family of solutions of the triangular Schlesinger system (2) with fixed exponentsas in Theorem 1, whose solutions moduli space is of dimension p ( p − N − / . In the case n < , Theorem 1 yields a ( p − N − -parameter family of solutions to such triangular Schlesinger system (2) . In particular, in the (2 × -case, p = 2 , all solutions of such a system are algebro-geometric.Proof of Theorem 2. First, let us denote j = l − k and reformulate the statement of the theorem inthe following way: Let n and m be comprime. If n is positive and an integer j is not divisible by m or if n is negative then there exists a ∈ D such that among the N differentialsΩ ( j ) i ( a ) = w jn dzz − a i , i = 1 , . . . , N, (9)any N − H ( X a ).Indeed, any N − B are linearly dependent over C if and only if rk B ( a ) < N − a ∈ D (this is due to that the columns of B are solutions of a completely integrable lineardifferential system). The latter holds if and only if any N − B ( a ) are linearly dependent,that is, a nontrivial linear combination of any N − ( j )1 ( a ) , . . . , Ω ( j ) N ( a ) has allits periods equal to zero, which is equivalent to being an exact differential.Let us now prove that among the differentials (9) any N − α , . . . , α N − ∈ C such that the following linear combination ϕ = N − X i =1 α i Ω ( j ) i ( a ) = N − X i =1 α i w jn dzz − a i (10)is an exact differential on the Riemann surface X a of the algebraic curve w m = P ( z ) (where P ( z ) =( z − a ) . . . ( z − a N ) with a , . . . , a N fixed).Denote J : ˆΓ a → ˆΓ a the symmetry of the underlying algebraic curve: J ( z, w ) = ( z, εw ) with ε beingan m th primitive root of unity: ε = e π i /m and consider separately the cases of positive and negativevalues of n . • Let n > . In this case we assume that j is not divisible by m . The following integral of theexact differential ϕ y ( z, w ) = Z ( z,w )( a N , ϕ (11)is a well-defined meromorphic function on X a . We have J ∗ ϕ = ε jn ϕ and J ∗ y = Z J ( z,w )( a N , ϕ = Z J ( z,w ) J ( a N , ϕ = Z ( z,w )( a N , J ∗ ϕ = ε jn y . Let k be the smallest integer such that km − jn > . If m > jn , then k = 1, otherwise k > . The following meromorphic function on the surface X a f = w km − jn y (12)11s invariant under the symmetry J . Therefore it descends to a meromorphic function of z definedon the base of the ramified covering z : X a → C P . Given that this function has the only poleat the point at infinity, we conclude that f ( z ) is a polynomial.Recall from Section 2.2 that s = N/N = m/m and each differential Ω ( j ) i ( a ) has s poles atpoints at infinity P , . . . , P s ∈ X a of order jnN + 1 and that the local parameter at each ofthese points is t = z − /m . Thus the differential ϕ has poles at the points P , . . . , P s of order atmost jnN + 1 and the poles of function y at those points are of order at most jnN . Given thatthe function w has a pole of order N at each of the points P , . . . , P s , we obtain that the polesof the function f = w km − jn y at the points P , . . . , P s are of the order at most kmN . Therefore f is a polynomial in z of degree at most kmN /m = kN . On the other hand, the polynomial f has N zeros at z = a i , i = 1 , . . . , N . Let us show thateach zero is of multiplicity at least k .Consider the function J ∗ y evaluated at a branch point ( a i ,
0) of the curve. On one hand, weknow that J ∗ y = ε jn y and therefore J ∗ y ( a i ,
0) = ε jn y ( a i , . On the other hand, we have J ∗ y ( a i ,
0) = Z J ( a i , a N , ϕ = Z ( a i , a N , ϕ = y ( a i , . The two above relations imply that ε jn y ( a i ,
0) = y ( a i ,
0) and, since jn is not a multiple of m (because j is not, and n , m are coprime), we conclude that y ( a i ,
0) = 0 (note that y does nothave a pole at ( a i ,
0) since ϕ vanishes there). The differential dy = ϕ vanishes at ( a i ,
0) to theorder jn − t i = ( z − a i ) /m ) .Thus we have that the function y vanishes at every finite ramification point to the order jn . Coming back to f ( z ) defined by (12) and considered as function on the z -sphere, we find thatit behaves as O (( z − a i ) k ) at the branch point z = a i and thus it has N zeros of order k at a , . . . , a N . We can now conclude that f ( z ) is proportional to P k ( z ) : w km − jn y = c P k ( z )with some constant c which may depend on the { a i } . From here we obtain y = c P k ( z ) w jn − km = c w jn and thus ϕ = dy = c dw jn . (13) • Let n < j is not a multiple of m . In this case the function w jn has a zero ateach of the points P , . . . , P s and therefore the differential ϕ does. Define y ( P ) = s X i =1 Z PP i ϕ , (14)which is a well-defined meromorphic function on X a , given that the differential ϕ is exact.12he symmetry J permutes the set of the points at infinity { P , . . . , P s } having the period s onthis set: J s ( P i ) = P i , i = 1 , . . . , s . We have the following behaviour under the symmetry J : J ∗ ϕ = ε jn ϕ and J ∗ y = s X i =1 Z J ( P ) P i ϕ = s X i =1 Z J ( P ) J ( P i ) ϕ = s X i =1 Z PP i J ∗ ϕ = ε jn y . Let k be the smallest integer such that − jn − km < X a : f = w − jn − km y . (15)Similarly to the previous case, this function is invariant under the symmetry J and thereforedescends to a meromorphic function of z defined on the base of the ramified covering z : X a → C P , having now poles at z = a i with i = 1 , . . . , N and a zero at z = ∞ .The function w − jn − km has a pole of order jn + km at ( a i ,
0) (with respect to the local parameter t i = ( z − a i ) /m ) and the function y has a pole of order j | n | at ( a i , ϕ , therefore f ( z ) defined by (15) and considered as function on the z -sphere,has a pole of order k at each point z = a i .Let us analyze the order of the zero of f ( z ) at the point z = ∞ . Consider the function ( J ∗ ) s y =( J s ) ∗ y evaluated at any point P i ∈ X a with i = 1 , . . . , s . On one hand, we know that ( J ∗ ) s y = ε jns y and therefore ( J ∗ ) s y ( P i ) = ε jns y ( P i ) . On the other hand, we have( J ∗ ) s y ( P i ) = ( J s ) ∗ y ( P i ) = y ( J s ( P i )) = y ( P i ) . The two above relations imply that ε jns y ( P i ) = y ( P i ) . Note that y does not have a pole at P i as it would lead to a pole of ϕ at P i . Therefore, given the assumption that j is not a multiple of m , we conclude that jns is not a multiple of m and thus y ( P i ) = 0 . The differential dy = s ϕ vanishes at P i to the order j | n | N − t = z − /m near thispoint). Thus we have that the function y vanishes at any point P i with i = 1 , . . . , s to the order j | n | N and therefore the function f = w − jn − km y vanishes at P i to the order kmN . Hence, asfunction on the z -sphere, f ( z ) has a zero of order kmN /m = kN at infinity.Thus we obtain, similarly to the case n > f = w − jn − km y = cP k ( z )with some constant c which may depend on the { a i } . From here we get y = c P − k ( z ) w jn + km = c w jn and thus ϕ = 1 s dy = c dw jn . (16) • Finally, let n < j is a multiple of m , that is there is an integer r such that j = rm . Denote h = ( r, s ) with s = hs and r = hr , where r and s are coprime. In this case,the surface X a can be seen as a ramified covering of the Riemann surface b X a of the algebraiccurve b w s = P ( z ) with b w = w hm . Differentials Ω ( j ) i ( a ) can be considered as being defined on b X a : Ω ( j ) i ( a ) = w jn dz ( z − a i ) = w hr m n dz ( z − a i ) = b w r n dz ( z − a i ) =: b Ω ( r ) i ( a )13nd differential ϕ (10) is also defined on b X a as a linear combination of b Ω ( r ) i ( a ) : ϕ = N − X i =1 α i Ω ( j ) i ( a ) = N − X i =1 α i b Ω ( r ) i ( a ) . Since ( r , s ) = 1 , by the previous case of j being non divisible by m and a negative n we have ϕ = c d b w r n = c dw jn . (17)Relations (13), (16) and (17) imply N − X i =1 α i w jn dzz − a i = c dw jn . Knowing that dw jn = jn w jn dww = jnm w jn dPP = jnm w jn N X i =1 dzz − a i , the previous equality becomes N − X i =1 α i dzz − a i = c jnm N X i =1 dzz − a i . Given that { a i } Ni =1 is an arbitrary set of distinct complex numbers, the above equality is only possibleif α i = c = 0 for all i = 1 , . . . , N − N − ( j )1 , . . . , Ω ( j ) N − are linearlyindependent in H ( X a ) . (cid:3) Our differentials Ω ( j ) i ( a ) defined on the compact Riemann surface X a have poles at points at infinityor at finite ramification points, depending on the sign of n . In general, the residues at these polesof Ω ( j ) i ( a ) are non-zero and, according to Theorem 1, give rise to solutions of the Schlesinger system(2). In this section we show that such solutions are polynomial in a , . . . , a N in the case of n > a , . . . , a N in the case of n <
0. This will lead us, in subsequent sections, to rationalsolutions of some Painlev´e VI equations and to algebraic solutions of some Garnier systems.
