On local Weyl equivalence of higher order Fucshian equations
aa r X i v : . [ m a t h . C A ] D ec ON LOCAL WEYL EQUIVALENCEOF HIGHER ORDER FUCSHIAN EQUATIONS
SHIRA TANNY AND SERGEI YAKOVENKO
Abstract.
We study the local classification of higher order Fuchsian lineardifferential equations under various refinements of the classical notion of the“type of differential equation” introduced by Frobenius. The main source ofdifficulties is the fact that there is no natural group action generating thisclassification.We establish a number of results on higher order equations which are similarbut not completely parallel to the known results on local (holomorphic andmeromorphic) gauge equivalence of systems of first order equations. Local classification of linear ordinary differential equations
Systems and higher order equations.
The local analytic theory of linearordinary differential equations exists in two parallel flavours, either that of systemsof several first order equations, or of scalar (higher order) equations. One canrelatively easily transform one type of objects to the other, yet this transformationloses some additional structures.Let k be a differential field, called the field of coefficients . We will be interestedalmost exclusively in the field M = M ( C ,
0) of meromorphic germs at the origin t = 0 on the complex line C = C , the quotient field of the ring O = O ( C ,
0) ofholomorphic germs at the origin. The standard C -linear derivation ∂ = ddt acts onboth O and M according to the Leibniz rule and extends on vector and matrixfunctions with entries in k un the natural way.Let A ∈ Mat( n, k ) be an ( n × n )-matrix function, called the coefficients matrix , A = k a ij ( t ) k ni,j =1 , a ij ∈ k . This matrix defines the homogeneous system of linearordinary equations ∂x = Ax, x = ( x , . . . , x n ) ∈ C n , t ∈ ( C , . (1)The system (1) only exceptionally rarely has a solution x ∈ k n . However, it alwayshas n linear independent solutions in the class of functions analytic in a smallpunctured neighborhood of the origin, which are multivalued (ramified) over thepoint t = 0. Assembling these solutions (as column vectors) into a multivaluedmatrix function X = X ( t ) whose determinant never vanishes for t = 0, we canwithout loss of generality reduce the system (1) to one matrix differential equation ∂X = AX . For instance, the trivial system is defined by the equation ∂X = 0, andany invertible constant matrix C ∈ GL( n, C ) is its solution.Alternatively, one may consider homogeneous linear ordinary differential equa-tions of the form a ∂ n u + a ∂ n − u + · · · + a n − ∂u + a n u = 0 , a , . . . , a m ∈ k , a = 0 . (2) Date : Decemeber 25, 2014.
Each equation (2) is a linear (over k ) relation between the unknown function u andits derivatives ∂ k u up to order k = n . Traditionally, such equations are writtenusing linear differential operators : if L = P n a i ∂ n − i ∈ k [ ∂ ] is the formal expression,then the above equation is written under the form Lu = 0. Elements of the field k are identified with “operators of zeroth order” u au , a ∈ k . The key feature ofdifferential operators is the possibility of their composition which equips the k -spaceof linear operators with the structure of (noncommutative infinite-dimensional) C -algebra, denoted by W . As before, generically solution exists only as a multivalued function defined for t = 0 and ramified over the origin.1.2. Mutual reduction.
One can easily transform the equation (2) to a system(1) by introducing the variables x k = ∂ k − u , k = 1 , . . . , n . The corresponding firstorder identities take the form ∂x k = x k +1 , k = 1 , . . . , n − , ∂x n = − a − ( a x n − + · · · + a n x ) . (3)Conversely, each of the variables u = x k of a solution x ( t ) to the system (1) satisfiesan equation of the form (2). To obtain this equation, note that all derivatives ∂ i u are k -linear combinations of the formal variables x , . . . , x n . Indeed, by induction,if ∂ i x = A i x , A i ∈ Mat( n, k ), A = E , A = A , then ∂ i +1 x = ( ∂A i ) x + A i A x = ( ∂A i + A i A ) x = A i +1 x, i = 1 , , . . . . (4)Taking the k th line of these identities yields the required liner combination. Sincethe space of combinations is n -dimensional (over k ), we conclude that n + 1 deriva-tives u, ∂u, ∂ u, . . . , ∂ n u are necessarily linear dependent over k (the order can beless than n ). This dependence is of the form (2), but the corresponding equationwill in general depend on the choice of k between 1 and n . Slightly modifying thisconstruction, one can produce a differential equation of order n , satisfied by all components x ij of any fundamental matrix solution X = k x ij k of the equation ∂X = AX .1.3. Gauge equivalence of linear systems. Equations of the same type.
The group G = GL( n, k ) of invertible matrix functions with entries in the field k acts naturally on the space of all linear systems of the form (1). Namely, if H = k h ij ( t ) k ni,j =1 , h ij ∈ k , is such a function with the inverse H − ∈ GL( n, k ), thenone can “change variables” in (1) by substituting y = Hx , y = ( y , . . . , y n ) ∈ C n .This substitution transforms (1) to the identity ∂y = ( ∂H ) x + H∂x = ( ∂H ) H − y + HAH − y , so that ∂y = By, B ∈ Mat( n, k ) , ( ∂H ) · H − + HAH − . (5)This differs from the conjugacy of linear operators by the logarithmic derivative ( ∂H ) · H − ; this term vanishes if H is constant.Two systems ∂x = Ax and ∂y = By are called gauge equivalent , if there exists anelement H ∈ G such that (5) holds. Since G is a group, this equivalence naturallyis reflexive, symmetric and transitive. Thus one can formulate the problem of classification : what is the simplest form to which a given linear system can be The classical Weyl algebra is generated over C by two elements t, ∂ with the commutativityrelation [ ∂, t ] = 1. It embeds naturally into the algebra k [ ∂ ] for the differential field of rationalfunctions k = C ( t ). OCAL WEYL EQUIVALENCE 3 transformed by a suitable gauge transformation? The corresponding theory is fairlywell established, see below for the initial results.
Remark . Systems of linear equations (1) can be considered geometrically as flatmeromorphic connections on a vector bundle over the (complex) 1-dimensionalbase. The gauge transform corresponds to the change of a tuple of horizontalsections locally trivializing this bundle. Such interpretation allows for global andmultidimensional generalizations, see [IY, Chapter III] and [NY].Unfortunately, the notion of gauge equivalence is too restricted to deal with highorder equations: indeed, since the unknown function is scalar, only the transfor-mations of the form u = hv , h ∈ k , can be considered, but one cannot expect thissmall group to produce a meaningful classification.Instead it is natural to consider k -linear changes of variables of a more generalform which involve the unknown function and its derivatives. More specifically, onecan choose a tuple of functions h = ( h , . . . , h n − ) ∈ k n and use it to change thedependent variable from u to v as follows, v = h ∂ n − u + h ∂ n − u + · · · + h n − ∂u + h n u. (6)The reason why derivatives of order n and may be omitted, is rather clear: if thetransformation (6) is applied to an equation (2) of order n , then all such higher orderderivatives can be replaced by k -linear combinations of the lower order derivativesby virtue of the equation.The new variable v also satisfies a linear differential equation which can bederived as follows (cf. with § v by virtue ofthe equation (2), one can see that all higher order derivatives ∂ i v can be expressedas linear combinations (over k ) of the formal derivatives ∂ j u , u = 0 , . . . , n −
1. Thespace of such combinations is n -dimensional, so no later than on the n th step therewill necessary appear an identity of the form b ∂ m v + b ∂ m − v + · · · + b m − ∂v + b m v =0, b = 0, b j ∈ k , m n , which is the transform of the equation (2) by the actionof (6). Classically, the initial equation and the transformed equation are called equations of the same type , see [S, T, O], but we would prefer to use the term“Weyl equivalence” (justifying the fact), with an intention to refine it by imposingadditional restrictions imposed on the transformation (6).In order for this change of variables to be “faithful”, one has to impose theadditional condition of nondegeneracy: no solution of (2) is mapped into identicalzero by the transformation (6). Indeed, if this extra assumption is violated, one caneasily transform the initial equation to the trivial (meaningless) form 0 = 0. Onthe other hand, accepting this condition guarantees (as can be easily shown) thatthe transformed equation has the same order m = n .Still a few questions remain unanswered by this na¨ıve approach. The transfor-mation (6), unlike the gauge transformation of linear systems, is rather problematicto invert: transition from u to v always has a nontrivial kernel (solutions of the cor-responding homogeneous equations). In addition, “restoring” u from v is in generala transcendental operation requiring integration of linear equations, and it is by nomeans clear how one should proceed.The algebraic nature of these questions was studies since 1880-ies by F. Frobe-nius, E. Landau, A. Loewy, W. Krull and culminated in the perfect form in thebrilliant paper by Ø. Ore [O]. The idea is to consider the noncommutative algebra S. TANNY AND S. YAKOVENKO of differential operators k [ ∂ ] with coefficients in k . The next section § Singularities, monodromy.
