Factoring Differential Operators in n Variables
aa r X i v : . [ c s . S C ] M a r Factoring Differential Operators in n Variables
Mark Giesbrecht
David R. Cheriton School ofComputer ScienceUniversity of Waterloo200 University Avenue WestWaterloo, Ontario, Canada [email protected] Albert Heinle
David R. Cheriton School ofComputer ScienceUniversity of Waterloo200 University Avenue WestWaterloo, Ontario, Canada [email protected] Viktor Levandovskyy
Lehrstuhl D für Mathematik,RWTH Aachen UniversityTemplergraben 64Aachen, Germany [email protected]
ABSTRACT
In this paper, we present a new algorithm and an experimen-tal implementation for factoring elements in the polynomial n th Weyl algebra, the polynomial n th shift algebra, and Z n -graded polynomials in the n th q -Weyl algebra.The most unexpected result is that this noncommutativeproblem of factoring partial differential operators can be ap-proached effectively by reducing it to the problem of solvingsystems of polynomial equations over a commutative ring.In the case where a given polynomial is Z n -graded, we canreduce the problem completely to factoring an element in acommutative multivariate polynomial ring.The implementation in Singular is effective on a broadrange of polynomials and increases the ability of computeralgebra systems to address this important problem. Wecompare the performance and output of our algorithm withother implementations in commodity computer algebra sys-tems on nontrivial examples.
Categories and Subject Descriptors
G.4 [
Mathematical Software ]: Algorithm design and anal-ysis; I.1.2 [
Symbolic and Algebraic Manipulation ]: Al-gorithms—
Factorization
General Terms
Algorithms, Design, Theory
Keywords
Factorization, linear partial differential operator,non-commutative algebra, Singular, algebra of operators,Weyl algebra
1. INTRODUCTION
In this paper we present a new method and an implemen-tation for factoring elements in the n th polynomial Weylalgebra A n and the n th polynomial shift algebra. An adap-tions of these ideas can also be applied to classes of polyno-mials in the n th q -Weyl algebra, which is also outlined here. Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00.
There are numerous important applications for this method,notably since one can view those rings as operator algebras.For example, given an element L ∈ A n and viewing L asa differential operator, one can derive properties of its so-lution spaces. Especially concerning the problem of findingthe solution to the differential equation associated with L ,the preconditioning step of factoring L can help to reducethe complexity of that problem in advance.The new technique heavily uses the nontrivial Z n -gradingon A n and, to the best of our knowledge, has no analoguesin the literature on factorizations for n ≥
2. However, for n = 1 it is the same grading that lies behind the Kashiwara-Malgrange V -filtration ([16] and [21]), which is a very im-portant tool in the D -module theory. Van Hoeij also madeuse of this technique in [30] to factorize elements in the firstWeyl algebra with power series coefficients. Notably, for n ≥
2, the Z n -grading we propose is very different fromthe mentioned Z -grading. Among others, a recent resultfrom [4] states that the Gel’fand-Kirillov dimension [11] ofthe 0 th graded part of Z -grading of A n is in fact 2 n − A n is, forcomparison, 2 n . The 0 th graded part of the Z n -grading wepropose has Gel’fand-Kirillov dimension n . Definition Let A be an algebra over a field K and f ∈ A \ K be a polynomial. A nontrivial factorization of f is a tuple ( c, f , . . . , f m ) , where c ∈ K \ { } , f , . . . , f m ∈ A \ { } are monic polynomials and f = c · f · · · f m . In general, we identify two problems in noncommutative fac-torization for a given polynomial f : (i) finding one factor-ization of f , and (ii) finding all possible factorizations of f .Item (ii) is interesting since factorizations in noncommu-tative rings are not unique in the classical sense (i.e., upto multiplication by a unit), and regarding the problem ofsolving the associated differential equation one factorizationmight be more useful than another. We show how to ap-proach both problems here.A number of papers and implementations have been pub-lished in the field of factorization in operator algebras overthe past few decades. Most of them concentrated on lineardifferential operators with rational coefficients. Tsarev hasstudied the form, number and properties of the factors of adifferential operator in [26] and [27], which extends the pa-pers [19] and [20]. A very general approach to noncommuta-tive algebras and their properties, including factorization, isalso done in the book by Bueso et al. in [8]. The authors pro-vide several algorithms and introduce various points of viewswhen dealing with noncommutative polynomial algebras.In his dissertation van Hoeij [28] developed an algorithmo factorize a univariate differential operator. Several pa-pers following that dissertation extend these techniques [29,30, 31], and this algorithm is implemented in the DETools package of