In this section we consider the case of n >
0, when differentials Ω ( j ) i ( a ) have poles at s points atinfinity, s being the greatest common divisor of m and N . Thus in the case of coprime m and N theresidue of Ω ( j ) i at its only pole vanishes. In the case of s >
1, however, Ω ( j ) i has s poles with possiblynon-zero residues, which leads to the following statement on polynomial solutions of the Schlesingersystem. 14 heorem 3 Let the eigenvalues of each matrix B ( i ) , i = 1 , . . . , N , have the same rational difference: β ji − β j +1 i = n/m , j = 1 , . . . , p − , with n > , m > coprime , and s = ( m, N ) > be the greatestcommon divisor of the integers m and N . If there is an integer j ∈ { , . . . , p − } such that sj/m ∈ Z ,while j/m Z , then the set of triangular solutions of system (2) contains a family of non-trivialpolynomial ones: • b kli ( a ) = c l − k P ( l − k ) i ( a ) , where c l − k ∈ C is an arbitrary constant and P ( l − k ) i is a non-zero polynomialof degree ( l − k ) nm N given by (19) , if l and k are such that ( l − k ) s/m ∈ Z and ( l − k ) /m Z ; • b kli ( a ) ≡ otherwise.Proof. As explained in Section 2.2 for n >
0, in the case N = sN , m = sm , where N , m arecoprime, each differential Ω ( j ) i ( a ) has s poles P , . . . , P s ∈ X a . In a local parameter t at each pole P α such that t ( P α ) = 0, the coordinate representation of Ω ( j ) i is of the form:Ω ( j ) i = ν α (1 − a t m ) jn/m . . . (1 − a N t m ) jn/m t jnN +1 (1 − a i t m ) dt, with ν α = − m e π i jn ( α − /s . Hence, Ω ( j ) i = ν α dtt jnN +1 ∞ X k =0 (cid:18) jn/mk (cid:19) ( − a t m ) k . . . ∞ X k N =0 (cid:18) jn/mk N (cid:19) ( − a N t m ) k N ∞ X q =0 ( a i t m ) q == ν α dtt jnN +1 ∞ X r =0 h X k + ... + k N + q = r ( − r − q (cid:18) jn/mk (cid:19) . . . (cid:18) jn/mk N (cid:19) a k . . . a k N N a qi i t rm , (18)where we use generalized binomial coefficients defined for any β ∈ R and j ∈ N by (cid:18) βj (cid:19) = β ( β − · · · ( β − j + 1) j ! , (cid:18) β (cid:19) = 1 . Thus, due to Theorem 1, the integration of Ω ( l − k ) i ( a ), i = 1 , . . . , N , along a small loop γ l − k encirclingany pole P α gives b kli ( a ) = c l − k res P α Ω ( l − k ) i ( a ) , c l − k ∈ C . As follows from (18), the residue of Ω ( l − k ) i ( a ) equals zero if ( l − k ) nN is not a multiple of m , whichis equivalent to l − k not being a multiple of m because m and N , as well as m and n , are coprime.Therefore, b kli ( a ) ≡ l − k ) /m = ( l − k ) s/m Z .In the case ( l − k ) s/m is an integer, denoting d := ( l − k ) nm s , we haveres P α Ω ( l − k ) i ( a ) = X k + ... + k N + q = N d ( − q (cid:18) d/sk (cid:19) . . . (cid:18) d/sk N (cid:19) a k . . . a k N N a qi (19)up to an overall constant factor, that is b kli ( a ) is a polynomial of degree N d = ( l − k ) nm N . However,this polynomial is identically zero if ( l − k ) /m ∈ Z , since the differential Ω ( l − k ) i ( a ) is exact in this case.This finishes the proof of the theorem. (cid:3) .2 Rational solutions of the Schlesinger system In this section we consider the case of n <
0, when the differentials Ω ( j ) i ( a ) have poles at the finiteramification points ( a , , . . . , ( a N , ∈ ˆΓ a . Contrary to the case of positive n , now the residues ofΩ ( j ) i ( a ) at their poles are non-zero only if j is a multiple of m and we have the following statement onrational solutions of the Schlesinger system. Theorem 4
Let the eigenvalues of each matrix B ( i ) , i = 1 , . . . , N , have the same rational difference: β ji − β j +1 i = n/m , j = 1 , . . . , p − , with n < , m > coprime. If there is an integer j ∈ { , . . . , p − } such that j/m ∈ Z , then the set of triangular solutions of system (2) contains a family of non-trivialrational ones: • b kli ( a ) = c l − k R ( l − k ) i ( a ) , if l and k are such that ( l − k ) /m ∈ Z , where c l − k ∈ C is an arbitraryconstant and R ( l − k ) i is a non-zero rational function given by (20) , for i = ν , and by (21) for i = ν ,with an arbitrary number ν ∈ { , . . . , N } initially chosen; • b kli ( a ) ≡ otherwise.Proof. We have the following parametrization of ˆΓ a near each ramification point ( a ν , z = a ν + t mν , w = t ν N Y h =1 ,h = ν ( a ν − a h + t mν ) /m , t ν → , whence the coordinate representation of Ω ( j ) i is of the form:Ω ( j ) i = w − j | n | z − a i dz = mt j | n |− m +1 ν ( a ν − a i + t mν ) N Y h =1 ,h = ν ( a ν − a h + t mν ) − j | n | /m dt ν . Hence, for i = ν one hasΩ ( j ) i = mt j | n |− m +1 ν ( a ν − a i ) − (cid:16) t mν a ν − a i (cid:17) − N Y h =1 ,h = ν ( a ν − a h ) − j | n | /m (cid:16) t mν a ν − a h (cid:17) − j | n | /m dt ν == m dt ν t j | n |− m +1 ν ∞ X r =0 h X k + ... + k N = r ( − k ν ( a ν − a i ) k ν +1 N Y h =1 ,h = ν (cid:0) − j | n | /mk h (cid:1) ( a ν − a h ) k h + j | n | /m i t rmν , while Ω ( j ) ν = mt j | n | +1 ν N Y h =1 ,h = ν ( a ν − a h ) − j | n | /m (cid:16) t mν a ν − a h (cid:17) − j | n | /m dt ν == m dt ν t j | n | +1 ν ∞ X r =0 h ′ X k + ... + k N = r N Y h =1 ,h = ν (cid:0) − j | n | /mk h (cid:1) ( a ν − a h ) k h + j | n | /m i t rmν , where the summation index k ν is missed in the above sum P ′ .Like in the previous theorem, the integration of Ω ( l − k ) i ( a ), i = 1 , . . . , N , along a small loop γ l − k encircling any pole ( a ν ,
0) gives b kli ( a ) = c l − k res ( a ν , Ω ( l − k ) i ( a ) , c l − k ∈ C .
