An algorithm for complete enumeration of all

An algorithm for complete enumeration of all factorizations of a linear ordinary dierential operator S.P.Tsarev Dept. Math. Krasnoyarsk State Pedagogical University, Lebedevoi, 89, 660049, Krasnoyarsk, Russia e-mail: [email protected] [email protected]

Abstract We discuss the problem of exhaustive enumeration of all possible factorizations for a given linear ordinary dierential operator. A theoretical investigation of topological and combinatorial obstacles to uniform description of factors which include arbitrary parameters and a complete algorithm for enumeration of all (discrete and parameterized) factorizations are given.

known procedures of construction of hyperexponential solutions [8, 20] of LODO obviously fail. Hence if the right factor Lr depends on parameters (they may be retrieved by the methods of [23]) we get the quotient L1 


 Lr;1 = L  L;r 1 depending on the same parameters and shall give the parameters some de nite values to proceed further obtaining only several (certainly not all in the general case) factorizations. Only in special cases the methods of [8, 10, 20] give all the possible factorizations for example if parameters do not appear or if L is completely reducible (see below sections 2, 3). An alternative approach proposed in [13] suers from the same problem. Fortunately according to results by Loewy [15, 16] all possible factorizations of a given (non-parametric) operator L have the same number of factors in dierent expansions L = L1 


 Lk = L1 


 Lr into irreducible factors and the factors Ls, Lp are pairwise "similar". (Hereafter we always suppose the order of factors to be greater than 0: ord(Li ) > 0, ord(Lj ) > 0). Still the problem of description of all possible factorizations was unsolved in the case of factors with parameters. In this paper we give the proper theoretical background (section 3, 4) for such exhaustive enumeration of factorizations using the Loewy-Ore [15, 16, 17, 18, 19] formal theory of LODO (section 2) and describe an algorithm for such enumeration (section 5). For simplicity we discuss here only the case of dierential operators, a generalization for the case of a general Ore ring (including dierence and q-dierence operators, see [3, 10, 9]) is straightforward. Many of the results of this paper may be easier proved within the framework of the Picard-Vessiot theory. But we follow the formal approach of Loewy and Ore in order to facilitate the aforementioned generalizations.

1 Introduction Factorization of linear ordinary dierential operators (LODO)

L = f0 (x)Dn + f1 (x)Dn;1 + : : : + fn (x); D = d=dx; (1) fs(x) belong to some dierential eld K, is a useful tool for computing a closed form solution of the corresponding LODE Ly = 0 as well as determining its Galois group [21, 22]. For simplicity and without loss of generality we suppose that operators are reduced (i.e. f0 (x)  1) un-

less otherwise stated explicitly. The known algorithms of factorization can provide a factorization of a LODO (over K = Q(x)). But as the well-known example D2 = D  D = (D +1=(x ; c))  (D ; 1=(x ; c)) shows some LODO may have essentially dierent factorizations with factors depending on some arbitrary parameters. The algorithms of [8, 10, 20] are based essentially on stepwise splitting of the right factors using the old method of Beke [5] which reduces the problem of construction of a right factor Lr = Dm + fr1 (x)Dm;1 +


+ frm (x) of order m to nding "hyperexponential" solutions of the so-called associated equation L(m) w = 0, i.e. solutions which have the property fr1 = ;Dw=w 2 K. This approach fails in the case when the coecients of L (and consequently of L(m) ) depend on parameters since the

2 Loewy-Ore formal theory of LODO Here we sketch the basics of this theory [15, 16, 17, 18, 19] necessary for sections 3, 4, 5. For any two LODO L and M from the existence of the Euclid algorithm one can determine their right greatest common divisor rGCD (L; M ) = G, i.e. L = L1  G, M = M1  G (the order of G is maximal) and their right least common multiple rLCM (L; M ) = K , i.e. K = M  L = L  M (the order of K is minimal) as well as

Proc. ISSAC'96, 24{26 July 1996, Zurich, p. 226{231. 1

transforming P1 


 Pk;1 ;! Q1  : : :  Qs  Pfp . Take the rst stage of it, when an operator in P1 


 Pk;1 , tagged by the same index as Pfp , is consecutively interchanged to the right of P1 