From this moment we focus on the special casewhere k = M is the differential field of meromorphic germs at the origin and denotefor brevity W = M [ ∂ ] the algebra of operators with meromorphic coefficients.For each linear system (1) or a high order equation (2) with meromorphic coef-ficients one can choose representatives of germs of all coefficients a ij ( t ), resp., a i ( t )in a punctured neighborhood of the origin ( C , r { } so small that all represen-tatives are holomorphic in this punctured neighborhood. The classical theorems ofanalysis guarantee that solutions of the system (resp., equation) are holomorphicon the universal cover of this punctured neighborhood, i.e., in the more traditionalterminology, are multivalued analytic functions on ( C ,
0) ramified at the origin.If the coefficients of the system (1) are holomorphic at the origin, i.e., A ∈ Mat( n, O ) ( Mat( n, M ), then for the same reasons solutions of the system areholomorphic (hence single-valued) at the origin. This case is called nonsingular ,and the corresponding matrix equation admits a unique solution X ∈ GL( n, O )with the initial condition X (0) = E (the identity matrix).Solution X of a general matrix equation ∂X = AX with A ∈ GL( M , n ) aftercontinuation along a small closed loop around the origin gets transformed intoanother solution X ′ = XM of the same equation. The monodromy matrix M ∈ GL( n, C ) depends on X .A homogeneous equation (2) defined by a linear operator L = P ni =0 a i ∂ n − i can always be multiplied by a meromorphic multiplier so that all its coefficientsbecome holomorphic and at least one of them is nonvanishing at the origin. Thereduction (3) shows that if it is the leading coefficient a that is nonvanishing, thenall solutions of the equation Lu = 0 are holomorphic at the origin (we call suchoperators nonsingular ), otherwise they may be ramified at the origin.Choose a neighborhood U = ( C ,
0) and meromorphic representatives of thegerms a j ( · ) which have no other poles in U expect for t = 0. If 0 = t ∈ U is any other point in the domain of the system (equation), then it is well knownthat germs of solutions of the system (equation) Lu = 0 form a C -linear subspacein Z L ⊂ O ( C , t ) of dimension dim C Z L exactly equal to n . After the analyticcontinuation along a small loop around the origin, this space is mapped into itselfby a linear invertible map called the monodromy transformation (monodromy, forshort): for any basis u , . . . , u n in the space of solutions (considered as a row vectorfunction), we have ∆ (cid:0) u · · · u n (cid:1) = (cid:0) u · · · u n (cid:1) M (7)for a suitable nondegenerate matrix M (depending on the basis { u i } ni =1 ).1.5. Different flavors of the gauge classification.
The gauge transformationgroup G = GL( n, M ) introduced above, may be too large for certain problemsof analysis, see § G h = GL( n, O ) of holomorphic matrix functions which are holomorphicallyinvertible. It is the semidirect product of GL( n, C ) and the group G of holomorphicmatrix germs H which are identical at the origin, G = { H ∈ G : H (0) = E } .Besides, one can identify two types of singularities of linear systems, character-ized by strikingly different behavior of solutions, called respectively regular (in full, OCAL WEYL EQUIVALENCE 5 regular singular, to avoid confusion with nonsingular systems) and irregular singu-larities. Recall [IY, Definition 16.1] that the system (1) is called regular if the norm | X ( t ) | of any its fundamental matrix solution grows no faster than polynomiallywhen approaching the singular point in any sector on the t -plane (more precisely,on the universal cover of ( C , r | X ( t ) | Ct − N ∀ t ∈ ( C , , α < Arg t < β, C > , N < + ∞ , (8)for some constants C, N depending on the sector (its opening and the radius).This condition is difficult to verify as it refers to the properties of solutions, butit is automatically satisfied for
Fuchsian systems, when the meromorphic matrixfunction A has a pole of at most first order [IY, Theorem 16.10 (Sauvage, 1886)]. Example 1. An Euler system is any system of the form ∂X = t − BX witha constant matrix B ∈ Mat( n, C ). Its fundamental matrix solution is given bythe (multivalued) matrix function X ( t ) = t B = exp( B ln t ), t ∈ ( C , B =diag( λ , . . . , λ n ) is a diagonal matrix with λ i ∈ C , then the solution is also diagonal, X ( t ) = diag( t λ , . . . , t λ n ). The monodromy matrix of this solution is exp 2 πiB ∈ GL( n, C ).In general if λ , . . . , λ n are the eigenvalues of the matrix B , then the correspond-ing Euler system is called resonant if some of the differences λ i − λ k are naturalnumbers (nonzero), otherwise the system is called nonresonant .The principal results on classification of linear systems are summarized in Ta-ble 1, based on [IY, § § equivalent to a condition on the order of the poles of the ratios a i /a ∈ M of thecoefficients of the equation (this condition is also called the Fuchsian condition).1.6. Goals of the paper and main results.
We study the classification of non-singular or Fuchsian (singular) equations with respect to the Weyl equivalence (for-mally introduced below).It can be easily shown (see below) that nonsingular equations are Weyl equivalentto the trivial equation ∂ n u = 0, whose solutions are polynomials of degrees n − Fuchsian equivalence , or F -equivalencefor short, using expansion of operators in noncommutative Taylor series. It turnsout that the corresponding classification of Fuchsian operators is very similar to theholomorphic classification of Fuchsian systems. In particular, in the nonresonantcase any Fuchsian equation is F -equivalent to an Euler equation, while resonantoperators are F -equivalent to operators with polynomial coefficients, i.e., from C [ t ][ ∂ ]. Finally, we show that any (resonant) Fuchsian operator is F -equivalent toan operator which is Liouville integrable , that is, whose solutions can be obtainedfrom rational functions by iterated integration and exponentiation.1.7.
Acknowledgements.
We are grateful to all our friends and colleagues whohelped us to identify classical sources and thus avoid re-inventing the bicycle. The
S. TANNY AND S. YAKOVENKO
Table 1.
Normal forms of linear systems.Type of singularity / Group Holomorphic G Meromorphic G Nonsingular Trivial TrivialFuchsian nonresonant EulerFuchsian resonant Polynomial integrable EulerRegular non-Fuchsian RationalIrregular nonresonant Formally diagonalizable, divergentIrregular resonant Ramified gauge transforms are required
Polynomial normal form.
The system takes the form ∂X = t − ( B + B t + B t + · · · + B p t p ) X , where p is the maximal integer difference between the eigenvalues of the Jordanmatrix B . The matrices B k may have nonzero entry in the ( i, j )th position only if λ i − λ j = k , that is, are very sparse. The system in the normal form can be explicitlysolved: there exists a fundamental matrix solution of the form X ( t ) = t P t Q with twoconstant matrices P, Q ∈ Mat( n, C ) not commuting between themselves. Rational normal norm.
In this case the normal form is rational and explicit but its de-scription is off the main track of this work.
Irregular systems.
For irregular systems with the matrix of coefficients represented by aLaurent series A ( t ) = t − r ( A + tA + · · · ), r >
2, the definition of resonance requiresthat the eigenvalues of the leading matrix coefficient A are pairwise different. In thenonresonant case one can find a formal matrix series H ( t ) = E + H t + H t + · · · whichreduces the system to a diagonal normal form ∂X = t − r D ( t ) X with a diagonal polynomialnormal form, D (0) = A , but this series almost always diverges, see [IY, § t . We will not deal with irregular systems orequations in this paper. primary gratitude goes to Yuri Berest, Gal Binyamini, Dima Gourevitch, DmitryNovikov, Michael Singer and Yuri Zarkhin.2. Algebras of differential operators
In this section we recall the basic facts about the algebra of differential operatorswith coefficients from a different field.2.1.
Noncommutative polynomials in one variable over a differential field.
Consider the C -algebra k [ ∂ ] generated by the differential field k and the symbol ∂ with the noncommutative multiplication satisfying the Leibniz rule, ∂ · a = a · ∂ + a ′ , a, a ′ ∈ k , a ′ = ∂a = the derivative of a. (9)This algebra can be considered as the algebra of differential operators acting on“test functions”, where elements from k act by multiplication u au and ∂ is thederivation. The operation corresponds to the composition of operators (and thedot will be omitted from the notation).Any operator from k [ ∂ ] can be uniquely represented under the “standard form” L = a ∂ n + a ∂ n − + · · · + a n − ∂ + a n , a , . . . , a n ∈ k , a = 0 (10) OCAL WEYL EQUIVALENCE 7 with the coefficients a i to the left from the powers of ∂ . The number n > L . The composition LM of two operators L and M = b ∂ m + · · · ∈ k [ ∂ ] of orders n and m is an operator of order n + m with the(nonzero) leading coefficient a b ∈ k .The key property of the algebra k [ ∂ ] is the possibility of division with remainder.Indeed, if n = ord L > m = ord M , then the difference L − a b − ∂ n − m M is anoperator with zero (absent) “leading coefficient” before ∂ n , i.e., is of order strictlyless than n . Iterating this order depression, one can find two operators Q, R ∈ k [ ∂ ]such that L = QM + R, ord Q = ord L − ord M, ord R < ord M. (11)When R = 0 we say that L is divisible by M .This construction allows to define for any two operators L, M ∈ k [ ∂ ] their greatestcommon divisor D = gcd( L, M ) as the operator of maximal order which dividesboth L and M (this operator is defined modulo a multiplication by an element from k ). The Euclid algorithm [O, Theorem 4] guarantees that for any L, M there exist
U, V ∈ k [ ∂ ] such that U L + V M = gcd(
L, M ) , ord U < ord M, ord V < ord L. (12)A less direct computation allows to construct the least common multiple lcm( L, M )which is by definition the smallest order operator divisible by both L and M (andalso defined modulo a nonzero coefficient from k ). Indeed, consider the operators M, ∂M, ∂ M, . . . , ∂ n M modulo L , i.e., their remainders after division by L , n =ord L . Since all these n + 1 remainders are of order n −
1, they must be lineardependent over k , that is, a certain linear combination ( c ∂ n + · · · + c n − ∂ + c n ) M = P M must be divisible by L : P M = QL , ord P ord L , ord Q ord M . There isan explicit formula expressing lcm( L, M ) through the operators appearing in theEuclid’s algorithm, see [O, Theorem 8].2.2.