16s follows from the above coordinate representation, the residue of Ω ( l − k ) i ( a ) equals zero if ( l − k ) n isnot a multiple of m , which is equivalent to l − k not being a multiple of m . Therefore, b kli ( a ) ≡ l − k ) /m Z .In the case ( l − k ) /m is an integer, denoting d := ( l − k ) | n | /m , we haveres ( a ν , Ω ( l − k ) i ( a ) = X k + ... + k N = d − ( − k ν ( a ν − a i ) k ν +1 N Y h =1 ,h = ν (cid:18) − dk h (cid:19) a ν − a h ) k h + d , i = 1 , . . . , N, i = ν, (20)up to an overall constant factor, andres ( a ν , Ω ( l − k ) ν ( a ) = ′ X k + ... + k N = d N Y h =1 ,h = ν (cid:18) − dk h (cid:19) a ν − a h ) k h + d . (21)This finishes the proof of the theorem. (cid:3) As is well known, in the case p = 2, N = 3 (assuming ( a , a , a ) = (0 , , x ), x ∈ C \ { , } ) theSchlesinger system for traceless (2 × B (1) ( x ), B (2) ( x ), B (3) ( x ), dB (1) dx = [ B (3) , B (1) ] x , dB (2) dx = [ B (3) , B (2) ] x − , B (1) + B (2) + B (3) = (cid:18) − β ∞ β ∞ (cid:19) (22)(if β ∞ = 0, the last matrix sum is a Jordan cell), corresponds to the sixth Painlev´e equationP VI ( α, β, γ, δ ) d ydx =12 (cid:18) y + 1 y − y − x (cid:19)(cid:18) dydx (cid:19) − (cid:18) x + 1 x − y − x (cid:19) dydx + y ( y − y − x ) x ( x − (cid:18) α + β xy + γ x − y − + δ x ( x − y − x ) (cid:19) . The parameters ( α, β, γ, δ ) of P VI are computed from the eigenvalues ± β i of the matrices B ( i ) , i =1 , ,
3, as follows: α = (2 β ∞ − , β = − β , γ = 2 β , δ = 12 − β . Namely, the function y ( x ) = xb b + (1 − x ) b , (23)where b i is a (1 , B ( i ) , satisfies the Painlev´e VI with the above parameters.In our triangular case, solutions B ( i ) = (cid:18) β i b i ( x )0 − β i (cid:19) , i = 1 , , , (24)17f the Schlesinger system (22) are hypergeometric. For example, as a consequence of the Schlesingerequations, the functions b and b satisfy the following linear differential system: b ′ = x (( β + β ) b + β b ) b ′ = x − ( β b + ( β + β ) b ) (25)and thus solve the hypergeometric linear differential equations of the form (see [18, Ch. 4, § b ′′ + (2 β + 2 β −
1) + (1 − β − β − β ) xx ( x − b ′ + 4 β ( β + β + β ) x ( x − b = 0 , (26) b ′′ + (2 β + 2 β ) + (1 − β − β − β ) xx ( x − b ′ + 4 β ( β + β + β ) x ( x − b = 0 , (27)while b = − b − b . This means that solutions of a triangular
Schlesinger system (22) always lead to hypergeometricsolutions of the corresponding sixth Painlev´e equation through (23). More precisely, from a generaltwo-parameter family of solutions of (26) linearly parameterized by constants c , c , one obtains b using the first equation of (25), and then b = − b − b . A particular one-parameter family of solutionsof the corresponding sixth Painlev´e equation parametrized by the ratio c /c is then obtained by (23).In the case we consider, the eigenvalues in (24) are given by β = β = β = n m , and β ∞ = − n m , with any coprime integers n > m > n < m > y ( x ) of the sixthPainlev´e equation P VI (cid:16) (3 n + m ) m , − n m , n m , m − n m (cid:17) : y ( x ) = xb b + (1 − x ) b , (28) b = I γ w n dzz + c I γ w n dzz , b = I γ w n dzz − x + c I γ w n dzz − x , c ∈ C , where γ , γ are suitable closed contours on the Riemann surface X x of the curve w m = z ( z − z − x )with the only variable branch point x ∈ C \ { , } (or, on the X x punctured at three points, the polesof the differentials w n dz/z , w n dz/ ( z − x ), w n dz/ ( z − P VI : a torus with three punctures In this section we consider the case (b) of Theorem 1 in the context of Painlev´e VI equations, that is,the case of n > m > p = 2 and s = ( m, N ) = ( m,
3) = 3. Let us analyze the requirements of18heorem 3 in this case and see when we can apply this theorem to obtain polynomial expressions forthe b i ’s.As s = 3, the requirement s/m ∈ Z of Theorem 3 implies that m = 3. Hence we deal with theRiemann surface X x of the curve w = z ( z − z − x )punctured at three points P , P , P at infinity. The genus of X x equals g = 12 (cid:0) ( m − N − − s + 1 (cid:1) = 1 , that is, this is a torus and there are four basic cycles on X x \ { P , P , P } .Computing the residues of the differentials w n dz/z , w n dz/ ( z − w n dz/ ( z − x ), say at the pole P ,directly or applying the formula (19) with N = 3, N = 1, d = n , s = 3, ( a , a , a ) = (0 , , x ) weobtain polynomial solutions (24) of the Schlesinger system (22), with β = β = β = n/ Z , β ∞ = − n/ b ( x ) = b P1 ( x ) = ( − n +1 n X j =0 (cid:18) n/ j (cid:19)(cid:18) n/ n − j (cid:19) x j ,b ( x ) = b P2 ( x ) = ( − n +1 n X j =0 (cid:18) n/ j (cid:19)(cid:18) n/ − n − j (cid:19) x j ,b ( x ) = b P3 ( x ) = ( − n +1 n X j =0 (cid:18) n/ − j (cid:19)(cid:18) n/ n − j (cid:19) x j . The functions b P1 and b P2 are related to each other by system (25). They give degree n polynomialsolutions to the hypergeometric equations (26) and (27), respectively. Furthermore, the polynomials xb P1 ( x ) and b P1 ( x ) + (1 − x ) b P3 ( x ) = − b P2 ( x ) − xb P3 ( x ) = ( − n n + 1) n n +1 X j =0 (cid:18) n/ j (cid:19)(cid:18) n/ n − j + 1 (cid:19) x j give, via (28), a rational solution to the Painlev´e VI equation with the parameters( α, β, γ, δ ) = (cid:18) (2 β ∞ − , − β , β , − β (cid:19) = (cid:18) ( n + 1) , − n , n , − n (cid:19) , and thus we obtain the following assertion. Theorem 5
For every positive integer n not divisible by , the polynomials P n +1 ( x ) = n +1 X j =1 (cid:18) n/ j − (cid:19)(cid:18) n/ n − j + 1 (cid:19) x j and Q n +1 ( x ) = − n + 1) n n +1 X j =0 (cid:18) n/ j (cid:19)(cid:18) n/ n − j + 1 (cid:19) x j of degree n + 1 define the rational solution y ( x ) = P n +1 ( x ) /Q n +1 ( x ) of the sixth Painlev´e equation P VI ( α, β, γ, δ ) with the parameters α = ( n + 1) , β = − n , γ = n , δ = 9 − n . M , M , M of the triangular Schlesinger isomonodromicfamily corresponding to the above b P i ’s, at the points z = 0, z = 1, z = x respectively, equals ± I , sincethe eigenvalues e ± π i β i = e ± π i n/ of each M i do not equal ±
1. Therefore, due to Lemma 3.3 from[41], the monodromy of this family is commutative. In fact, the commutativity of the monodromy ofa Schlesinger isomonodromic family is a general necessary condition for the corresponding solution ofthe sixth Painlev´e equation to be rational, see Remark 6 below.
Example 1
Let us compute degree n polynomial solutions to the hypergeometric equations (26),(27), with β = β = β = n/
6, and a rational solution to the corresponding Painlev´e VI equation inthe case n = 1, n = 2, and n = 4.1. For n = 1, we obtain the following linear functions: b P1 ( x ) = x + 13 ; b P2 ( x ) = x −
23 ; b P3 ( x ) = − x + 13 , where b P1 satisfies (26) and b P2 satisfies (27), with β = β = β = 1 /
6. The correspondingrational solution of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) is given by y ( x ) = x ( x + 1)2 x − x + 2 .
2. For n = 2, we obtain the functions b P1 ( x ) = 19 (cid:0) x − x + 1 (cid:1) ; b P2 ( x ) = 19 (cid:0) x + 2 x − (cid:1) ; b P3 ( x ) = 19 (cid:0) − x + 2 x + 1 (cid:1) , leading to the following rational solution of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) : y ( x ) = x ( x − x + 1)2 x − x − x + 2 .
3. For n = 4, we get the functions b P1 ( x ) = − (cid:0) x − x + 12 x − x + 5 (cid:1) ; b P2 ( x ) = − (cid:0) x − x − x + 20 x − (cid:1) ; b P3 ( x ) = − (cid:0) − x + 20 x − x − x + 5 (cid:1) , leading to the following rational solution of the sixth Painlev´e equation P VI (cid:0) , − , , − (cid:1) : y ( x ) = x (5 x − x + 12 x − x + 5)10 x − x + 10 x + 10 x − x + 10 . Remark 5
The polynomials b P1 ( x ) and Q n +1 ( x ) are both reciprocal. Let us recall that a polynomial ofdegree n of the form P nj =0 a j x j is reciprocal if a j = a n − j , for all j = 0 , . . . , n . Thus, for n = 2 k +1 odd,the polynomial P n +1 ( x ) has zeros at , − and k pairs of zeros z j , z − j , while the polynomial Q n +1 ( x ) has k + 1 pairs of zeros w j , w − j . For n = 2 k even, the polynomial P n +1 ( x ) has a zero at and k pairsof zeros z j , z − j , while the polynomial Q n +1 ( x ) has a zero at − and k pairs of zeros w j , w − j . Since,the polynomials have all coefficients real, this implies that the roots of each polynomial are situatedsymmetrically with respect to the real axis and all roots different from are placed symmetrically withrespect to the unit circle in the sense of inversion. Some initial numerical experiments show quiteinteresting behaviour of the zeros of the polynomials P and Q . Figures 1 and 2 show the distributionof zeros of P and Q with n = 25 and n = 28 . These nice numerical patterns deserve further studies. P and Q .Figure 2: Distribution of zeros for P and Q . Remark 6