 Pk;1 (and becomes Pfp ). After this interchange Pfp and Pk in L. We have made the rst stage of "bubble sort" interchanges transforming L = P1 


 Pk;1  Pk ;! Qe1  : : :  Qek;2  Pek  P p . By induction we have another series of "bubble sort" interchanges transforming Qe1  : : :  Qek;2  Pek ;! P 1 


 P p;1 so we get the complete series of "bubble sort" interchanges P1 


 Pk;1  Pk ;!

their "left" analogues lGCD and lLCM. Operator equations X  L + Y  M = B; L  Z + M  T = C (2) with unknown operators X , Y , Z , T are solvable i respectively rGCD (L; M ) divides B on the right and lGCD (L; M ) divides C on the left. We say that operator L is (right) transformed into L1 by an operator (not necessary reduced) B B , and write L ;! L1 , if rGCD (L; B ) = 1 and K = rLCM (L; B ) = L1  B = B1  L. In this case any solution of Ly = 0 is mapped by B into a solution By of L1 y = 0. Using (2) one may nd with rational algebraic operations B1 an operator B1 such that L1 ;! L, B1  B = 1(modL). Operators L, L1 will be also called similar or of the same kind (in the given dierential eld K ). So for similar operators the problem of solution of the corresponding LODE's Ly = 0, L1 y = 0 are equivalent. Below we will also use the notion of left-hand transformation of L by B into L1 : K = lLCM (L; B ) = B  L1 = L  B1 . Obviously left- and right-hand transformations are connected via the adjoint operation. Also ([19]) two operators are left-hand similar i they are right-hand similar. A (reduced) LODO is called prime or irreducible (in the given dierential eld K) if it has no nontrivial factors aside from itself and 1. Every LODO similar to a prime LODO is also prime. Two (prime for simplicity) LODO P and Q are called interchangeable in the product P  Q and this product will be called interchangeable as well if P  Q = Q1  P1, Q1 6= P , P1 6= Q. In this case P Q is similar to P1 , Q is similar to Q1 and P1 ;! P. Theorem 1 Any two dierent decompositions of a given LODO L into products of prime LODO L = P1 


 Pk = P 1 


 P p have the same number of factors (k = p) and the factors are similar in pairs (in some transposed order). One decomposition may be obtained from the other through interchanges of contiguous factors (in the pairs Pi  Pi+1 ). If we will tag the prime factors by their numbers in some xed "initial" nonparametric decomposition L = P1 


 Pk and transpose tags during interchanges of factors then (compare our proof below with the proofs in [15, 16, 19]) the prime factors in any other decomposition L = A  B 


 Z may be tagged A = P i1 , B = P i2 , : : : , Z = P ik , in such a way that interchanges of factors transforming the initial decomposition into the second L = A 


 Z  P i1 


 P ik imitate the "bubble sort" method: rst we consecutively interchange Pik in L = P1 


 Pk to the right until it reaches the right end of the decomposition, then Pik;1 (more precisely the operator tagged with the number ik;1 since if ik;1 > ik , Pik;1 will change after interchanging places with Pik ) will be moved to the last but one place, etc. Proof. Theorem 1 is proved inductively. If ord(L) = 1 then the statement of Theorem 1 as well as the "bubble sort" are trivial. Provided that we have proved this for operators L of order  n take an operator L = P1 


 Pk = P 1 


P p of order n+1. If Pk and P p coincide we can delete them reducing the order of L, which gives the desired result. If Pk and P p do not coincide take R = rLCM (Pk ; P p ) = Pfp  Pk = Pek  P p . Then R divides L: L = Q  R and we have L = P1 


 Pk;1  Pk = Q  R = Q1  : : :  Qs  Pfp  Pk = Q1  : : :  Qs  Pek  P p = P 1 


 P p;1  P p where Qi give a decomposition of Q into irreducible factors. Since the orders of P1 