Algebra vs. analysis.
Denote by W the local Weyl algebra k [ ∂ ] in the casewhere k = M is the differential field of meromorphic germs.If an operator L is divisible by M in W , then their spaces of solutions Z L ,resp., Z M , are subject to the inclusion Z M ⊆ Z L . Conversely, if for two operators L, M ∈ W we have Z M ⊆ Z L , then L is divisible by M . Indeed, otherwise theremainder of division of L by M would be an operator of order strictly less thanord M , whose solutions form the space of superior dimension dim Z M = ord M . Interms of solutions, D = gcd( L, M ) ⇐⇒ Z D = Z L ∩ Z M ,P = lcm( L, M ) ⇐⇒ Z P = Z L + Z M (13)(the sum of linear subspaces in O ( C , t ) is assumed).Thus two equations Lu = 0 and M v = 0 are of the same type in the sense of § H ∈ W which maps Z L to Z M isomorphically: for any u such that Lu = 0, the function v = Hu is annulledby M . Definition 1.
Two operators
L, M ∈ W of the same order n are called Weylequivalent (or Weyl conjugate), if there exist two operators
H, K ∈ W of order n −
1, such that
M H = KL, gcd(
L, H ) = 1 , ord H, K < ord
L, M. (14)
S. TANNY AND S. YAKOVENKO
The operator H is said to be the conjugacy between L and M . Remark . Ø. Ore uses the notation M = lcm( L, H ) H − = HLH − to denote thefact of conjugacy to stress its resemblance with the “similarity” in the noncommuta-tive algebra W . It has its mnemonic advantages, although the formal constructionof (noncommutative) field of ratios for W requires additional efforts [O, p. 487].We will abbreviate the words “Weyl equivalence” (resp., conjugacy) to W -equivalence (conjugacy) for simplicity. Theorem 1. W -conjugacy is indeed an equivalence relation : it is reflexive, tran-sitive and symmetric.Proof. It is obvious that this relationship is reflexive (suffices to choose H = K = 1).To prove its transitivity, assume that L is W -conjugate with L , and L with L .This means that there exist operators H i , K i ∈ W , i = 1 ,
2, of order n − L H = K L and L H = K L . Then L H H = K K L . To produce apair ( H ′ , K ′ ) conjugating L with L , it suffices to define H ′ = H H mod L : theorder of this remainder will not exceed n − H ′ , L ) = 1, but this is obvious: if u is a nontrivial solution of L u = 0and gcd( H , L ) = 1, then v = H u is a nontrivial solution of L v = 0, hence H v = 0. Replacing H H by its remainder modulo L cannot change the fact that H H u = 0 for any solution of L u = 0.The symmetry is less trivial, see [O, Theorem 13]. For the reader’s conveniencewe provide here a short direct proof due to Yu. Berest. It is convenient to formulateit as a separate lemma. (cid:3) Lemma 1.
For any two operators
L, M satisfying (14) , there exists a pair of operators
V, W ∈ W such that LV = W M and gcd(
V, M ) = 1 .Proof.
By (12), the condition gcd(
L, H ) = 1 implies that there exist
U, V ∈ W such UL + V H = 1. Multiplying this identity by L from the left, we see that ( LU − L = LV H ,that is, the operator Q , expressed by each side of the identity, is divisible by both H and L . This means that the operator LV H is divisible by P = lcm( H, L ), which in turnhas two representations, P = MH = KL as in (14). The last divisibility means that LV H = W P = W MH in W . Yet since W has no zero divisors (the leading coefficient ofany composition is nonzero), we can cancel H and arrive at the identity LV = W M . It isa simple exercise to see that gcd(
V, M ) = 1. (cid:3)
Nonsingular operators.
An operator L ∈ W of the form (10) is referred toas nonsingular , if all its coefficients are holomorphic, a i ∈ O ( C , a (0) = 0. Nonsingular operators can be reduced by thetransformation (3) to a holomorphic (nonsingular) system of first order equations.An immediate conclusion is that the corresponding equation Lu = 0 has onlyholomorphic solutions, and a fundamental system of solutions { u k } nk =0 can alwaysbe chosen so that u k ( t ) = t k + · · · where the dots stand for terms of order greaterthan k .2.4. Fuchsian operators.
There exists another special subclass of linear operators L ∈ W with the property that the respective linear equations Lu = 0 enjoy acertain regularity, namely, all their solutions grow moderately when approachingthe singular point at the origin. Unlike the general linear systems (1), such operatorsadmit precise algebraic description. It can be given in several equivalent forms. OCAL WEYL EQUIVALENCE 9
Note that together with the “basic” derivation ∂ any other element a∂ ∈ W isalso a derivation of the field M ( C -linear self-map satisfying the Leibniz rule). Itcan be used as the generator of the algebra W . We will be mostly interested in the Euler derivation ǫ = t∂ ∈ W with the commutation rule ǫ = t · ∂, ǫ · t m = t m · ( ǫ + m ) , ∀ m ∈ Z , (15)cf. with (9). Though the derivations ∂ and ǫ are very simply related, their alge-braic nature is radically different. Restricted on the finite-dimensional subspace ofpolynomials of any finite degree, the standard derivation ∂ is nilpotent, while theEuler derivation is semisimple ( ǫ is diagonal in the monomial basis, ǫ ( t m ) = mt m ).For any polynomial w ∈ C [ ǫ ] in the variable ǫ denote by w [ j ] , j ∈ Z , the shift ofthe argument: w w [ j ] , w [ j ] ( ǫ ) = w ( ǫ + j ) , j ∈ Z . (16)This operator preserves the degree of the polynomial, and using it one can rewritethe commutation rule (15) as follows, ∀ w ∈ C [ ǫ ] , ∀ j ∈ Z , wt j = t j w [ j ] . (17)Substituting ∂ = t − ǫ and re-expanding terms, any operator L ∈ W can berepresented under the form L = r ǫ n + r ǫ n − + · · · + r n − ǫ + r n , r i ∈ M , r = 0 . (18) Definition 2.
An operator L is called Fuchsian , if in the representation (18) allcoefficients r i are holomorphic and the leading coefficient r is invertible (nonvan-ishing): r , r , . . . , r n ∈ O ( C , , r (0) = 0 . (19)The operator is pre-Fuchsian, if it has a form hL with any nonzero h ∈ M ; withoutloss of generality one may assume that h = t k , k ∈ Z .An operator is called Eulerian , if all coefficients r , . . . , r n ∈ C are constant. Remark . In the classical literature the notion of Fuchsian operators is not de-fined, only the notion of a (homogeneous) Fuchsian equation of the form Lu = 0is discussed. Clearly, two operators L ∈ W and hL , h ∈ M , define the samehomogeneous equation. For operators written in the form (18), the correspondinghomogeneous equation will be Fuchsian if and only if all ratios r i /r ∈ M are actu-ally holomorphic at t = 0 for all i = 1 , . . . , n . Our choice may seem to be artificial,yet it is justified by subsequent computations.We will denote by F ⊂ W the set of all Fuchsian operators. It is convenientto assume that holomorphically invertible germs and meromorphic germs belong in F , resp., pre- F as “differential operators of zero order”.A Fuchsian differential equation Lu = 0 with L as in (18) can be reduced to aFuchsian system in the sense (1.5) by slightly modifying the computation (3): onehas to introduce the new variables as follows, x = u , and then ǫx k = x k +1 , k = 1 , . . . , n − , ǫx n = − r − ( r x n − + · · · + r n x ) (20)(recall that r is invertible hence r − ∈ O ), or in the matrix form, ǫx = Rx , withthe holomorphic matrix R ∈ Mat( n, O ) of coefficients. This computation explainsthe relation of two Fuchsian objects of different nature. However, unlike the case ofsystems, in the case of scalar equations the Fuchsian condition is not only sufficient,but also necessary for the regularity (moderate growth of solutions). Theorem 2 (L. Fuchs (1868), see [IY, Theorem 19.20]) . The operator L ∈ W is pre-Fuchsian if and only if all solutions of the equation Lu = 0 and all theirderivatives grow at most polynomially in any sector with the vertex at the origin inthe sense (8) . (cid:3) First results on W -classification. The initial results on W -equivalence arecompletely parallel to G h -classification of nonsingular systems and G -classificationof regular systems: even the ideas of the proofs remain the same. Theorem 3.