All Painlev´e VI equations which have non-degenerate rational solutions were classifiedby Marta Mazzocco [40] who proved that they occur if and only if for the corresponding Schlesingersystem (22) there holds β ∞ + ε β + ε β + ε β ∈ Z , for some choice of ε i ∈ {± } and at least one β i ∈ Z . The monodromy of the correspondingSchlesinger isomonodromic family is necessarily commutative. As stated in [41], all such rationalsolutions are equivalent, via Okamoto’s birational canonical transformations [45] and up to symmetries,to the following solutions: y ( x ) = x (1 + 2 β ) + (1 + 2 β ) x , β ∞ + β + β + β = 0 , β = 12 ; (29) y ( x ) = 2( β + β x ) − β − β x (2 β + 2 β − β + β x ) , β ∞ + β + β + β = 0 , β = − . (30) The mentioned symmetries are given by i) x − x, y − y, β ↔ β , ii) x x , y y , β ∞ ↔ β + 12 , iii) x xx − , y y − x − x , β ↔ β . As an illustration, we see that the solution obtained in Example 1 for n = 1 is equivalent to (30) with β = − , β = β = 1 / , β ∞ = 2 / by the symmetry ii).However, given that the birational canonical transformations are not easy to apply, obtaining rationalsolutions of Painlev´e VI equations with an arbitrary degree of the numerator and denominator from We will refer to a more recent arXiv version [41], where some instances of [40] were formulated differently. he basic ones (29) and (30) is not a simple task. Rational solutions of Painlev´e equations typically canbe expressed in terms of logarithmic derivatives of special polynomials that are defined through secondorder recursion relations, possess a determinant structure and whose zeroes have a highly symmetricand regular behaviour. For the Painlev´e equations II–V these issues are better understood, see PeterClarkson’s expositions [7], [8] and references therein. The sixth Painlev´e equations seem to standapart and to be less studied in this sense. We would mention Gert Almqvist’s contribution [2] asan example of research done in this direction. We don’t discuss here the possibility of including ourrational solutions in this context. We just point out that Theorem 5 gives a simple auxiliary methodfor calculating explicitly a sequence of rational solutions of Painlev´e VI equations.It also turns out, as we will see in the next two sections, that particular Painlev´e VI equations possess one-parameter families of rational solutions, not only isolated ones. In our understanding, the emer-gence of such one-parameter families is not clarified in [41]: on one hand, they occur under the actionof particular birational canonical transformations on the degenerate solutions; on the other hand, thesolutions (29) , (30) are included in one-parameter rational families for particular values of the param-eters β i ’s. This issue is discussed in Section 4.4, where we explain in more detail Okamoto’s birationalcanonical transformations and their action on the degenerate solutions of Painlev´e VI equations. P VI : a sphere with three punctures Now we consider the case (c) of Theorem 1, that is, the case of n < m > b i ’s by Theorem 4in this case, one requires 1 /m ∈ Z , that is, m = 1. Hence we deal with the Riemann surface X x = C P of the curve w = z ( z − z − x )punctured at the three points (0 , , (1 , , ( x, X x \{ (0 , , (1 , , ( x, } and the integration of the triple w n dz/z , w n dz/ ( z − w n dz/ ( z − x ) along these very cycles, dueto Theorem 2, gives us two basic elements ( b R1 ( x ) , b R2 ( x ) , b R3 ( x )) and (˜ b R1 ( x ) , ˜ b R2 ( x ) , ˜ b R3 ( x )) in the two-dimensional space of triangular solutions (24) of the Schlesinger system (22), with β = β = β = n < , β ∞ = − n . These basic solutions are rational, with explicit expressions given by Theorem 4 and presented below.In turn, the pairs b R1 , ˜ b R1 and b R2 , ˜ b R2 are basic solutions of the corresponding hypergeometric equations(26) and (27), which are thus solvable in rational functions.Taking two basic cycles on X x \ { (0 , , (1 , , ( x, } encircling, for example, the points ( a ,
0) = (0 , a ,
0) = (1 ,
0) and computing the residues of the differentials w n dz/z , w n dz/ ( z − w n dz/ ( z − x )22irectly or applying the formulae (20), (21) with N = 3, d = | n | , ( a , a , a ) = (0 , , x ) we obtain b R1 ( x ) = ( − n x | n | | n | X j =0 (cid:18) −| n | j (cid:19)(cid:18) −| n || n | − j (cid:19) x j ,b R2 ( x ) = ( − n x | n |− | n |− X j =0 (cid:18) −| n | − j (cid:19)(cid:18) −| n || n | − − j (cid:19) x j ,b R3 ( x ) = ( − n x | n | | n |− X j =0 (cid:18) −| n | j (cid:19)(cid:18) −| n | − | n | − − j (cid:19) x j . and ˜ b R1 ( x ) = 1(1 − x ) | n |− | n |− X j =0 (cid:18) −| n | − j (cid:19)(cid:18) −| n || n | − − j (cid:19) (1 − x ) j , ˜ b R2 ( x ) = 1(1 − x ) | n | | n | X j =0 (cid:18) −| n | j (cid:19)(cid:18) −| n || n | − j (cid:19) (1 − x ) j , ˜ b R3 ( x ) = 1(1 − x ) | n | | n |− X j =0 (cid:18) −| n | j (cid:19)(cid:18) −| n | − | n | − − j (cid:19) (1 − x ) j . Again, for any n < , according to (28) the functions cb R1 ( x ) + ˜ b R1 ( x ) and cb R3 ( x ) + ˜ b R3 ( x ), c ∈ C , givea rational solution to the Painlev´e VI equation with parameters( α, β, γ, δ ) = (cid:18) (2 β ∞ − , − β , β , − β (cid:19) = (cid:18) (3 n + 1) , − n , n , − n (cid:19) , and thus we obtain the following theorem. Theorem 6
For every negative integer n , the functions y ( x ) = x ( c b R1 ( x ) + ˜ b R1 ( x )) c b R1 ( x ) + ˜ b R1 ( x ) + (1 − x )( c b R3 ( x ) + ˜ b R3 ( x )) , c ∈ C , give a one-parameter family of rational solutions of the sixth Painlev´e equation P VI ( α, β, γ, δ ) with theparameters α = (3 n + 1) , β = − n , γ = n , δ = 1 − n . Like in the previous section, the monodromy of the triangular Schlesinger isomonodromic family cor-responding to the above b P i ’s is commutative, since the eigenvalues e ± π i β i = e ± π i n of each monodromymatrix M i coincide (and all M i ’s may be chosen triangular). Example 2
Let us compute two basic rational solutions to the hypergeometric equations (26), (27)with β = β = β = n/ < n = − n = −
2, and n = − n = −
1, we obtain b R1 ( x ) = 1 + xx , ˜ b R1 ( x ) = 11 − x ,b R2 ( x ) = − x , ˜ b R2 ( x ) = x − − x ) ,b R3 ( x ) = − x , ˜ b R3 ( x ) = 1(1 − x ) , where b R1 and ˜ b R1 satisfy (26) and b R2 , ˜ b R2 satisfy (27) with β = β = β = − /
2. The cor-responding family of rational solutions of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) is givenby y ( x ) = 12 (1 − c ) x + c (1 − c ) x + c , c ∈ C .
2. For n = −
2, we obtain b R1 ( x ) = 3 + 4 x + 3 x x , ˜ b R1 ( x ) = − x (1 − x ) ,b R2 ( x ) = − xx , ˜ b R2 ( x ) = 10 − x + 3 x (1 − x ) ,b R3 ( x ) = − xx , ˜ b R3 ( x ) = − x (1 − x ) , where b R1 and ˜ b R1 satisfy (26) and b R2 , ˜ b R2 satisfy (27) with β = β = β = −
1. The correspondingfamily of rational solutions of the sixth Painlev´e equation P VI (cid:0) , − , , − (cid:1) is given by y ( x ) = 15 (1 − c ) x (3 x −
5) + c (3 − x )(1 − c ) x ( x −
2) + c (1 − x ) , c ∈ C .
3. For n = −
3, we obtain b R1 ( x ) = 10 + 18 x + 18 x + 10 x x , ˜ b R1 ( x ) = 28 − x + 10 x (1 − x ) ,b R2 ( x ) = − x + 10 x x , ˜ b R2 ( x ) = − −
56 + 84 x − x + 10 x (1 − x ) ,b R3 ( x ) = −
10 + 12 x + 6 x x , ˜ b R3 ( x ) = 28 − x + 6 x (1 − x ) , where b R1 and ˜ b R1 satisfy (26) and b R2 , ˜ b R2 satisfy (27) with β = β = β = − /
2. The corre-sponding family of rational solutions of the sixth Painlev´e equation P VI (cid:0) , − , , − (cid:1) is givenby y ( x ) = 14 (1 − c ) x (14 − x + 5 x ) + c (5 − x + 14 x )(1 − c ) x (7 − x + 2 x ) + c (2 − x + 7 x ) , c ∈ C . .3 Further rational solutions of P VI The solutions obtained in Sections 4.1, 4.2 can be rewritten using the hypergeometric power series F ( a , b , c , x ) (solutions of the corresponding hypergeometric equations) that, for particular values ofthe parameters a , b , c , reduce to polynomials. For instance for n >
0, up to a constant factor, we have b P1 ( x ) = F ( − n, − n/ , − n/ , x ) = ∞ X j =0 ( − n ) j ( − n/ j (1 − n/ j j ! x j = n X j =0 ( − n ) j ( − n/ j (1 − n/ j j ! x j , where ( θ ) j = θ ( θ + 1) . . . ( θ + j − θ ) = 1, for any θ ∈ C . Recall that the polynomial b P1 is asolution of the hypergeometric equation x (1 − x ) b ′′ + [ c − ( a + b + 1) x ] b ′ − ab b = 0 , (31)where a = − β + β + β ) = − n, b = − β = − n/ , c = 1 − β + β ) = 1 − n/ β = β = β = n/ > Z ).In a similar way, the rational functions b R1 ( x ), ˜ b R1 ( x ) from the previous section, up to a constant factor,are expressed via reduced hypergeometric series as follows: b R1 ( x ) = x − | n | F ( −| n | , | n | , − | n | , x ) = 1 x | n | | n | X j =0 ( −| n | ) j ( | n | ) j (1 − | n | ) j j ! x j , ˜ b R1 ( x ) = (1 − x ) − | n | F (1 − | n | , | n | , − | n | , − x ) = 1(1 − x ) | n |− | n |− X j =0 (1 − | n | ) j (1 + | n | ) j (2 − | n | ) j j ! (1 − x ) j . These two functions are basic solutions of the hypergeometric equation (31), where a = − β + β + β ) = 3 | n | , b = − β = | n | , c = 1 − β + β ) = 1 + 2 | n | (33)(that is, solutions of (26), where β = β = β = n/ <
0) which is in agreement with a general factthat the hypergeometric functions x − c F ( b − c + 1 , a − c + 1 , − c , x ) , (1 − x ) c − a − b F ( c − a , c − b , − a − b + c , − x )are basic solutions of the hypergeometric equation (31) for generic values of the parameters.These new forms provide a pattern which gives a hint on how to use polynomial and rational solu-tions of hypergeometric equations coming from hypergeometric power series solutions, to constructcorresponding rational solutions of Painlev´e VI equations. In this way we obtain two statements. Thefirst one is a generalization of Theorem 5 for a much larger set of parameters a , b , c than the abovediscrete set of the form (32) which originates from the algebro-geometric approach of Section 4.1. Theorem 7
For every positive integer n and any b ∈ C , c ∈ C \ {− , . . . , − n + 1 } , c = b + 1 , thepolynomials P n +1 ( b , c , x ) = xb P1 ( b , c , x ) and Q n +1 ( b , c , x ) = b P1 ( b , c , x ) + (1 − x ) b P3 ( b , c , x )25 f degree n + 1 with b P1 , b P3 given by b P1 ( b , c , x ) = n X j =0 (cid:18) − b j (cid:19)(cid:18) c + n − n − j (cid:19) x j ,b P3 ( b , c , x ) = − x b − c db P1 dx − b b − c b P1 , define the rational solution y = P n +1 /Q n +1 of the sixth Painlev´e equation P VI ( α, β, γ, δ ) with theparameters α = ( n + 1) , β = − (1 + b − c ) , γ = (1 − n − c ) , δ = 1 − b . Proof.