 Pk;1 = Q1  : : :  Qs  Pfp and Q1  : : :  Qs  Pek = P 1 


 P p;1 are  n, we have s = k ; 2, p = k, and we can nd a "bubble sort" series of transpositions

Qe1  : : :  Qek;2  f Pp  Pk ;! Qe1  : : :  Qek;2  Pek  P p ;! P 1 


 P p;1  P p which completes the proof. Thus we need at most p(p ; 1)=2 interchanges. Parame-

ters in the second decomposition may appear in the process of interchanging of prime neighbors of the same kind (see below), one parameter may appear after every such an interchange if the sub eld k of constants of the dierential eld K is algebraically closed. Corollary 1 If the sub eld k of constants of the dierential eld K is algebraically closed the total number of parameters in an arbitrary prime decomposition does not exceed p(p ; 1)=2. Here p is the number of prime factors. We will say that the decompositions L = A 


 Z  Pi1 


Pip are roughly enumerable by their tag sets (i1 ; : : : ; ip ) 2 Sp , i.e. by the elements of the symmetric group Sp . Certainly not all elements of Sp may correspond to some factorizations of a given operator L due to impossibility of interchanges of some factors. If P and Q are interchangeable then P  Q = Q1  P1 = rLCM (Q; P1 ) = lLCM (P;Q1 ) and vice versa. It is useful to study decompositions of operators which are LCM of prime operators. Such operators will be called completely reducible. Loewy's results and methods [15, 16] give us the tool for exhaustive enumeration of all possible factorizations of completely reducible LODO. If L = rLCM (P1; : : : ; Pp) where Ps are prime, ord(L) = ord(P1 ) +


+ ord(Pp ), then L = P 1 


 P p;1  Pp, P s are similar to Ps and it is possible to nd Pes such that L = lLCM (Pe1 ; : : : ; Pep ). Theorem 2 If L = rLCM (P1 ; : : : ; Pp ) has an irreducible right factor Lr which does not coincide with P1; : : : ; Pp, then among Ps one can nd at least two operators similar to Lr . Corollary 2 If all P1 ; : : : ; Pk are of dierent kinds then L has only these as right factors (i.e. there are no parameters in the right factors). Theorem 3 The necessary and sucient conditions that a LODO be completely reducible is that two arbitrary factors in any arbitrary decomposition be interchangeable. Corollary 3 If all factors Ps in L = rLCM (P1; : : : ; Pk ) are of dierent kinds then there exist exactly k! dierent factorizations of L. All of them are nonparametric. Now let us suppose that some Ps are similar (but dierent), namely P1 , : : : , Pm , m  k are similar to some L0 , and Pk , k > m are not similar to L0 . B1 Bm P . Then the Theorem 4 Let L ;! P , : : : , L ;! m 0

1

0

most general right factor P of L = rLCM (P1 ; : : : ; Pk ) similar to L0 is the result of transformation of L0 by B = B1 E1 + B2 E2 +


+ Bm Em where Es are "Eigenwurzeln" Es L0 , ord(Es ) < ord(L0 ). of L0 , i.e. such that L0 ;!

2

3

Parameterization of inner factors of completely reducible LODO For simplicity we will assume hereafter that the sub eld k of constants of the dierential eld K is algebraically closed. First let us suppose that V = rLCM (P1 ; : : : ; Pm ), such that ord(L) = ord(P1 ) +


+ ord(Pm ), and all Ps are simBi ilar to some L0 , L0 ;! Pi . Then for any factorization V = Pe1 


 Pem the products Vi = Pei 


 Pem are also comBs pletely reducible Pm k([19]), Vi = rLCM (P 1 ;


; P m ), L0 ;! P s , B s = k=1 es Bk . According to the results of section 2, Vi is completely determined by the (m ; i + 1)-dimensional subspace1 Fm;i+1m  km spanned by ~ei = (e1i ; : : : ; em i ), : : : , ~em = (em ; : : : ; em ), and do not depend on the choice of the particular basis f~esg in Fm;i+1 . So we have the following Theorem 5 Every factorization V = Pe1 