A nonsingular operator is W -conjugate to the operator M = ∂ n by a nonsingular operator H of order n − .Proof. Any nonsingular equation Lu = 0 of order n always admits n linear indepen-dent solutions of the form u k ( t ) = t k − (1 + · · · ), k = 1 , . . . , n . Indeed, one shouldlook for solutions of the companion system (3) with a suitable initial condition x k (0) = 1, x j (0) = 0 for all j = k .A linear operator H transforming solutions v k = t k − of the equation ∂ n = 0to solutions of the equation Lu = 0 by the formulas (6) can be obtained by themethod of indeterminate coefficients: H = h ∂ n − + · · · + h n − ∂ + h n . The equations Hv k = u k , k = 1 , . . . , n correspond to a system of linear algebraic equations over O for the unknown coefficients h i : (cid:0) h n h n − · · · h (cid:1) t t · · · t n − t · · · ( n − t n − n − = (cid:0) v v · · · v n (cid:1) The matrix J of coefficients, the companion matrix of the tuple of solutions v = 1, v = t, . . . , v n = t n − , is holomorphic and invertible (it is upper triangular withnonzero diagonal entries). A simple inspection shows that the leading coefficient h cannot vanish at t = 0, hence the operator H will be nonsingular. (cid:3) A minor modification of this argument proves the following general result.
Theorem 4.
Any ( pre- ) Fuchsian operator is W -equivalent to an Euler operatorfrom C [ ǫ ] .Proof. Let, as before, J denote the Euler-companion matrix of n linear independentsolutions u , . . . , u n of the equation Lu = 0: unlike the usual companion matrix, itis obtained by applying the iterated Euler derivation ǫ instead of ∂ to the functions u i : J = J ( t ) = ǫ ... ǫ n − · (cid:0) u u . . . u n (cid:1) = u u . . . u n ǫu ǫu . . . ǫu n ... ... . . . ... ǫ n − u ǫ n − u . . . ǫ n − u n Unlike in the nonsingular case, we cannot guarantee anymore that J ( t ) is holomor-phic and invertible: its entries are in general multivalued and grow moderately atthe origin. The companion matrix J ( t ) has a monodromy factor C ∈ GL( n, C ):∆ J ( t ) = J ( t ) C exactly as in (7) which applies to each row of the matrix J . Yet onecan always find an Euler equation whose tuple of solutions v = ( v , · · · , v n ) will OCAL WEYL EQUIVALENCE 11 exhibit exactly the same monodromy matrix factor C , see [IY, Proposition 19.29]:∆ v = vC . The corresponding linear system of algebraic equations takes the form (cid:0) h n h n − · · · h (cid:1) J ( t ) = (cid:0) v v · · · v n (cid:1) The solution is given by the product (cid:0) h n h n − · · · h (cid:1) = (cid:0) v v · · · v n (cid:1) · J − ( t ). This product is single-valued: after analytic continuation around the originwe have∆ (cid:0) h n · · · h (cid:1) = (cid:0) v · · · v n (cid:1) C · C − J − ( t ) = (cid:0) h n · · · h (cid:1) . Because of the moderate growth assumption, the coefficients h j of the conjugatingoperator H must be meromorphic germs at the origin, thus H ∈ W . (cid:3) Remark . There is no reason to expect that the operator H conjugating two Fuchsianoperators is necessarily (pre)-Fuchsian. Indeed, let L be any Fuchsian operator and H anirregular conjugacy. Applying H to any basis tuple of solutions u , . . . , u n of Lu = 0, weobtain another tuple v i = Hu i which also grow moderately and have the same monodromyas u i . By the Fuchs theorem, they satisfy a Fuchsian equation Mv = 0. Thus, the twoFuchsian operators L, M are conjugated by a (unique for the reasons of order/dimension)irregular operator H . In other words, the (general) Weyl classification of (pre)-Fuchsian operators co-incides with the classification of their monodromy matrices, very much like themeromorphic gauge classification of linear systems (1).3.
Fuchsian equivalence
It appears that a comprehensive analog of the holomorphic gauge equivalencebetween Fuchsian linear systems is the Fuchsian equivalence of Fuchsian operators:modulo technical details, this equivalence means the Weyl conjugacy (14) by aFuchsian operator H subject to certain nondegeneracy constraints. We start withdeveloping the formal theory of such equivalence via noncommutative formal powerseries .3.1. Noncommutative Taylor expansions for Fuchsian operators.
Togetherwith the representation of differential operators from the ring W = M [ ǫ ] as poly-nomials in ǫ ∈ W with coefficients in M , we can expand them in convergent noncommutative Laurent series in the variable t ∈ ( C ,
0) with (right) coefficientsfrom the (commutative) ring C [ ǫ ]. Any operator L ∈ W of order n = ord L can beexpanded under the form L = + ∞ X k = − N t k p k ( ǫ ) , max k deg ǫ p k = n, N < + ∞ . The operator is Fuchsian if and only if all powers are nonnegative and the leadingcoefficient p is of the maximal degree: L ∈ F if and only if L = ∞ X k =0 t k p k ( ǫ ) , p k ∈ C [ ǫ ] , deg p k n, deg p = n. (21)The differential operator p ∈ C [ ǫ ] ⊂ F is called the Euler part of L , or its Euler-ization (in analogy with linearization) and denoted by E ( L ).Very informally, an operator with holomorphic coefficients can be considered asa small perturbation of its Eulerization. The Fuchsian condition means that this perturbation is nonsingular, i.e., it does not increase the order of the Euler part,in the same way as the nonsingularity condition means that the operator can beconsidered as a small nonsingular perturbation of the operator p ( ∂ ).The key tool used in this paper will be a systematic use of the Taylor expansion(21) in exactly the same way the theory of formal series with matrix coefficientsof the form H ( t ) = P ∞ k =0 t k H k , H ∈ Mat( n, C ), is used in the theory of formalnormal forms of vector fields [IY, § § H k commutewith the variable t but in general do not commute between themselves. In theoperator case the polynomial coefficients p k ∈ C [ ǫ ] commute between themselvesbut do not commute with t .Together with the convergent noncommutative Taylor series, it is convenient tointroduce the class of formal Fuchsian operators.
Definition 3.
A formal Fuchsian operator is a formal series of the form (21) with-out any convergence assumption. The set of formal Fuchsian operators is denotedby ˆ F . Remark . For any two Fuchsian operators
L, M ∈ F their composition is again aFuchsian operator of order ord L + ord M . If ord M ord L , then the incompleteratio Q as in (11) is a Fuchsian operator of order ord L − ord M . The same appliesto ˆ F . This follows from direct inspection of the division algorithm.However, the set F is not a subalgebra of W : the sum of two Fuchsian operatorsmay well be non-Fuchsian. Hence the remainder R as in (11) after the incompletedivision may well turn non-Fuchsian (the leading coefficient may vanish). Yet forany two given Fuchsian operators L, M of degrees n > m one can construct a relaxeddivision with remainder L = Q ′ M + R ′ with ord Q ′ = n − m and ord R ′ = m and Q, R
Fuchsian. Indeed, it suffices to modify the standard division with remainder L = QM + R with ord R n − Q, R with holomorphic coefficients)and replace Q ′ = Q − R ′ = M + R : the latter operators will be automaticallyFuchsian.3.2. Main definition.Definition 4.
Two operators
L, M ∈ W of the same order n are called Fuchsianequivalent (or F -equivalent), if there exist two Fuchsian operators H, K ∈ F suchthat M H = KL (exactly as in Definition 1), but with the additional propertythat the Euler parts of H and L are mutually prime (have no common roots), i.e.,gcd( E ( H ) , E ( L )) = 1 ∈ C [ ǫ ].Two formal Fuchsian operators L, M ∈ ˆ F are called formally F -equivalent ( ˆ F -equivalent in short) if there exist H, K ∈ ˆ F such that M H = KL and the Eulerparts of H, L are mutually prime.We expect that the Fuchsian classification (and its formal counterpart) for ar-bitrary operators from W will be a very challenging problem with the Stokes phe-nomenon [IY, §
20] manifesting itself in a new way. However, everywhere below wewill deal only with the F -equivalence between Fuchsian operators .Note that we dropped the condition on the order of
H, K which can now be higher than n . Besides, in this definition we replaced the condition gcd( H, L ) = 1 ∈ W from (14) by the stronger condition on the mutual primality of the Eulerizations. OCAL WEYL EQUIVALENCE 13
Theorem 5. ˆ F -conjugacy is indeed an equivalence relation : it is reflexive, sym-metric and transitive. Reflexivity is obvious: each operator L is F -equivalent to itself by admissibleconjugacy H = 1 (which is a zero order Fuchsian operator).The transitivity is even simpler compared to the proof of Theorem 1: we do notreplace the composition H H of F -conjugacies, which is always Fuchsian, by itsremainder mod L , which may be non-Fuchsian.However, the proof of the symmetry, given in Lemma 1 relies on the possibilityof representing the identical operator 1 by a combination 1 = U L + V H withFuchsian operators
U, V ∈ ˆ F . Simple example shows that even under the strongerassumption gcd( E ( L ) , E ( H )) = 1, this representation is not always possible withoperators of the minimal order n − Fuchsian invertibility.