The polynomial b P1 ( b , c , x ) is, up to a constant factor, a reduced hypergeometric series F ( − n, b , c , x ) = ∞ X j =0 ( − n ) j ( b ) j ( c ) j j ! x j = n X j =0 ( − n ) j ( b ) j ( c ) j j ! x j , which satisfies the hypergeometric equation (31) with a = − n or, equivalently, the hypergeometricequation (26) with β = (1+ b − c ) / β = ( n + c − / β = − b /
2. Then the expression for b P3 ( b , c , x )follows from (25) and the relation b = − b − b . This triple b P1 , b P2 , b P3 determines the polynomialsolution (24) of the Schlesinger system (22) with the above β , β , β , β ∞ = − β − β − β = − n/ VI ( α, β, γ, δ ) with parameters( α, β, γ, δ ) = (cid:18) (2 β ∞ − , − β , β , − β (cid:19) = (cid:18) ( n + 1) , − (1 + b − c ) , (1 − n − c ) , − b (cid:19) . Note that in our triangular case, β ∞ + β + β + β = 0 ∈ Z and β ∞ = − n/ ∈ Z , in agreement withMazzocco’s theorem. (cid:3) Remark 7
Making straightforward calculations for the polynomial b P1 + (1 − x ) b P3 = 11 + b − c (cid:16) (1 − c ) b P1 − x db P1 dx + b x b P1 + x db P1 dx (cid:17) one can obtain explicit formulae for the polynomials P n +1 , Q n +1 similar to those of Theorem 5: P n +1 ( b , c , x ) = n +1 X j =1 (cid:18) − b j − (cid:19)(cid:18) c + n − n − j + 1 (cid:19) x j ,Q n +1 ( b , c , x ) = − n + 11 + b − c n +1 X j =0 (cid:18) − b j (cid:19)(cid:18) c + n − n − j + 1 (cid:19) x j . Note that the reciprocity of the polynomials b P1 and Q n +1 still holds, if c + n − − b . The next statement in a similar way generalizes Theorem 6 for a larger set of integer parameters a , b , c than that of the form (33) originating from the algebro-geometric approach of Section 4.2 and has asimilar proof. 26 heorem 8 For any integers c > , b > and a , such that a > c and c − a < b < c − , the rationalfunctions b R1 ( a , b , c , x ) = 1 x c − c − b − X j =0 (cid:18) − b j (cid:19)(cid:18) c − a − c − b − − j (cid:19) x j , b R3 ( a , b , c , x ) = − x b − c db R1 dx − b b − c b R1 , ˜ b R1 ( a , b , c , x ) = 1(1 − x ) a + b − c a − c X j =0 (cid:18) − b j (cid:19)(cid:18) b − ca − c − j (cid:19) (1 − x ) j , ˜ b R3 ( a , b , c , x ) = − x b − c d ˜ b R1 dx − b b − c ˜ b R1 define a one-parameter family of rational solutions y ( x ) = x ( c b R1 ( x ) + ˜ b R1 ( x )) c b R1 ( x ) + ˜ b R1 ( x ) + (1 − x )( c b R3 ( x ) + ˜ b R3 ( x )) , c ∈ C , of the sixth Painlev´e equation P VI ( α, β, γ, δ ) with the parameters α = ( a − , β = − (1 + b − c ) , γ = (1 + a − c ) , δ = 1 − b . Let us mention that the monodromy of the families obtained in Theorems 7 and 8 are commutative,as it was the case for the families obtained in Theorems 5 and 6. P V I rationalsolutions
Birational canonical transformations (of the first kind ) of Painlev´e VI equations, as they were definedby Kazuo Okamoto [45], act on the pair, the initial unknown y and its conjugated momentum p , withrespect to which the sixth Painlev´e equation P VI ( α, β, γ, δ ) can be rewritten as a first order system: y ′ = y ( y − y − x ) x ( x − (cid:16) p − β − y − x − β y − β y − (cid:17) p ′ = − x ( x − (cid:16) [3 y − x + 1) y + x ] p + [(2 − β − β − β ) y ++2 β + 2 β − β + 2 β ) x ] p + κ (cid:17) , (34)where κ = ( β + β + β − β ∞ )( β + β + β + β ∞ − = β + β , b = β − β , b = β + β ∞ − , b = β − β ∞ , Okamoto defines the following affine transformations on their space C : w : (b , b , b , b ) (b , b , b , b ) ,w : (b , b , b , b ) (b , b , b , b ) ,w : (b , b , b , b ) (b , b , b , b ) ,w : (b , b , b , b ) ( − b , − b , b , b ) , w : (b , b , b , b ) (b , b , − b − , − b − w is inducedby a birational transformation ( y, p ) ( y w , p w ) of the Painlev´e system (34) via the formula F [b] (cid:18) yy ( y − p (cid:19) + g [b] = F [ w (b)] (cid:18) y w y w ( y w − p w (cid:19) + g [ w (b)] , (35)where b = (b , b , b , b ) and F [b] = (cid:18) − h + σ ′ [b] − b − b σ ′ [b] h − σ ′ [b] − h + b b (cid:19) , g [b] = (cid:18) − σ [b] − σ [b] h + σ [b] (cid:19) . (36)In the above formulae, σ k [b] denotes the elementary symmetric polynomial of degree k in four variablesb , b , b , b , and σ ′ k [b] denotes the elementary symmetric polynomial of degree k in three variablesb , b , b . The polynomial h = h ( y, p ) is given by the formula h = − y ( y − p + (cid:0) y − (b + b ) (cid:1) p − b . Remark 8
Originally, the transformation w was defined by Okamoto in the form w : (b , b , b , b ) (b , b , − b , − b ) . Note that there is a misprint in the image of this w , where the last two coordinates are − b , − b in[45]. Another misprint in [45] is the absence of the factor / in the second coordinate of the vector g [b] in (36) . The last misprint is not essential for calculating y w , p w if w is a transposition, but itcan have an effect in the case when w contains the change of sign or a translation, like w does. As can be easily seen, the birational transformation associated with w does not change y nor p , since F [ w (b)] = F [b] and g [ w (b)] = g [b] in this case. The birational transformations associated with w , w , and w do not change y but change p (they correspond to the change of sign β ↔ − β , β ↔ − β ,and β ↔ − β , respectively).Birational transformations associated with w or with those containing w as a factor, change both y and p , and they are of a particular interest for us. We will study the action of w w w on the degenerate solutions of the sixth Painlev´e equation P VI ( α, β, γ, δ ). The latter are:i) y ( x ) ≡ ∞ for α = 0 (that is, for β ∞ = 1 / y ( x ) ≡ β = 0 (that is, for β = 0);iii) y ( x ) ≡ γ = 0 (that is, for β = 0);iv) y ( x ) ≡ x for δ = 1 / β = 0).This set is invariant under the action of the symmetries mentioned in Remark 6 (birational transfor-mations of the second kind , as they change the independent variable x ). Note that for any solution y ( x ) different from i)–iv), its conjugated momentum p ( x ) is uniquely determined by the first equationof the Painlev´e system (34), in particular, p is rational if y is. On the other hand, for each of thesolutions i)–iv), its conjugated momentum is a one-parameter family of solutions of the correspondingRiccati equation coming from the second equation of (34). Therefore for such a pair ( y, p ), the image y w of y under the birational transformation associated with w = w w w : (b , b , b , b ) (b , b , b , b ) , one-parameter family of solutions of the corresponding sixth Painlev´e equation. Letus explain this in more detail in the case of the sixth Painlev´e equation P VI ( α, , γ, δ ) possessing thedegenerate solution y ≡ β = 0, one has b = − b and the polynomial h is equal to h = − y ( y − p + 2b yp − b . Taking into consideration the equalities g [ w (b)] = g [b] and σ ′ k [ w (b)] = σ ′ k [b] for w = w w w , oneobtains from (35) the formula (cid:18) y w y w ( y w − p w (cid:19) = 1 h + b (cid:18) h + b b − b h + b (cid:19) (cid:18) yy ( y − p (cid:19) (see Example 2.1 on p. 356 in [45]), which implies y w = y + (b − b )( y − − ( y − p + 2b , p w = ( y − − y ( y − p + 2b yp − b + b ) − ( y − p + 2b y w ( y w − . The above formulae give explicitly the action of the birational canonical transformation associatedwith w = w w w on the Painlev´e system (34) with β = b + b = 0. Note that det F [ w (b)] = 0 for y = 0 since in this case h + b = 0, but the final expressions for y w , p w are defined also for y = 0. Wethus have a prolongation of the birational canonical transformation to the degenerate solution y ≡ VI ( α, , γ, δ ): (0 , p ) (cid:16) b − b p + 2b , − (b + b )( p + 2b ) p + b + b (cid:17) , (37)where p is the general solution of the Riccati equation − x ( x − p ′ = xp + (2b x + b + b ) p + (b + b )(b + b ) . (38)Therefore, if the parameters in equation (38) were such that its general solution was a rational function,we would obtain the transformation of the degenerate solution y ≡ VI ( α, , γ, δ ) to a one-parameter family of rational solutions of the transformed Painlev´e VI equation, under the actionprovided by (37). We give the following example. Example 3