 Pem of a completely reducible operator V = rLCM (P1 ; : : : ; Pm ) with all similar Ps bijectively corresponds to a complete ag in km , i.e.m an m-tuple of vector subspaces F1  F2 


 Fm = k , dim Fi = i. Topologically the ag manifold (the set of all complete

ags) is a compact smooth algebraic variety of dimension m(m ; 1)=2. But since it is not isomorphic to km(m;1)=2 (and is much more complicated than the projective space) it is again topologically impossible to give a complete uniform parameterization of all factorizations as expressions V = Pe1 


 PeMm with Pes depending on some parameters (t1 ; : : : ; tM ) 2 k , M = m(m ; 1)=2. One of the possible enumerations of all parameterized factorizations may be obtained from the known decomposition of the ag manifold into Schubert cells (see [6]) | nite number of nonintersecting subsets isomorphic to kj , j  M . In this case the dependence of Pes on parameters in each subcase of this enumeration can be made rational: in fact the Schubert cell decomposition gives us a rational parameterization of the basis vectors ~e1 ;~e2 ; : : : ;~em , Fm;i+1 being spanned by ~ei ; : : : ;~em . Thus a rational parameterization of B i = P eki Bk as well as Pei and P i is obtained from Bi with rational operations. The case of a more general completely reducible L = rLCM (P1 ; : : : ; PM ) where P1 , : : : , Pm1 are similar to L1 , Pm1 +1 ; : : : ; Pm2 are similar to L2 , etc., can be easily studied along the same line: all (parameterized in general) factorizations of L correspond to cartesian ags F1  F2 


 FM = kM = km1  km2 ;m1 


kM ;ms;1 , dim Fi = i, each Fi being a direct product Fi = Fi;1 


 Fi;si , Fi;j  k(mj ;mj;1 ) , ms = M , m0 = 0. Each kmj ;mj;1 corresponds to factors similar to Lj (i.e. to Pmj;1 +1 , : : : , Bj;s Pmj , Lj ;! Ps ) and Fi+1 is spanned by Fi and a vector : : : ; emi+1j ;mj;1 ) corresponding to the right fac~ei+1 = (e1i+1 ; P

The following Propositions were not explicitly stated in [15, 16, 17, 18, 19] though they are easily proved following the guidelines of these papers. Proposition 1 If the sub eld k of constants of the dierential eld K is algebraically closed then Ei = i Id, i 2 k. Proof. Since E = Ei transforms the space V of solutions of L0 y = 0 (in a proper dierential extension K) into itself we have a nite dimensional linear operator on V . Let  2 k be its eigenvalue. Then (E ; Id)y = 0 has at least one common solution with L0 y = 0 so rGCD (L0 ; E ; Id) is nontrivial which is impossible in view of irreducibility of L0 and ord(E ) < ord(L0 ), so E ; Id = 0, E = Id. Proposition 2 Two factors P and P obtained from L0 by B = 1 B1 +


+ m Bm and B = 1 B1 +


+ m Bm respectively, s ; s 2 k, coincide i s  c
s , c 6= 0. (The sub eld k of constants of the dierential eld K need not to be algebraically closed.) Proof. If s  c
s , c 6= 0 then B and B transform the space of solutions of L0 y = 0 into the same space so the transformed operators coincide. Let now s and s be nonproportional. Let us complete the vectors ~e1 = (1 ; : : : ; m ), ~e2 = (1 ; : : : ; m ) to the basis (~e1 ; : : : ;~em ) in km , ~es = Bs (e1s ; : : : ; em s ), B s = e1s B1 +


+ em s Bm , L0 ;! P s , i > 2. Form L1 = rLCM (P; P; P 3 ; : : : ; P m ). L1 divides L since the spaces of solutions of P s y = 0 are contained in the space of solutions of L0 y = 0. It is possible to express back Bs as linear combinations of B , B and B p , so L0 divides L1 as well. Thus L0 = L1 so P and P are dierent. The most general right factor of L = rLCM (P1 ; : : : ; Pk ) similar to L0 is parameterized by the points of the (m ; 1)-dimensional projective space kP m;1 . The procedure of B P involves only computation of the transformation L0 ;! rational operations so we see that for B = 1 B1 +