It will be convenient to introduce the following no-tation: ∀ L, H ∈ F gcd ( L, H ) = gcd( E ( L ) , E ( H )) ∈ C [ ǫ ] . (22)Using this notation, the second condition of F -equivalence can be shortened togcd ( L, H ) = 1.As follows from the proof of Lemma 1, the key step is to show that if H is aFuchsian operator such that gcd ( L, H ) = 1, then there exist two Fuchsian oper-ators
U, V ∈ F such that U L + V H = 1 ∈ F and gcd ( V, L ) = 1. Recall that if p, q ∈ C [ ǫ ] are two relatively prime polynomials of respective degrees n, m , then thelinear Sylvester map from C m × C n to C m + n S = S p,q : ( u, v ) pu + qv, deg u m − , deg v n − , (23)is injective and surjective (here we identify C m and C n with the linear spaces ofpolynomials of degree m −
1, resp., n − C [ ǫ ]of the form up + vq = r, deg r deg p + deg q − , is solvable with respect to u, v constrained as above.The following result is the analog of the implicit function theorem for differentialoperators. Lemma 2. If L, M ∈ F are two Fuchsian operators with gcd ( L, M ) = 1 , thenfor any operator R = P t k r k ( ǫ ) of order ord L + ord M − with holomorphiccoefficients the equation U L + V M = R is solvable with respect to the operators U, V of orders ord M − and ord L − respectively, also with holomorphic coefficients. Note that we do not assume R Fuchsian, nor claim the Fuchsianity of U and V . Proof.
The proof is achieved by inductive determination of the coefficients of theunknown operators
U, V .Substitute the expansions for L = P ∞ t k p k and M = P ∞ t k q k and the unknownoperators U = P ∞ t k u k , V = P ∞ t k v k , p k , q k , u k , v k ∈ C [ ǫ ] into the equation U L + V M = R :( u + tu + t u + · · · )( p + tp + t p + · · · )+ ( v + tv + · · · )( q + tq + · · · ) = r + tr + t r · · · . Using the commutation rules (16), we reduce this operator identity to an infiniteseries of identities in C [ ǫ ], u p + v q = r ,u [1] p + u p + v [1] q + v q = r ,u [2] p + u [1] p + u p + v [2] q + v [1] q + v q = r , . . . . . . . . . . . . . . . . . . . . . . . . · · · + u k p + v k q = r k , ∀ k > . This system has a “triangular” form: each left hand side is the sum of the term u k p + v k q = S ( u k , v k ) and terms involved shifted polynomials u [ j ] i , v [ j ] i with i, j 1, deg v k deg p − (cid:3) Remark . The proof of the convergence of the series for U and V can be obtaineddirectly by control over the growth of the polynomial coefficients.However, a simpler argument works. Expanding U, V as polynomials of ǫ with analyticcoefficients from O ( C , U = X k a k ( t ) ǫ k , V = X j b j ( t ) ǫ j , we see that the operator equation UL + V M = R reduces to a system of linear nonhomoge-neous algebraic equations with respect to the unknown coefficients a ( t ) , b ( t ): in a symbolicway, this system can be written as C ( t ) z = f ( t ), where C ( t ) is an ( n + m ) × ( n + m )-matrixwith holomorphic entries (produced from the coefficients of the operators L and M andtheir ǫ -derivatives), and f ( t ) is an ( n + m )-dimensional holomorphic vector function.One can easily see that the condition gcd ( L, M ) = 1 implies that the matrix C (0)is nondegenerate and the system has a holomorphic solution. The formal computationamounts to the formal inversion of the corresponding matrix C ( t ) without even explicitlywriting it down. Unfortunately, the goal of solving the equation U L + V H = 1 in the class ofFuchsian operators cannot be achieved using only this Lemma: indeed, there is noway to ensure that the polynomial v = E ( V ) has the maximal degree equal toord V . The way out is to look for a solution of higher order.We look for a Fuchsian solution of the equation U L + V H = 1 in the class ofoperators ord U ord H = m , ord V ord L = n as follows, U = H + U m − , V = − L + V n − , ord U m − m − , ord V n − n − . Substituting these formulas into the original equation, we transform it to the equa-tion U m − L + V n − H = 1 − [ H, L ] , [ H, L ] = HL − LH. (24)The commutator [ L, H ] of the two Fuchsian operators possesses two obvious prop-erties. It is an operator of order no greater than ord L + ord H − symbols of operators cancel each other whencomputing the commutator). On the other hand, its Euler part vanishes. OCAL WEYL EQUIVALENCE 15 Thus the equation is solvable by virtue of Lemma 2, and E ( U m − ) E ( L ) + E ( V n − ) E ( H ) = 1 ∈ C [ ǫ ] . In other words, gcd ( V n − , L ) = 1. The operator V = − L + V n − is Fuchsian (since L is Fuchsian of order n ), and gcd ( V, L ) = gcd ( V n − , L ) = 1.This completes the proof of the symmetry of the F -equivalence.4. Formal F -classification of Fuchsian operators This and the next section contain the main results of the paper. They areestablished on the formal level, yet at the end we will show that any ˆ F -conjugacybetween convergent Fuchsian operators in fact converges.4.1. Nonresonant case: Eulerization. We start by establishing an analog of thelinearization theorem for nonresonant systems, cf. with the second line in Table 1. Definition 5. A Fuchsian operator L ∈ F is nonresonant, if no two roots of E ( L ) ∈ C [ ǫ ] differ by a positive integer number (multiple roots are allowed). Proposition 1. A nonresonant Fuchsian operator is F -equivalent to its Eulerpart.Proof. Consider the expansion of the operator: L = P ∞ j =0 t j p j ( ǫ ), p = E ( L ). Welook for an operator H = P t j h j ( ǫ ) which would solve (together with some otherFuchsian operator K = P t j k j ( ǫ ) ∈ F ) the operator equation p ( ǫ ) H = KL . Aftersubstituting the expansions and using the commutation rule (15), we obtain in theleft hand side the operator p ( ǫ ) H = p h + tp [1] h + · · · + t j p [ j ] h j + · · · , cf. with the notation (16)–(17). In the right hand side the expansion for KL = ( k + tk + t k + · · · )( p + tp + t p + · · · )will have more complicated form: the term proportional to t j has the form t j ( k j p + k [1] j − p + k [2] j − p + · · · + k [ j ] p j ) . The operator equation thus splits into an infinite number of polynomial equationsinvolving the known polynomials p j and unknown h j , k j as follows, p h = k p ,p [1] h = k p + k [1] p ,p [2] h = k p + k [1] p + k [2] p , . . . . . . . . . . . . . . . . . . . . . . . . p [ j ] h j = k j p + k [1] j − p + · · · + k [ j ] p j , . . . . . . . . . . . . . . . . . . . . . . . . (25)This system can be solved inductively: on the first step we choose h = k anypolynomial of degree n − p . The remaining equations allhave the common structure: p [ j ] h j − p k j = u j , (26)where u j ∈ C [ ǫ ] is a polynomial of degree n − k , . . . , k j − and known p , . . . , p j . If L is nonresonant, no two roots of p differ by a positive integer j , hencegcd( p , p [ j ] ) = 1 for all j = 1 , , . . . and any such equation is (uniquely) solvable bya suitable pair ( h j , k j ) of polynomials of degree n − 1. Thus the entire infinitesystem admits a formal solution ( H, K ).It remains to show that if the series for L = P t j p j was convergent, so will bethe series for L and K . This can be done by the direct estimates, yet we give ageneral proof avoiding all computations later, in § (cid:3) F -normal form and apparent singularities. Some properties of solutionscan be easily described in terms of F -equivalence. Recall that a singular pointof a differential equation is called apparent , if all solutions of this equation areholomorphic at this point. Proposition 2. A Fuchsian operator has only meromorphic solutions if and onlyif it is F -equivalent to an Euler operator L = E ( L ) = p ( ǫ ) with integer pairwisedifferent roots, p ( ǫ ) = Q ni =1 ( ǫ − λ i ) , λ i ∈ Z , λ i = λ k for i = k .A Fuchsian operator has only holomorphic solutions, if and only if it is F -equivalent to an Euler operator as above, with nonnegative pairwise distinct roots, λ i ∈ Z + .Proof. In one direction both statements are obvious. We show that Fuchsian op-erators with only meromorphic (resp., holomorphic) solutions are F -equivalent toan Euler equation as above.One can easily show that any n -dimensional C -linear subspace ℓ in M ( C , O ( C , f i = t λ i u i ( t ) with pairwise different integer (resp., nonnegative integer) powers λ i , u i ∈ O ( C , 0) and u i (0) = 1. Indeed, we can start with any C -basis f , . . . , f n in ℓ and normalize themso that each function has a monic leading term t λ i (1 + · · · ). If there are two equalpowers among the initial collection, λ i = λ k , then their difference (which cannotbe identically zero by linear independence) has the leading term proportional to t µ , λ i = λ k < µ ∈ Z . Repeating this procedure finitely many steps, one can alwaysachive the situation when λ i = λ k .Now we construct explicitly the Fuchsian operator H = P t j h j ( ǫ ) which wouldtransform the monomials t λ i , i = 1 , . . . , n , to the functions c i f i for suitable co-efficients c