Consider the set of parameters b = (b , b , b , b ) = ( − , , ,
1) and, consequently, β = b + b , β = b − b − , β = b + b + 12 = 1 , β ∞ = b − b + 12 = 0 . The corresponding sixth Painlev´e equation P VI (cid:0) , , , − (cid:1) possesses the degenerate solution y ≡ p is the general solution of the Riccati equation (38) − x ( x − p ′ = xp + ( − x + 1) p, that is, p ( x ) = 2 x ( x − x + c , c ∈ C . w w w (b) = (0 , , − , y ≡ VI (cid:0) , , , − (cid:1) is mapped, according to (37), to the one-parameter family of rationalsolutions y w ( x ) = − x ( x − x + c − x + cx + c (39)of the corresponding sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) . This family has been already obtainedin Example 2.Concluding this section we note that it also would be natural to call such one-parameter familiesof rational solutions of Painlev´e VI equations degenerate , as they are obtained from the degeneratesolutions. They do not participate in Mazzocco’s classification of rational solutions. On the other hand,Mazzocco’s basic rational solutions (29), (30) themselves, for some values of the parameters β i ’s, canbelong to one-parameter families of rational solutions of the corresponding Painlev´e VI equations.For example, when β = 1 / β = − / β = 1 / β ∞ = − /
2, solution (29) of P VI (cid:0) , − , , (cid:1) is y ( x ) = x/
2, which belongs to the family (39) and thus can be obtained from the degenerate solution y ≡ VI (cid:0) , , , − (cid:1) via a birational transformation. Similarly, when β = β = β = − β ∞ = 3,solution (30) of P VI (cid:0) , − , , − (cid:1) is y ( x ) =
15 3 x +4 x +31+ x , which belongs to the second rational familyof Example 2 (formally, it corresponds to the value c = ∞ of the family parameter).The reasoning above raises the following question: for which values of the parameters β i ’s the cor-responding basic rational solution (29) or (30) is isolated and for which values it belongs to a one-parameter rational family, thus being the candidate for being birationally equivalent to a degeneratesolution. This shows, in our understanding, that the problem of the classification of rational solutionsis not completely closed. Remark 9
We also mention the paper [51] with the classification of rational solutions of Painlev´eVI equations. However, it was observed in [1] that Theorem 4.2. from [51] states that y ( x ) is a non-constant rational solution of the sixth Painlev´e equation P VI ( α, β, γ, δ ) if and only if its conjugatedmomentum p ≡ , that is, if and only if y ( x ) solves the corresponding Riccati equation ( the firstequation of the Painlev´e system (34) with p ≡ . This is not always the case, as we could see inExample 3: the rational solution y w ( x ) given by (39) of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) solves the algebraic first order ODE of the third degree in y rather than the Riccati equation, since theconjugated momentum p w ( x ) of y w ( x ) , according to (37) , is p w ( x ) = p ( x ) − p ( x ) − , p ( x ) = 2 x ( x − x + c . P VI Let us return to the hypergeometric equation (31) with a = − n a negative integer or, equivalently, tothe hypergeometric equation (26) with β = (1 + b − c ) / β = ( n + c − / β = − b / b ′′ + ( b − n + 1) x − c x ( x − b ′ − n b x ( x − b = 0 . (40)As we have seen, this equation possesses a degree n polynomial solution b P1 ( b , c , x ). Therefore, itssecond basic solution is also Liouvillian (we explain this in detail below after Remark 10) and the30quation is thus solvable in the Liouvillian sense (that is, two basic solutions of the equation areexpressed in terms of elementary or algebraic functions and their integrals, speaking informally) . Remark 10
There is the Schwarz–Kimura list [28] of hypergeometric linear ODEs that are solvablein the Liouvillian sense ( see also [54, Ch. 12] ) . Its first part consists of 15 families of equations, eachfamily being parametrized by a triple ( λ, µ, ν ) = (1 − c , b − a , c − a − b ) . These include 14 discrete families that belong to Schwarz’s list of hypergeometric linear ODEs integrablein algebraic functions. The fifteenth family is continuous, it extends the corresponding family ofSchwarz’s list. The second part of the Schwarz–Kimura list consists of equations such that at least oneof the four numbers λ + µ + ν , − λ + µ + ν , λ − µ + ν , λ + µ − ν is an odd integer.In our case, λ + µ + ν = 1 − a = 1 + 2 n is an odd integer. Let us recall how to find a second basic solution of the equation (40). We know its polynomial solution b P1 ( b , c , x ). Then a second basic solution b L1 ( b , c , x ) satisfies the equality b P1 ( b L1 ) ′ − ( b P1 ) ′ b L1 = W ( b P1 , b L1 ) = exp (cid:16) − Z ( b − n + 1) x − c x ( x − dx (cid:17) = x − c ( x − c − b + n − , where W ( · , · ) is the Wronskian of a pair of functions. Hence, one finds b L1 as a solution of a first orderlinear inhomogeneous ODE, b L1 ( b , c , x ) = b P1 ( b , c , x ) Z x − c ( x − c − b + n − b P1 ( b , c , x ) dx. (41)Defining b L3 ( b , c , x ) = − x b − c db L1 dx − b b − c b L1 , (42)we come to the following assertion extending Theorem 7. Theorem 9
For every positive integer n and any b ∈ C , c ∈ C \ {− , . . . , − n + 1 } , the sixth Painlev´eequation P VI (cid:16) ( n +1) , − (1+ b − c ) , (1 − n − c ) , − b (cid:17) possesses a one-parameter family of Liouvillian solu-tions of the form y ( x ) = P n +1 ( b , c , x ) + cx b L1 ( b , c , x ) Q n +1 ( b , c , x ) + c (cid:0) b L1 ( b , c , x ) + (1 − x ) b L3 ( b , c , x ) (cid:1) , c ∈ C , where P n +1 , Q n +1 are polynomials of degree n + 1 from Theorem 7 and b L1 , b L3 are the Liouvillianfunctions given by (41) , (42) , respectively. Remark 11
Note that all classical non-algebraic solutions of the sixth Painlev´e equations wereclassified by Humihiko Watanabe [50]. They occur if and only if, for the corresponding Schlesingersystem (22) either i) β ∞ + ε β + ε β + ε β ∈ Z , In the sense of Hiroshi Umemura [49]. The Liouvillian solutions are classical. or some choice of ε i ∈ {± } , or ii) β ∈ Z , or iii) β ∈ Z , or iv) β ∈ Z , or v) β ∞ ∈ Z . Moreover, the number of one-parameter families of classical non-algebraic solutions to a fixed equationcan be equal to zero, one, two, three or four, according to the number of the fulfilled conditions among i), ii), iii), iv), v) . If all five conditions are satisfied, there are still only four one-parameter families. Allthese classical solutions were described in [42] via solutions of hypergeometric and Riccati equations, upto symmetries and birational canonical transformations. On the other hand, if none of the conditions i) – v) holds, the equation still can have particular algebraic solutions not forming a one-parameterfamily. The problem of the classification of all algebraic solutions to the sixth Painlev´e equations,after a long period of works by Nigel Hitchin, Boris Dubrovin and Marta Mazzocco, Fedor Andreev,Alexey Kitaev, and Philip Boalch, has been entirely closed by Oleg Lisovyy and Yuriy Tykhyy [36] ( seereferences therein ) .As in our case exactly two conditions, i) and v) , hold ( if b , c are non-integer ) , the sixth Painlev´e equa-tion with the parameters from Theorem 9 possesses two one-parameter families of classical solutions,and we propose one of them generated by two polynomials and two quadratures. Example 4
Taking into consideration the computations of Example 1, let us write down the familiesfrom Theorem 9 in the case n = 1, b = − / c = 1 / n = 2, b = − / c = − / n = 1, b = − / c = 1 /
3, we have the following one-parameter family of Liouvilliansolutions of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) : y ( x ) = x (cid:0) x + 1 + c b L1 ( x ) (cid:1) x + 1 + c b L1 ( x ) + (1 − x ) (cid:0) − x + c b L3 ( x ) (cid:1) , c ∈ C ,b L1 ( x ) = ( x + 1) Z ( x − x ( x + 1) dx, b L3 ( x ) = (1 − x ) Z ( x − x ( x + 1) dx − x ( x − x + 1 .