+ m Bm the coecients of the general right factor P similar to L0 depend rationally and homogeneously on 1 ; : : : ; m . The actual number of parameters due to this homogeneity is m ; 1, but if we will eliminate one of s (say 1 ) setting it to 1, we will miss operators with 1 = 0. This gives the rst (quite trivial for the right factors) topological obstruction to uniform complete parameterization for all possible P . We shall either leave one additional homogeneous parameter in the coecients of P or split the general parameterized case into subcases: a) 1 6= 0, b) 1 = 0, 2 6= 0, : : : , m) 1 = 2 =


= m;1 = 0, m = 21. This problem was already visible for our rst example D = D  D = (D + 1=(x ; c))  D (D2 ; 1=(x ; c)). As we see from the example D2 + 1 ;! D + 1 the statement of the Proposition 1 is not true for non-algebraically closed constant sub elds k. In this case the number of parameters in the right factor may exceed m ; 1. The study of the right factors of a general (not completely reducible) operator is easily reduced to that for completely reducible operators: let us take the set (possibly in nite) of all prime right factors of L and let V be its rLCM . Since ord(V )  ord(L), V is generated by a nite number of prime right factors, let k be their minimal number. Then V = rLCM (P1 ; : : : ; Pk ) and all right factors of L are right factors of V so the most general right factor of L similar to some L0 homogeneously depends on m parameters, m being the number of prime factors in V = rLCM (P1 ; : : : ; Pk ) similar to L0 .

tor P i+1 , Lj

esi+1 Bj;s s ;!

P i+1 ).

4 Interchangeability of parameterized factors in arbitrary operators Let us now turn to the general case of a not completely reducible LODO. All its factorizations are obtained from a xed initial one through interchanges of products of factors Pi  Pi+1 . If all Pi are of dierent kinds the problem is reduced to checking interchangeability of two given nonparametric prime operators: Pi  Pi+1 = P i+1  P i = K . 3

Corollary 4 Assume that in the product L  L  L we

This is easy to do checking with any of factoring algorithms [8, 10, 20] whether K has a right factor not equal to Pi+1 . Alternatively one may try to nd the transformation opB P ), ord(B ) < ord(P ) = n , in the erator B (in Pi ;! i i i Pi+1 B following way: since Pi ;! P i and P i ;! Pi , we have +1 B Pi Pi;! Pi so Pi+1  B = 
Id(mod Pi), for some nonzero B P and (due to the fact that the composition of Pi ;! i Pi+1 P i ;! Pi gives a nonzero mapping of the solution space of Pi into itself) constant . So we can change B ! B= and get Pi+1  B = Id(mod Pi ), that is Pi+1  B + C  Pi = Id: (3) with unknown operators B = b0 (x)Dni;1 +