i ∈ C . Note that each monomial t λ i is an eigenfunction for any Euleroperator, in particular, h j ( ǫ ) t λ i = h j ( λ i ) t λ i , and therefore Ht λ i = ϕ i ( t ) t λ i , ϕ i ( t ) = X j > t j h j ( λ i ) . The equations Ht λ i = t λ i ( c i + c i t + c i t + · · · ) are thus transformed to the infinitenumber of interpolation problems, h ( λ i ) = c i , h j ( λ i ) = c ij , i = 1 , . . . , n, j = 1 , , . . . Such problems are always solvable by polynomials h j ∈ C [ ǫ ] of degree n − 1, andsince c i = h ( λ i ) = 0, we have gcd( h , p ) = 1. By a suitable (generic) choice ofthe constants c i = 0, one may guarantee that deg h = n − 1, that is, H is indeeda Fuchsian operator, as required for the F -equivalence. (cid:3) Note that in both cases the normal form is maximally resonant: all differencesbetween the roots of the Euler part are integer. OCAL WEYL EQUIVALENCE 17 Remark . This results shows to what extent the F -equivalence is more fine thanthe W -equivalence. Indeed, given the trivial monodromy, all operators having onlymeromorphic solutions, are W -equivalent to the same Euler operator t − n ∂ n = ǫ ( ǫ − · · · ( ǫ − n + 1). On the other hand, two different Euler operators are never F -equivalent: if gcd( p , h ) = 1, then the identity p h = q k in C [ ǫ ], the firstline from (25), implies that p = q and h = k .4.3. Resonant case: Homological equation and its solvability. If some ofthe roots of the Euler part p differ by a natural number, then the correspond-ing equations (26) may become unsolvable and in general transforming a resonantFuchsian operator L ∈ F to its Euler part E ( L ) ∈ C [ ǫ ] is impossible. However, onecan use F -equivalence to simplify Fuchsian operators.If a Fuchsian operator H = P t j h j ( ǫ ) conjugates L with another operator M = P t j q j ( ǫ ) ∈ F , then the left hand side of the identity p ( ǫ ) H = KL should bereplaced by M H = ( p + tq + t q + · · · )( h + th + t h + · · · )= p h + t ( q h + p [1] h ) + · · · + t j ( q j h + q j − h [1] + · · · + p h [ j ] j ) + · · · , (27)and, accordingly, the equations (26) should be replaced by the equations p [ j ] h j − p k j + q j h = v j , j = 1 , , . . . , (28)where, as before, v j ∈ C [ ǫ ] is a polynomial of degree n − q i , h i , k i with smaller indices 0 < i < j and p , . . . , p j .First, we use the fact that although some of the equations (28) may be non-solvable, they are always solvable for sufficiently large orders. Proposition 3. Let L = p + tp + · · · ∈ F be a Fuchsian operator and N themaximal natural difference between the roots of p = C n [ ǫ ] .Then L is F -equivalent to the polynomial operator M obtained by truncation ofthe Taylor series at the order N , M = N X j =0 t j p j ( ǫ ) = n X k =0 b k ( t ) ǫ n − k ∈ C [ t, ǫ ] with polynomial coefficients b k ∈ C [ t ] of degree deg t b k N , obtained by truncationof the analytic coefficients a k ∈ O ( C , of the initial operator L at the order N .Proof. First we find a pair of operators H , K ∈ W of order n − L with M in the form H = 1 + P j>N t j h j , K = 1 + P j>N t j k j , so that M H = K M . Substituting these expansions inthe equations (28), we see that all equations of order j = 0 , , . . . , N are satisfiedautomatically if we set q j = p j and 0 = h j = k j for all j = 1 , . . . , N .The operators H , K are (usually) non-Fuchsian, since 0 = ord h < ord H = n − 1. However, the operators H = H + L and K = K + M are Fuchsian,satisfy the identity M H = M ( H + L ) = K L + M L = ( K + M ) L = KL and thenondegeneracy condition gcd ( L, H ) = gcd( p , p + 1) = 1 is satisfied. (cid:3) Integrable normal form. The polynomial normal form established in Propo-sition 3, lacks any integrability properties. Yet using the same method, one canconstruct a Liouville integrable F -normal form for any Fuchsian operator. Proposition 4. A Fuchsian operator = p + tp + · · · ∈ F with the Eulerization p ∈ C n [ ǫ ] as above, is F -equivalent to an operator M ∈ F of the form M = ( ǫ − λ + r ) · · · ( ǫ − λ n + r n ) , r i = r i ( t ) ∈ C [ t ] , r i (0) = 0 . (29) In other words, M is a ( noncommutative ) product of polynomial operators of order . The degrees of the polynomials r i ( t ) are explicitly bounded, deg t r i ( t ) N , where N , as before, is the maximal order of resonance between roots of p .Remark . If λ i − = λ i is a multiple root of p , still the polynomials r i − and r i in general will be different. Lemma 3. Any analytic Fuchsian operator L ∈ F can be factorized as L = ( ǫ − λ + R ) · · · ( ǫ − λ n + R n ) , R i = R i ( t ) ∈ O ( C , , R i (0) = 0 , (30) with analytic ( rather than polynomial ) functions R , . . . , R n .Proof of the Lemma. Consider an eigenfunction u ( t ) of the monodromy operator,associated with the equation Lu = 0: the corresponding eigenvalue is nonzero,hence ∆ u = e πiλ u for some λ ∈ C . Then u = t λ v ( t ), where v is a meromorphicgerm, and modulo replacing λ by λ + j for some j ∈ Z , we may assume that v isholomorphic invertible, v ∈ O ( C , v (0) = 0. Applying ǫ to this function, we seethat ǫu = λt λ v + t λ ( ǫv ) = λu + Ru , R = ǫvv ∈ O ( C , u satisfies aFuchsian equation of the first order and L is divisible from the right by ǫ − λ − R ( t ).The quotient is again a Fuchsian operator of order n − 1, and the process can becontinued by induction. (cid:3) Proof of the Proposition 4. Consider the factorization (30) of the operator L asprovided by Lemma 3, and replace each analytic function R i by its polynomialtruncation r i to order N , so that ord t =0 ( r i − R i ) > N . The (polynomial) operator M thus obtained has the same N -jet with respect to t as the initial operator L . ByProposition 3, M is F -equivalent to L . (cid:3) The normal form established by Proposition 4 has an advantage of being Liou-ville integrable. Each linear equation of the first order is explicitly solvable “inquadratures”. In particular, the homogeneous equation Lu = 0 , L = ǫ − λ + r ( t ) , r ∈ C [ t ] , r (0) = 0 , has a 1-dimensional space of solutions u ( t ) = Ct λ exp ρ ( t ), where ρ ( t ) = − R r ( t ) t d t is a polynomial in t .To solve the nonhomogeneous equations, the method of variation of constantscan be used to produce a particular solution using operations of integration (com-putation of the primitive), exponentiation and the field operations in the field C ( t )of rational functions (the details are left to the reader). Iterating this computa-tion, one can find a general solution of the equation M u = 0 with a completelyreduced operator M as in (29): if M = L L · · · L n , ord L i = 1, then solution ofthe equation M u = 0 amounts to solving a chain of equations of order 1, L u = 0 , L u = u , . . . , L n u n = u n − , u = u n . (31) OCAL WEYL EQUIVALENCE 19 Corollary 1. Any Fuchsian operator is ˆ F -equivalent to a Liouville integrable op-erator. (cid:3) Non-Eulerizability of resonant Fuchsian equations. The explicit inte-grability of the factorized equations allows to show that the resonant Fuchsianequations, “as a rule”, are even not W -equivalent to their Euler part. Example 2. Consider the Fuchsian operator L = ( ǫ + t )( ǫ − 1) = E ( L ) + t ( ǫ − ∈ F , E ( L ) = ǫ ( ǫ − . The Euler part of L has simple integer roots, hence the trivial monodromy. On theother hand, the equation Lu = 0 can be explicitly solved. One solution, u ( t ) = t ,satisfying the equation ( ǫ − u = 0, is obvious. The equation ( ǫ + t ) v = 0 hassolution v ( t ) = e − t , and another solution u ( t ) of the linear non-homogeneousequation ( ǫ − u ( t ) = e − t , can be found by the method of variation of constants, u ( t ) = t Z e − t t − d t . The monodromy transformation of the pair of solutions( u , u ) is given by the non-identical matrix (cid:18) π i1 (cid:19) . This means that the fulloperator is even not W -equivalent to its Euler part.5. Minimal normal form The polynomial normal forms established in the preceding section are of ratherlimited interest: indeed, no attempt was made to modify the lower order terms ofthe Taylor expansion of the resonant Fuchsian operators.The system of equations (28) can be solved recursively with respect to h j , k j even in the resonant case gcd( p , p [ j ] ) = 1, provided that q j are chosen in a suitableway : the difference v j − q j h should belong to the image of the Sylvester map S j = S p ,p [ j ]0 , cf. with (23). This image consists of all polynomials of degree n − w j = gcd( p , p [ j ] ) ∈ C [ ǫ ]. In this section we describe possible choicesfor the terms q j .5.1. Abstract normal forms. Denote by P = C [ ǫ ] / h p i ≃ C n − [ ǫ ] the quotientalgebra: as a C -space it is n -dimensional and can be identified with the residuesmodulo p , polynomials of degree n − n roots λ , . . . , λ n ∈ C of p are simple, can be identified with the C -algebra of functions on n points Λ = { λ , . . . , λ n } ⊆ C : P = { ϕ : Λ → C } ≃ C × · · · × C : any such function canbe represented as the restriction of a polynomial h ∈ C n − [ ǫ ] of degree n − h | Λ = ϕ . The functions ϕ i equal to 1 at one point λ i ∈ Λ and vanishing at all otherpoints λ k = λ i , form a natural basis of P . Remark . In the general case where the roots λ i may have nontrivial multiplicities µ i ∈ N , p ( ǫ ) = Y i ( ǫ − λ i ) µ i , X i µ i = deg p = n, the quotient algebra P is naturally isomorphic to the direct sum of the local algebras J i ≃ C [ ǫ ] / ( ǫ − λ i ) µ i +1 of dimension µ i : each element of P = L i J i can be identified witha multijet , a collection of µ i -jets (Taylor polynomials of order µ i ) at the points λ i ∈ Λ ⊂ C . For any polynomial s ∈ C [ ǫ ] the multiplication by s is an endomorphism of thealgebra P . It is invertible (automorphism of P ) if and only if gcd( p , s ) = 1.The equations (28) induce the equations in the algebra P : p [ j ] h j + q j h = v j , j = 1 , , . . . (32)They can be re-written in the operator form as P j h j + H q j = v j (33)where P j , H are endomorphisms (self-maps) of P , induced by multiplication, P j : h p [ j ] h, H : q j h q j . (34)The endomorphisms commute between themselves and H is invertible. Definition 6. An affine normal form for the polynomial p is a family of subspaces V j ⊆ P (not necessarily subalgebras) such that V j is complementary to the imageof P j , P j P + V j = P j = 1 , , . . . . (35)The affine normal form is minimal , if dim V j = dim Ker P j .Without loss of generality we may assume that V j = 0 for all sufficiently largevalues of j (for minimal normal forms this condition is automatically satisfied).Note that the choice of an affine normal form is by no means unique: moreover,being a normal form is an open property (small perturbation of the subspaces V j does not violate the property (35).These definitions are tailored to make the following statement trivial. Theorem 6. Let { V j } be an abstract affine normal form for the polynomial p ∈ C n [ ǫ ] .Then any Fuchsian operator L = p ( ǫ ) + tp ( ǫ ) + · · · is F -equivalent to anoperator M = p + P Nj =1 t j q j ( ǫ ) with q j ∈ V j .Proof. By invertibility of H , we have H − P = P = H P . By (35), P j H − P + V j = P . Applying to both parts of the latter equality the operator H and using thecommutativity, we see that P j P + H V j = P , that is, each homological equation (33), regardless of the right hand side v j , admitsa solution h j ∈ P , q j ∈ V j . This solution generates (by definition of P ) a solution( h j , k j ) of (28). (cid:3) Minimal affine normal form. One possibility to chose an affine normal formis to stick to the polynomials of minimal degree modulo p . Denote by w j ∈ C [ ǫ ] thegreatest common divisor w j = gcd( p , p [ j ] ); this is a polynomial of degree ν j n − Proposition 5. Any Fuchsian operator L = p ( ǫ ) + tp ( ǫ ) + · · · is F -equivalentto a polynomial operator of the form M = p + P j t j q j ( ǫ ) with deg q j ν j − . Inparticular, q j = 0 for all nonresonant orders.Proof. It suffices to note that the subspace V j ≃ C ν j − [ ǫ ] ⊆ C n − [ ǫ ] ≃ P is natu-rally complementary to the image P j P which consists of all polynomials of degree n − 1, divisible by w j . This follows from the division with remainder by w j in P ≃ C n − [ ǫ ]. (cid:3) Note that the family of the subspaces V j ≃ C ν j − [ ǫ ] is a minimal normal form. OCAL WEYL EQUIVALENCE 21 Example 3. Assume that the operator L has a single resonance, i.e., only onepair of roots of p differs by an integer k . Then the operator L is F -equivalent to p ( ǫ ) + ct k , c ∈ C .5.3. Separation of resonances. A different strategy of choice of the subspaces { V j } constituting a normal form, is to reproduce the strategy which results in thePoincar´e–Dulac normal form for Fuchsian systems with diagonal residue matrix A ∈ Mat( n, C ). Recall that in this case instead of solving the homological equation(28), one has to solve matrix equations of the form [ A, H ] + jH = B j , where B j aregiven matrices from Mat( n, C ), cf. with [IY, Theorem 16.15]. The operator takinga matrix H into the twisted commutator as above, is diagonal in the natural basisof matrices having only one nonzero entry, and kernel of this operator is naturallycomplementary to its image.An analogous construction can be applied in the case of operators P j if thepolynomial p = E ( L ) has simple roots. Then multiplication by any polynomial,including w j , is diagonal, hence one can choose V j = Ker P j . The polynomials q j which appear in the corresponding normal form, will be vanishing at all roots of p /w j , hence divisible by the latter polynomial (recall that we consider polynomialsof degree deg p − λ i of p does not appearin any resonance, then all polynomials q j in the normal form will be divisible by ǫ − λ i , and therefore the operator M in the normal form established in Theorem 6will be divisible (from the right) by the first order Euler operators ǫ − λ i ∈ F .This claim gives a partial effective factorization of the normal form (29), whichallows to identify factors with r i = 0. In the next section we explain how one cangive an accurate description of the factors in (29) in general.5.4. Completely reducible minimal normal form. Occurrence of resonancesbetween the roots Λ = { λ , . . . , λ n } ⊂ C of the polynomial p ∈ C n [ ǫ ] allows tointroduce certain combinatorial structures. First, the (natural linear) order on Z induces a partial order on the roots: λ i > λ k ⇐⇒ λ i − λ k ∈ Z . Remark . If the Euler part has multiple roots, then the set Λ contains repeti-tions. To simplify the subsequent arguments, it is convenient to extend the partialorder to a full order as follows. The roots of p are subdivided in resonant groups in such a way that inside each group all roots have integer differences (and henceare comparable in the sense of the partial order). Different resonant groups canbe arranged between themselves in any way. The corresponding order is conve-niently represented by the enumeration of the set of all roots Λ = { λ i } in thenon-decreasing order. This will make Λ into an ordered set naturally isomorphic to { , , . . . , n } : multiple roots of p occupy consecutive positions in this list. We callthis order a natural order on Λ (it is not unique, since different resonant groupscan be transposed, but is convenient for the formulations).Second, for each order we can list all roots which produce this resonances of thisorder. Given a natural index j ∈ N , we define Λ j = { λ ∈ Λ : λ + j ∈ Λ } ⊂ Λ, j = 1 , , . . . . (36)This definition, unambiguous in the case where p has only simple roots, shouldbe modified as follows: if µ + j = κ are two roots in resonance and the multiplicitiesof µ, κ in Λ (the list which now may have repetitions) are m, k respectively, then µ enters Λ j with the multiplicity equal to min( m, k ), that is, with its multiplicity asthe root of the polynomial w j = gcd( p , p [ j ] ).Together with the sets Λ j ⊂ Λ it is convenient to consider also their fully or-dered counterparts (cf. with Remark 10), the sets of the corresponding indices I j ⊂ { , , . . . , n } . In the case where λ a root of p with multiplicity m > p [ j ] with multiplicity k < m , we include in I j the last k instances where λ enters Λ (out of the total m ).The dual description can be given by the sets J ( λ ), which for any root λ ∈ Λ consists of the natural numbers j ∈ N such that λ + j ∈ Λ . The case of multipleroots needs no special treatment.Recall that support (or the Newton diagram) of a polynomial r = P c k t k ∈ C [ t ]is the set of indices k ∈ N such that the corresponding coefficient c k is nonzero:supp r = { k : c k = 0 } ⊂ N . Theorem 7. Any Fuchsian operator is F -equivalent to a completely reducible op-erator of the form L = ( ǫ − λ + r ( t )) · · · ( ǫ − λ n + r n ( t )) ,r i ∈ C [ t ] , supp r i ⊆ J ( λ i ) , i = 1 , . . . , n. (37) In particular, deg r i max { j ∈ N : λ i + j ∈ Λ } . The rest of this section contains the proof of this theorem.5.5. Expansion of noncommutative products. From that moment we assumethat the roots λ i are labeled in a natural order, see Remark 10.Consider the operators E ij ∈ F of the form (37) in the case where only onepolynomial r i is different from zero and is itself a monomial of degree j : E ij = ( ǫ − λ ) · · · ( ǫ − λ i − )( ǫ − λ i + t j )( ǫ − λ i +1 ) · · · ( ǫ − λ n ) , i = 1 , . . . , n. After complete expansion of E ij we obtain E ij = p ( ǫ ) + t j p ij ( ǫ ) , p ij ∈ C [ ǫ ] , i = 1 , . . . , n,p ij = ( ǫ − λ + j ) · · · ( ǫ − λ i − + j )( ǫ − λ i +1 ) · · · ( ǫ − λ n ) . (38)In other words, p ij is obtained by shifting the argument by j in the first i − p , while keeping the last terms the same asin p . Accordingly, the roots of p ij are obtained by shifting the first i − Λ to the left by j units, removing the i th root from the list and keeping