2. For n = 2, b = − / c = − /
3, we have the following one-parameter family of Liouvilliansolutions of the sixth Painlev´e equation P VI (cid:0) , − , , (cid:1) : y ( x ) = x (cid:0) x − x + 1 + c b L1 ( x ) (cid:1) x − x + 1 + c b L1 ( x ) + (1 − x ) (cid:0) − x + 2 x + 1 + c b L3 ( x ) (cid:1) , c ∈ C ,b L1 ( x ) = ( x − x + 1) Z x ( x − ( x − x + 1) dx,b L3 ( x ) = ( − x + 2 x + 1) Z x ( x − ( x − x + 1) dx − x ( x − x − x + 1) . Here we consider Garnier systems G M ( θ ) (a multidimensional generalization of Painlev´e VI equations)depending on M + 3 complex parameters θ , . . . , θ M +2 , θ ∞ . These are completely integrable PDEs32ystems of second order [19], [20]. They can be written in a Hamiltonian form obtained by K. Okamoto[44], ∂u i ∂a j = ∂H j ∂v i , ∂v i ∂a j = − ∂H j ∂u i , i, j = 1 , . . . , M, (43)for the unknown functions ( u, v ) = ( u , . . . , u M , v , . . . , v M ) of the variable a = ( a , . . . , a M ), where theHamiltonians H j = H j ( a, u, v, θ ) are rational functions of their arguments (see also [26] and Example5 below).Let us recall how the Garnier system is determined by the Schlesinger system for M + 2 traceless (2 × B (1) ( a ) , . . . , B ( M +2) ( a ) depending on the variable a (here a M +1 = 0, a M +2 = 1 arefixed) which belongs to a disc D of the space ( C \ { , } ) M \ S i = j { a i = a j } .Let ± β i be the eigenvalues of the matrix B ( i ) ( a ) = (cid:0) b kli ( a ) (cid:1) k,l , i = 1 , . . . , M + 2, and M +2 X i =1 B ( i ) ( a ) = diag( − β ∞ , β ∞ ) . Since P M +2 i =1 b i ( a ) ≡
0, the numerator of the fraction M +2 X i =1 b i ( a ) z − a i is a polynomial of degree M in z . If one denotes its zeros by u ( a ) , . . . , u M ( a ) and defines v j ( a ) = M +2 X i =1 b i ( a ) + β i u j ( a ) − a i , j = 1 , . . . , M, (44)then the pair ( u, v ) = ( u , . . . , u M , v , . . . , v M ) satisfies the Garnier system (43) with parameters( θ , . . . , θ M +2 , θ ∞ ) = (2 β , . . . , β M +2 , β ∞ − u ( a ) , . . . , u M ( a ) depend on the b i ’s algebraically, for M > Z of thespace ( C \ { , } ) M \ S i = j { a i = a j } . However, some information concerning the elementary symmetricpolynomials in the coordinates u , . . . , u M can be obtained in this context (see, for example [24]).As we have seen in Section 4, solutions of the Schlesinger system for triangular traceless (2 × M = 1 variable always lead to solutions of the corresponding sixth Painlev´eequation that are expressed rationally via a logarithmic derivative of solutions of a hypergeometriclinear ODE. This fact admits a generalization for the multivariable case of triangular traceless (2 × M > via logarithmic derivatives of solutionsof a Lauricella hypergeometric PDE. Before exposing this in more detail, let us recall that the latteris a system of linear PDEs of the second order of the form(1 − a i ) M X j =1 a j ∂ u∂a i ∂a j + ( κ − ( α + 1) a i ) ∂u∂a i − µ i M X j =1 a j ∂u∂a j − αµ i u = 0 , i = 1 , . . . , M, ( a i − a j ) ∂ u∂a i ∂a j + µ i ∂u∂a j − µ j ∂u∂a i = 0 , i, j = 1 , . . . , M, u of M variables a , . . . , a M , where α, µ , . . . , µ M , κ are complex parameters.Its solution space is ( M + 1)-dimensional, as follows from the proof of Prop. 9.1.4 in [26]. Now, in thetriangular case, a solution B ( i ) ( a ) = (cid:18) β i b i ( a )0 − β i (cid:19) , i = 1 , . . . , M + 2 , M +2 X i =1 B ( i ) ( a ) = diag( − β ∞ , β ∞ ) , (45)of the Schlesinger system determines the polynomial P M ( z, a ) = ( z − a ) . . . ( z − a M +2 ) M +2 X i =1 b i ( a ) z − a i of degree M in z with zeros u ( a ) , . . . , u M ( a ). Then due to (44), the pair( u, v ) ε = ( u , . . . , u M , v ε , . . . , v εM ) , (46)where v εj ( a ) = M +2 X i =1 (1 + ε i ) β i u j ( a ) − a i , with ε = ( ε , . . . , ε M +2 ) ∈ {± } M +2 , satisfies the Garnier system (43) with parameters( θ , . . . , θ M +2 , θ ∞ ) = (2 ε β , . . . , ε M +2 β M +2 , β ∞ − . (47)Then, introducing new independent variables t = ( t , . . . , t M ) with t i = a i a i − , i = 1 , . . . , M, and functions q i ( t ) = a i ( a i − u ) . . . ( a i − u M ) Q M +2 j =1 ,j = i ( a i − a j ) , i = 1 , . . . , M, one has the following expressions for the latter: q i ( t ) = t i ( t i − β ∞ − (cid:18) − β i t i − f ∂f∂t i (cid:19) , where f is a solution of the Lauricella hypergeometric equation with parameters( α, µ , . . . , µ M , κ ) = (cid:0) β M +2 , − β , . . . , − β M , − M +1 X j =1 β j (cid:1) (see [26, Th. 9.2.1]).After mentioning these general relations between triangular Schlesinger (2 × β = . . . = β M +2 = n m , β ∞ = − ( M + 2) n m , n > m > n < m >
0. Applying Theorems 1 and 2 we obtainalgebro-geometric expressions for an ( M + 1)-parameter family of solutions of the corresponding tri-angular Schlesinger (2 × M + 1)-parameter family of the coefficients ofthe polynomial P M ( z, a )): b i ( a ) = c I γ w n dzz − a i + . . . + c M +1 I γ M +1 w n dzz − a i , i = 1 , . . . , M + 2 , (48)where γ , . . . , γ M +1 are suitable closed contours on the Riemann surface X a of the curve w m = z ( z − z − a ) . . . ( z − a M )(or, on the X a with punctures at the poles of the differentials w n dz/z , w n dz/ ( z − w n dz/ ( z − a ) , . . . , w n dz/ ( z − a M ), depending on which of the cases (a), (b), (c) of Theorem 1 holds). Theseexpressions lead to an M -parameter families of algebro-geometric solutions (46) of the Garnier systemswith parameters (47): ( θ , . . . , θ M +2 , θ ∞ ) = (cid:16) ± nm , . . . , ± nm , − ( M + 2) nm − (cid:17) , the signs being independent.Like for the Painlev´e VI equations, let us study in more detail the cases when Theorems 3 and 4can be applied to obtain polynomial and rational expressions for b i ’s and, as a consequence, algebraicsolutions of particular Garnier systems. m punctures In this section we consider the case of n > m >
1. The requirement s/m ∈ Z of Theorem 3 for s = ( m, M + 2) implies that s = m and m is a divisor of the integer M + 2. Hence we deal with theRiemann surface X a of the curve w m = z ( z − z − a ) . . . ( z − a M )punctured at m points P , . . . , P m at infinity. The genus of X a equals g = 12 (cid:0) ( m − M + 1) − m + 1 (cid:1) = 12 ( m − M, thus there are ( m − M + 1) basic cycles on X a \ { P , . . . , P m } .Further, formula (19), where d = n , s = m , M = ( M + 2) /m and the role of N being played by M + 2, gives us the following polynomial solutions (45) of the Schlesinger (2 × β = . . . = β M +2 = n/ m > b i ( a ) = X k + ... + k M + k M +2 + q = M n ( − q (cid:18) n/mk (cid:19) . . . (cid:18) n/mk M (cid:19)(cid:18) n/mk M +2 (cid:19) a k . . . a k M M a qi (49)(recall that a M +1 = 0, a M +2 = 1). Hence, the coefficients of the corresponding polynomial P M ( z, a )are also polynomials (in a , . . . , a M ) in this case, and thus we come to the following assertion concerningalgebraic solutions of Garnier systems. 35 heorem 10 For any coprime integers n > , m > such that m is a divisor of the integer M + 2 ,the Garnier system G M ( θ ) with parameters ( θ , . . . , θ M +2 , θ ∞ ) = (cid:16) ± nm , . . . , ± nm , − ( M + 2) nm − (cid:17) ( the signs are independent ) possesses an algebraic solution, which can be computed explicitly. Example 5