+ bni ;1 (x), n ; 1 i +1 C = c0 (x)D +


+ cni+1 ;1 (x). This equation gives a system of LODO's for bi (x), cj (x). Using the known algorithms [1, 2, 3, 4, 7] one can look for its solutions (in K). If it exists it is unique: (Pi+1 B1 +C1 Pi );(Pi+1 B2 +C2 Pi ) = 0, Pi+1  (B1 ; B2 ) = (C2 ; C1 )  Pi which is only possible for B1 ; B2 = 0, C2 ; C1 = 0 due to irreducibility and non-similarity of Pi+1 , Pi and ord(Bs ) < ord(Pi ), ord(Cs ) < ord(Pi+1 ), so it is impossible (Theorem 1, section 2) to have two such decompositions K = Pi+1  (B1 ; B2 ) = (C2 ; C1 )  Pi . Some further results on the properties of the equation (3) see in [12, 18]. The main diculty of enumeration of all factorizations certainly lies in the case of parametric factors Pi  Pi+1 . Parameters appear in the process of (previous) interchanges of factors of the same kind: if Pi  Pi+1 = P i+1  P i = K = rLCM (Pi+1 ; P i ) and Pi , Pi+1 are similar, then the most general P i will depend on an additional parameter if this parameter was not already included in Pi . On the other hand if Pi, Pi+1 already include some parameters it may occur that they are interchangeable only for some values of the parameters. Remark. A good source of examples of interchangeable and noninterchangeable operators is given by rst order x)Id Pi LODO: Pi = D + pi (x). It is easy to show that Pi f (;! i P i = D +pi (x), pi (x) = pi (x);D log f , and Pi Pi+1 (with the operators of the same or dierent kind) is interchangeable Pi  Pi+1 = P i+1  P i i pi (x) = pi+1 (x) ; D log R ; R, R = pi+1 (x) ; pi (x) 2 K. In particular if Pi, Pi+1 are similar,Rpi (x) = pi+1 (x) ; D log Q, then they are interchangeable i Q dx 2 K. For example it is easy to construct operators L1 , L2 , L3 , such that in the product L1  L2  L3 operator L2 is interchangeable with both L1 , L3 : L1  L2  L3 = L1  Le3  Le2 = L2  L1  L3 but neither L1  L3 nor L1  Le3 are interchangeable:   L1 = D +2=x ,  take for example 21 x;1 1 L2 = L3 = D; L1  L2 = D + x(1 x;1)  D ; x(1 x;1) , L2  L3 = (D + (x;12 ) )  (D ; (x;12 ) ).

1

Corollary 5 If P

2

1

3

1 B1 +


+m Bm

;! P 1 then the parameterized P 1 is interchangeable with some xed operator P2 in the product P 1  P2 for the set of the values of the parameters (1 ; : : : ; m ) 2 km forming a linear subspace of km . 1




+m Bm P , P 1 B1 +;!


+m Bm P , P ;! C If P1 1 B1 +;! 3 1 4 1 P5 , C = (1 B1 +


+ m Bm ) + (1 B1 +


+ m Bm ) (we x now the i , i ) and P3  P2 , P4  P2 are interchangeable, we will prove that also P5  P2 is interchangeable. Take L = rLCM (P3 ; P4 )  P2 = P 3  P4  P2 = P 4  P3  P2 = P 3  Pe2  Pe4 = P 4  Pb2  Pe3 . So P 3 is interchangeable in succession with P4 , P2 in P 3  P4  P2 and as we have shown above L = rLCM (P2 ; Pe4; Pe3 ). So L is completely reducible which implies that P5 is interchangeable with P2 (Theorem 3, section 2) since L has P5 as its factor in some decomposition of rLCM (P3; P4 ) = P6  P5 (see section 2), which nishes the proof. Proposition 4 Factors in all decompositions L = P i1 


 P ik corresponding to a given element (i1 ; : : : ; ik ) 2 Sk in our rough enumeration (we recall that the initial non-parametric factorization L = P1 


 Pk corresponds to (1; : : : ; k)) Bis are built by transformations Pis ;! P is with coecients of B is including constants s;k parameterized by some ane algebraic varieties of dimension  k(k ; 1)=2, s;k appear in B is rationally. Proof.

Starting from the initial xed L = P1 


 Pk we obtain the rst parameters through interchanges of similar factors, the conclusion of the Proposition holds in this case. Suppose now that we have (after several interchanges of factors) two operators Pei  Pej satisfying the formulated Bj e Bi e Pi, Pj ;! Pj , B p are parameterproperty. So Pi ;! ized (with some algebraic constraints for the parameters), ord(Bp ) < np , np = ord(Pp). Since we can use the left-hand transformation instead of the usual right one (section 2) we can nd some B j (obtained from B j with rational operations) such that B j  Pej = Pj  C , C is parameterized as Pj;1  Bj  Pei . Looking for an interchange Pei  Pej = Pbj  Pbi for

Proof.