theremaining (larger) roots in place. Speaking informally, p ij has a gap on i th placein the (partially) ordered set Λ . Lemma 4. For each j > the polynomials p j , . . . , p nj ∈ C n − [ ǫ ] are linear inde-pendent over C .Proof. A vanishing C -linear combination of polynomials p j , . . . , p nj after divisionby p would result in a vanishing C -linear combination between the correspondingrational functions. However, this is impossible, since the first fraction will have apole at λ , and in a similar way p ij p will have either a pole at λ i , or (if λ i = λ i − wasa multiple root of p ) the order of the pole will increase compared with the previousfraction. Since the roots were ordered, this new poles appear at the points whereall previous ratios were holomorphic, which means that no linear combination canarise in the process. (cid:3) OCAL WEYL EQUIVALENCE 23 Corollary 2. For any j the polynomials p j , . . . , p nj span C n − [ ǫ ] .Proof. Since these polynomials are linear independent, they span an n -dimensional C -subspace in C n − [ ǫ ] which for the reasons of dimension must coincide with C n − [ ǫ ]. (cid:3) A minor modification of this argument proves a similar statement. Lemma 5. Let j ∈ N be a natural number and w j = gcd( p , p [ j ] ) . Then thepolynomials p ij for i ∈ I j are linear independent modulo w j . Note that the polynomials p ij and p i +1 ,j in general are different even if λ i = λ i +1 . Proof. Arguing as before, consider the rational fractions p ij w j . Since the roots of w j constitute only a proper subset of Λ , then not all of these fractions have either anew pole at λ i , or a pole of larger order. On the other hand, if λ i ∈ Λ j , then thismeans that one or more (depending on multiplicity) of the larger roots when shiftedby j will coincide with λ i and hence create a pole of w j of the corresponding order.In the case where λ i is a multiple root, we have to consider the fractions p ij w j for i ∈ I j This means that in the ordered subsequence p ij w j , i ∈ I j , the behavior of the poleswill be as before (either a new pole appears or the order of the previous pole isincreased). In both cases the linear dependence is impossible. (cid:3) Corollary 3. The linear span V j of polynomials { p ij : i ∈ I j } ⊂ C n − [ ǫ ] ≃ C [ ǫ ] mod p is a linear subspace transversal to the image of the operator P j from (35) , and hence these subspaces form a minimal abstract normal form in the senseof Definition 6.Proof. This follows from the linear independence above and the fact that the num-ber of these polynomials is equal to the codimension of the image (which consistsof polynomials of degree n − w j ). (cid:3) Proof of Theorem 7. Assume (by way of induction) that the a Fuchsianoperator L ∈ F is already shown to be F -equivalent to an operator L j − ∈ F whose ( j − L j − = ( ǫ − λ + r ,j − ) · · · ( ǫ − λ n + r n,j − ) + t j v j ( ǫ ) + · · · ,r i,j − ∈ C [ t ] , supp r i ⊆ J ( λ i ) ∩ [1 , j − , i = 1 , . . . , n. We will show that there exists an operator L j of the same form but with supp r i,j ∈ J ( λ i ) ∩ [1 , j ], which is F -equivalent to L j − . Indeed, adding monomials of order j to the polynomials r i,j − , r i,j = r i,j − + c i t j , c i = 0 ⇐⇒ j ∈ J ( λ i ) , i = 1 , . . . , n will affect only terms of order j and higher after the expansion: the (polynomial)coefficient v j will be replaced by v j + P c i p ij by definition (38) of the polynomials p ij . By a suitable choice of the coefficients c i for i ∈ I j , one can bring this sum intothe range of the homological operator P j , as follows from Corollary 3.For this choice the homological equation (28) will be solvable with respect to h j , k j by setting h = 1, q j = − P c i p ij . Continuing this way, we eventually reachthe values of j which exceed the maximal order N of possible resonances. Thecorresponding operator L N , by construction F -equivalent to the initial operator L , is F -equivalent to its product part Q ni =1 ( ǫ − λ i + r iN ( t )) with supp r iN ∈ J ( λ i )by Proposition 3. (cid:3) Remark . The same argument allows to construct an effective factorization ofany Fuchsian operator. Indeed, by Corollary 2, one can always construct a linearcombination P ni =1 c i p ij ∈ C [ ǫ ] which cancels the term v j . Proceeding this way, onecan construct the formal factorization L = Q ni =1 ( ǫ − λ i + ˆ r i ( t )), ˆ r i ∈ C [[ t ]]. Onecan show that in the Fuchsian case this factorization is always converging.5.7. Concluding remarks. The minimality of the normal form (37) does not im-ply that coefficients of the first order factors r , . . . , r n ∈ C [ t ] are F -invariant.Nevertheless, one can expect that for operators of sufficiently high order there willappear moduli (numeric invariants) of F -classification: for holomorphic gauge clas-sification of Fuchsian systems this was discovered by V. Kleptsyn and B. Rabinovich[KR]. 6. Convergence of the formal series Here we prove that the formal and analytic Fuchsian classifications for Fuchsianoperators coincide.More precisely, assume that two formal operators H, K ∈ ˆ F , H = P n − k =0 u k ( t ) ǫ k , K = P n − k =0 v k ( t ) ǫ k with formal coefficients u k , v k ∈ C [[ t ]] conjugate two Fuchsianoperators L = P nk =0 a k ( t ) ǫ k , M = P nk =0 b k ( t ) ǫ k with analytic coefficients a k , b k ∈ O ( C , a n (0) b n (0) = 0. Theorem 8. The formal series for the coefficients u k , v k ∈ C [[ t ]] necessarily con-verge, hence H, K ∈ F .Proof. One possibility of proving this result is to control explicitly the growth rate ofthe Taylor coefficients. However, a simple strategy is to use the fact that a (vector)formal Taylor series which solves a holomorphic Fuchsian system of equations, isnecessarily convergent.The conjugacy equation M H = KL takes the form of a noncommutative identity (cid:18) n X k =0 b k ( t ) ǫ k (cid:19)(cid:18) n − X k =0 u k ( t ) ǫ k (cid:19) = (cid:18) n − X k =0 v k ( t ) ǫ k (cid:19)(cid:18) n X k =0 a k ( t ) ǫ k (cid:19) . (39)We claim that this identity implies that the coefficients u k ( t ) of the operator H ,after passing to a companion form (20), together satisfy a Fuchsian system of linearordinary differential equation. This follows from the direct inspection of the waythe highest order derivatives of u k enter the expressions in (39).The identity (39), using the commutation relationship in the Weyl algebra ǫf = f ǫ + g, g = ǫ ( f ) ∈ O ( C , 0) the Euler derivative of f , (40)can be rewritten as equality of two differential operators n − X j =0 l j ǫ j = n − X j =0 r j ǫ j of order 2 n − 1, implying the identical coincidence of their coefficients, l j = r j . Thuswe have a system of 2 n linear ordinary differential equations of order n involving2 n unknown functions u k , v k and their derivatives. We will show that this systemcan be reduced to a Fuchsian system of n differential equations of order 1. OCAL WEYL EQUIVALENCE 25 One can instantly verify that these equations have the following structure.(1) All expressions for l j are linear with respect to the functions u k = u k, and their iterated Euler derivatives u ki = ǫu k,i − of orders 1 i n withholomorphic coefficients.(2) All expressions for r j are linear with respect to the functions v k with holo-morphic coefficients.It is rather easy to control the coefficients with which the highest order derivatives u kn and v k enter these equations.The coefficients with which the variables v k enter the linear forms r j , form an“upper triangular” n × n -matrix with the same invertible diagonal entry a n : thehighest number forms r n − , . . . , r n − k depend only on the variables v n − , . . . , v n − k ,and the variable v n − k enters with the coefficient a n for all k = 1 , . . . , n .The coefficients with which the highest order derivatives u kn enter the linearforms l j , are zero for l n − , . . . , l n and form a “diagonal” n × n -matrix with thesame invertible diagonal entry b n in the forms l n − , . . . , l . Indeed, the formu-las (40) imply that a highest order derivative u kn can appear only after iteratedtransposition with the term b n ǫ n and only before the powers of the type ǫ j − n .Together these two observations imply that the system of the linear equations l j = r j , j = 2 n − , . . . , , u kn , v k ,in particular, u kn ( t ) = n − X i =0 n − X j =0 c knij ( t ) u ij ( t ) , c knij ∈ O ( C , , k = 0 , . . . , n − v k ).This system of n linear ordinary differential equations of order n with respectto the functions u k = u k ( t ) is explicitly resolved with respect to the highest orderderivatives, hence is a Fuchsian system of n first order equations in exactly thesame way as in (20).It remains only to refer to the well-known fact: any formal solution of a Fuchsiansystem converges, see [IY, Lemma 16.17 and Theorem 16.16]. 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