Consider some examples of bivariate Garnier systems in the variables a , a . The system G ( θ , θ , θ , θ , θ ∞ ) has the form ∂u ∂a = ∂H ∂v , ∂u ∂a = ∂H ∂v , ∂u ∂a = ∂H ∂v , ∂u ∂a = ∂H ∂v ,∂v ∂a = − ∂H ∂u , ∂v ∂a = − ∂H ∂u , ∂v ∂a = − ∂H ∂u , ∂v ∂a = − ∂H ∂u , with the Hamiltonians H = − Λ( a ) T ′ ( a ) X j =1 T ( u j )( u j − a )Λ ′ ( u j ) (cid:20) v j − (cid:16) θ − u j − a + θ u j − a + θ u j + θ u j − (cid:17) v j + κ u j ( u j − (cid:21) ,H = − Λ( a ) T ′ ( a ) X j =1 T ( u j )( u j − a )Λ ′ ( u j ) (cid:20) v j − (cid:16) θ u j − a + θ − u j − a + θ u j + θ u j − (cid:17) v j + κ u j ( u j − (cid:21) , where κ = (cid:0) ( θ + θ + θ + θ − − θ ∞ (cid:1) ,Λ( x ) = ( x − u )( x − u ) , T ( x ) = x ( x − x − a )( x − a ) . The polynomial P M ( z, a ) = P ( z, a , a ) equals P ( z, a , a ) = (cid:0) b + a b + a b (cid:1) z + (cid:0) a a ( b + b ) + a ( b + b ) + a ( b + b ) (cid:1) z − a a b in this case. As M + 2 = 4, there are two divisors of M + 2: m = 2 and m = 4.1. Let m = 2 and n = 1. Then M = ( M + 2) /m = 2 and, due to (49), b ( a , a ) = X k + k + k + q =2 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k + q a k = 3 a − a a − a − a + 2 a − ,b ( a , a ) = X k + k + k + q =2 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k + q = 3 a − a a − a + 2 a − a − ,b ( a , a ) = X k + k + k =2 (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k = − a + 2 a a − a + 2 a + 2 a − ,b ( a , a ) = X k + k + k + q =2 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k = − a + 2 a a − a − a − a + 3(up to a common constant factor 1 / P ( z, a , a ) defines analgebraic function, two branches u ( a , a ), u ( a , a ) of which give us the algebraic solution( u , u , v ε , v ε ) , v εj ( a , a ) = 14 (cid:16) ε u j − a + 1 + ε u j − a + 1 + ε u j + 1 + ε u j − (cid:17) , ε i ∈ {± } , of the Garnier system G (cid:0) ε , ε , ε , ε , − (cid:1) . 36. Let m = 4 and n = 1. Then M = ( M + 2) /m = 1 and, due to (49), b ( a , a ) = X k + k + k + q =1 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k + q a k = − a + a + 1 ,b ( a , a ) = X k + k + k + q =1 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k + q = a − a + 1 ,b ( a , a ) = X k + k + k =1 (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k = a + a + 1 ,b ( a , a ) = X k + k + k + q =1 ( − q (cid:18) / k (cid:19)(cid:18) / k (cid:19)(cid:18) / k (cid:19) a k a k = a + a − − / P ( z, a , a ) simi-larly determines the algebraic solutions of the Garnier systems G (cid:0) ± , ± , ± , ± , − (cid:1) . M + 2 punc-tures In this section we consider the case of n < m > /m ∈ Z of Theorem 4 implies that m = 1 and we deal with the Riemannsurface X a = C P of the curve w = z ( z − z − a ) . . . ( z − a M )punctured at the points ( a , , . . . , ( a M , , (0 , , (1 , M +1 basic cycles on X a \{ ( a , , . . . , ( a M , , (0 , , (1 , } and the integration of the vector (cid:16) w n dzz − a , . . . , w n dzz − a M , w n dzz , w n dzz − (cid:17) along these very cycles, due to Theorem 2, gives us M + 1 basic elements in the ( M + 1)-dimensionalspace of triangular solutions (45) of the Schlesinger (2 × β = . . . = β M +2 = n/ <
0. These basic solutions are rational, with explicit expressions given by Theorem 4, whichimplies the existence of an M -parameter family of algebraic solutions of the corresponding Garniersystem. Theorem 11
For any integer n < , the Garnier system G M ( θ ) with parameters ( θ , . . . , θ M +2 , θ ∞ ) = ( ± n, . . . , ± n, − ( M + 2) n − the signs are independent ) possesses an M -parameter family of algebraic solutions, which can becomputed explicitly by using Theorem 4. Example 6
Let us illustrate the above theorem by computing two-parameter families of algebraicsolutions of bivariate Garnier systems G ( ± , ± , ± , ± ,
3) (the case of M = 2, n = − a , a , a ,
0) = (0 ,
0) of the differentialsΩ = dzw ( z − a ) , Ω = dzw ( z − a ) , Ω = dzwz , Ω = dzw ( z − ,
37e have three linear independent vector functions, respectively, (cid:0) b ( j )1 ( a , a ) , b ( j )2 ( a , a ) , b ( j )3 ( a , a ) , b ( j )4 ( a , a ) (cid:1) , j = 1 , , , where b ( j ) i = res ( a j , Ω i . Using the explicit formula (20) leads to the following expressions: b (1)2 = 1( a − a ) a ( a − , b (1)3 = 1( a − a ) a ( a − , b (1)4 = 1( a − a ) a ( a − ,b (1)1 = − b (1)2 − b (1)3 − b (1)4 ; b (2)1 = 1( a − a ) a ( a − , b (2)3 = 1( a − a ) a ( a − , b (2)4 = 1( a − a ) a ( a − ,b (2)2 = − b (2)1 − b (2)3 − b (2)4 ; b (3)1 = 1 a a , b (3)2 = 1 a a , b (3)3 = − a a + a + a a a , b (3)4 = 1 a a . Then, like in Example 5, we consider the polynomial P ( z, a , a ) = (cid:0) b + a b + a b (cid:1) z + (cid:0) a a ( b + b ) + a ( b + b ) + a ( b + b ) (cid:1) z − a a b , with b i = c b (1) i + c b (2) i + b (3) i , which contains two free parameters c , c ∈ C and thus defines atwo-parameter family of algebraic functions, two branches u ( a , a , c , c ), u ( a , a , c , c ) of whichgive us the two-parameter family of algebraic solutions( u , u , v ε , v ε ) , v εj ( a , a , c , c ) = 12 (cid:16) − ε u j − a + − ε u j − a + − ε u j + − ε u j − (cid:17) , ε i ∈ {± } , of the Garnier system G (cid:0) ε , ε , ε , ε , (cid:1) . Remark 12
Classical solutions of Garnier systems were studied and partially described in [29] andin [42], mainly in terms of the monodromy of a Fuchsian family (3) that is governed by the corre-sponding Schlesinger (2 × -system ( though, there is no full classification of classical solutions hereyet, in contrast to sixth Painlev´e equations ) . In particular, if the monodromy of the Fuchsian familyis triangular, the corresponding Garnier system G M ( θ ) possesses an M -parameter family of classicalsolutions expressed via Lauricella hypergeometric functions ( Theorem 6 in [42] ) . Our case, that of atriangular Schlesinger system, is certainly included in that context of triangular monodromy, however,Theorem 10 provides us with an explicit form of algebraic solutions to particular Garnier systems and,moreover, Theorem 11 presents some cases when algebraic solutions of a Garnier system form an M -parameter family.Concerning the problem of classification of algebraic solutions to Garnier systems itself, it is obvi-ously more recent than the analogous one for Painlev´e VI equations and is still open. Due to Ga¨elCousin [10], algebraic solutions correspond to finite braid group orbits on the character variety ofthe ( M + 3) -punctured Riemann sphere (i. e. , on the moduli space of its rank two linear monodromyrepresentations ) . With respect to this correspondence, in the case of a non-degenerate linear mon-odromy ( that is, neither finite, nor dihedral, nor triangular ) , algebraic solutions were partially classifiedin [13] for an arbitrary M and in [6] for M = 2 . For a non-abelian triangular linear monodromy, theclassification of Schlesinger isomonodromic (2 × -families leading to algebraic solutions of Garnier ystems, was done in [11]. Note that the monodromy of the triangular Schlesinger isomonodromicfamilies corresponding to algebraic solutions from Theorems 10, 11 is abelian , similarly to the linearmonodromy of the rational solutions to the Painlev´e VI equations from Theorems 5, 6, 7 and 8. For a dihedral linear monodromy, there are families of algebraic solutions obtained in [21], for M = 2 andin [32] for an arbitrary even M . Earlier, algebraic solutions of some particular Garnier systems werealso proposed in [48], by applying birational canonical transformations to a fixed algebraic solution,without appealing to Schlesinger isomonodromic deformations though. Acknowledgements.
We thank Vladimir Leksin who had drawn attention of the second author tothe paper [16], which has led to the present work, as well as Irina Goryuchkina for helping us to verifyby Maple the solutions of Example 5 (they indeed satisfy the corresponding Garnier systems!).V.S. is grateful to the Natural Sciences and Engineering Research Council of Canada for the financialsupport through a Discovery grant. The research of V.D. was partially supported by the SerbianMinistry of Education, Science, and Technological Development, the Science Fund of Serbia and byThe University of Texas at Dallas.
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