L3 e e Proposition 3 If L  L ;! L  L and all Li are irre1

2

can interchange L1 with L2 and L3 in succession and let the result be L1  L2  L3 = Le2  Le3  Le1 . Then the interchangeability of L2  L3 is equivalent to the interchangeability of Le2  Le3 . e2  L1  L3 = Le2  Le3  Le1 . In Proof. We have L1  L2  L3 = L e L1 e e order to show that L2  L3 ;! L2  L3 we have to prove that Q = rGCD (L2  L3 ; Le1 ) = 1. Let on the contrary ord(Q) > 0. Since Le3  Le1 is interchangeable, rGCD (L3 ; Le1 ) = 1 and ([19, Theorem 11, p. 489]) since the product L2  L3 is L3 divisile by Q, L2 is divisible by some Q, (Q ;! Q), Q being the right factor of L1 , which contradicts to the fact rGCD (L2 ; L1 ) = 1 (interchangeability of L1  L2 = Le2  L1 ).

2

ducible then L1  L2 is interchangeable i Le1  Le2 is interchangeable.

If L1  L2 is interchangeable then K = rLCM (L1  L2 ; L3 ) = L3  L1  L2 = L3  L2  L1 = Le1  Le2  L3 so K = rLCM (L2 ; L3 ; L1 ) and according to Theorem 3 (section 2) Proof.

Pe

X b j e at least some values of parameters we set Pi ;! Pi ;! Pi , so for some Y , Pej  X = B i + Y  Pi , ord(X )  ni ; 1, ord(Y )  nj ; 1. Multiplying on the left by B j we get

all factors in all decompositions are interchangeable. The converse is proved analogously.

4

B j  Pej  X = Pj  C  X = B j B i + B j  Y  Pi. Introducing new unknown operators X = C  X , Y = ;Bj  Y , ord(X )  ni + nj ; 2, ord(Y )  2nj ; 2 we have

decomposition of L into "greatest completely reducible factors" Vs ([16]). Checking interchangeability of prime factors on the "border" between Vi and Vi+1 using the technique of section 4 and only "bubble sort" transpositions we also get the complete enumeration of all the possible factorizations. If necessary (and possible) one may split the algebraic varieties restricting the values of parameters in the factors into cells parameterized by non-restricted parameters obtaining a " ner" (in comparison with the "rough enumeration") enumeration. Possible further eectivisations of factorization algorithms [8, 10, 20] crucial in the algorithm may be obtained using the techniques of [23].

Pj  X + Y  Pi = B j  B i (4) with xed Pi , Pj , i.e. a system of LODO (for the unknown coecients of X , Y ) with xed coecients and parame-

terized right hand sides. Using the standard technique and improvements of [1, 2, 3, 4] we can solve (4) providing additional algebraic constraints necessary and sucient for its solvability. Provided some parameterized solutions of (4) Y 0 and X 0 are found, the general form of the solution of (4) is X = X 0 + X1  Pi , Y = Y 0 ; Pj  X1 for the case of Pi , Pj of dierent kinds and X = X 0 + X1  Pi ; T , T P, Y = Y 0 ; Pj  X1 + W for similar Pi , Pj , Pi ;! j Pj  T = W  Pi (see [18] and section 2 for the proof). Since Y = ;B j  Y applying the adjoint operation we obtain the following system of linear algebraic (not dierential ) equations for the coecients of X1 , Y  (set  = 0 for non-similar Pi , Pj ):   X1  Pj ; Y   B j = Y 0 + W   which is always possible to solve due to rGCD (Pj ; B j ) = lGCD (Pj ; B j ) = 1 (see section 2). The solution is unique in virtue of the limitations on the orders of X1 ,Y . Substituting (parameterized) X1 into X = X C = X 0 +Pi X1 ;T  and solving it with respect to X (we have again a linear algebraic not dierential system) we again obtain algebraic  constraints for the parameters in X 0 , X1 and these constraints as well as the constraints originating from the solution of (4) give the necessary and sucient conditions for interchangeability of Pei  Pej as well as some parameterization of the interchanged factors Pbj  Pbi. The additional  may be retained or disappear in the process of solution of the above equations.

6 Conclusions Since it was possible to parameterize all factorizations of completely reducible operators rationally (using for example Schubert cells) a natural conjecture arises:

Conjecture. Algebraic relations binding the parameters in all factorizations obtained by the algorithm of section 5 have rational complete solutions splitting the respective algebraic varieties into non-intersecting cells with rational paramiterizations. Also the theoretical study of the general structure of possible interchangeable pairs has to be continued. A large amount of information on a special case of interchangeable pairs (commuting LODO) may be found in the theory of nonlinear integrable partial dierential equations ("soliton theory", see for example [14]), an integration of this theory and the Loewy-Ore formal theory of LODO may be of great advantage for both of them. For example, a trivial fact from the theory of commuting LODO L1  L2 = L2  L1 states that L1 and L2 necessarily satisfy a polynomial identity P (L1 ; L2 ) = 0, P being some constant coecient polynomial in two variables, which is not obvious from the point of view of the Loewy-Ore formal theory of LODO. Another possibility is (though now very far from being developed even in the basics) the possibility of extension of the formal theory to the case of partial linear dierential operators. Now we have some rst interesting results about commuting partial linear dierential operators (see, for example, [11]). Also a challenging connection with the classical Riquier-Janet-Thomas theory of compatibility of overdetermined systems of partial linear dierential equations (at the least in the very important for the applications in solitonics case of systems of linear equations) may be foreseen in some distant future. Today this theory is transformed and developed along several lines, for example as the celebrated D-module theory. As the practice shows, all these theoretical results are very inecient. It would be really interesting to develop the theory of factorization for partial linear dierential operators to reduce the complexity of systems appearing during the transformation of an overdetermined linear system to the "passive" (also called "involutive") form.

5 The algorithm for complete enumeration of all possible factorizations of LODO Using the technique of [8, 10, 20] one may nd one xed prime factorization L = P1 


 Pk . Given an element I = (i1 ;


; ik ) 2 Sk ("rough enumeration") we can build the unique series of "bubble sort" transpositions of (1;


; k) into I . Using the technique of section 4 we can check all the needed transposition for real interchangeability of factors and get the nal required tagged factorization L = P i1 


 Pik rationally parameterized by some constants i;j bound by algebraic equations. Remark. Note that we can not guarantee that all (parameterized) decompositions constructed by this algorithm will produce dierent factorizations for dierent elements I = (i1 ;


; ik ) 2 Sk unlike the case of completely reducible operators (section 3). For completely reducible operators we get just the Schubert cell decomposition into ane subspaces ks . Among possible ways of eectivization we may try the following: using the existing algorithms for splitting of the right factors we split o the (parameterized) rLCM of these factors, the found factor V1 will not include parameters (they are "hidden" inside this greatest rLCM, see section 2 and [16]). Splitting o V1 , we repeat the procedure obtaining

7 Acknowledgements The author wishes to express his special gratitude to the referees whose remarks helped to improve the exposition of the results of the paper. The research described in this article was made during my visit to the University of Paderborn (Germany) and was supported in part by DAAD short-term academic visitor's program and the Russian Foundation for Basic Research grant No 95{011{168. The author enjoys the 5

[16] Loewy, A. U ber vollstandig reduzible lineare homogene Dierentialgleichungen. Math. Annalen 62 (1906), 89{117. [17] Ore, O. Formale Theorie der linearen Dierentialgleichungen (Erster Teil). J. fur die reine und angewandte Mathematik 167 (1932), 221{234. [18] Ore, O. Formale Theorie der linearen Dierentialgleichungen (Zweiter Teil). J. fur die reine und angewandte Mathematik 168 (1932), 233{252. [19] Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933), 480{508. [20] Schwarz, F. Ecient factorization of liner ODE's. SIGSAM Bulletin 28, 1 (1994), 9{17. [21] Singer, M. F. Testing reducibility of linear dierential operators: a group theoretic perspective. submitted to Applicable Algebra in Engineering, Communication and Computing, 1993. [22] Singer, M. F., and Ulmer, F. Galois groups of second and third order linear dierential equations. Journal of Symbolic Computation 16 (1993), 9{36. [23] Tsarev, S. P. On some problems in factorization of linear ordinary dierential operators. Programming & Computer Software 20, 1 (1994), 27{29.

occasion to thank Professor B.Fuchssteiner and the MuPAD group for their hospitality.

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