Maple [23] as the standard algorithm for factor-ization of these operators.In the
REDUCE -base computer algebra system
ALL-TYPES , Schwarz and Grigoriev [25] have implemented thealgorithm for factoring differential operators they introducedin [12]. As far as we know, this implementation is solely ac-cessible as a web service. Beals and Kartashova [6] considerthe problem of factoring polynomials in the second Weylalgebra, where they are able to deduce parametric factors.For special classes of polynomials in operator algebras,Foupouagnigni et al. [10] show some unexpected resultsabout factorizations of fourth-order differential equationssatisfied by certain Laguerre-Hahn orthogonal polynomials.From a more algebraic point of view, and dealing onlywith strictly polynomial noncommutative algebras, Melenkand Apel [22] developed a package for the computer alge-bra system
REDUCE . That package provides tools to dealwith noncommutative polynomial algebras and also containsa factorization algorithm for the supported algebras. It is,moreover, the only tool besides our implementation in
Sin-gular [9] that is capable of factoring in operator algebraswith more than one variable. Unfortunately, there are nofurther publications about how the implementation worksbesides the available code.The above mentioned algorithms and implementations arevery well written and they are able to factorize a large num-ber of polynomials. Nonetheless, as pointed out in [13, 14],there exists a large class of polynomials, even in the firstWeyl algebra, that seem to form the worst case for thosealgorithms. This class is namely the graded (or homoge-neous) polynomials in the sense of the Z n -graded structureon the n th Weyl algebra. Using our techniques, we are ableto obtain a factorization very quickly utilizing commutativefactorization and some combinatorics. Those techniques arediscussed for the first ( q -)Weyl algebra in detail in [14].Factorization of a general non-graded polynomial is muchmore involved. The main idea lies in inspecting the high-est and the lowest graded summands of the polynomial tofactorize. Any factorization corresponds respectively to thehighest or the lowest graded summands of the factors. Sincethe graded factorization appears to be easy, we are able tofactorize those summands and obtain finitely many candi-dates for highest and lowest summands of the factors. Ob-taining the rest of the graded summands is the subject con-sider in this paper.An implementation dealing with the first Weyl algebra,the first shift algebra, and graded polynomials in the first q -Weyl algebra, was created by Heinle and Levandovskyywithin the computer algebra system Singular [9]. For thelatter algebra, the implementation in
Singular is the onlyone available that deals with q -Weyl algebras, to the knowl-edge of the authors. The code has been distributed sinceversion 3-1-3 of Singular inside the library ncfactor.lib ,and received a major update in version 3-1-6.The new approach described in this paper will soon beavailable in an upcoming release of
Singular . There arenew functions for factoring polynomials in the n th poly-nomial Weyl algebra, homogeneous polynomials in the n thpolynomial q -Weyl algebra and the n th polynomial shift al-gebra. The remainder of this paper is organized as follows. Therest of this section is dedicated to providing basic notions,definitions and results that are needed to describe our ap-proach. Most of the results are well-known, but have notbeen used for factorization until now.Section 2 contains a methodology to deal with the factor-ization problem for graded polynomials, while in Section 3we utilize this methodology to factorize arbitrary polynomi-als in the n th Weyl algebra. In Section 5 we evaluate ourexperimental implementation on several examples in Section4 and compare the results to other commodity computer al-gebra systems. By K we always denote a field of characteristic zero (thoughsome of the statements also hold for some finite fields). Fornotational convenience we write n for { , . . . , n } and θ for θ , . . . , θ n for n ∈ N throughout. Definition The n th q -Weyl algebra Q n is defined as Q n := K (cid:28) x , . . . , x n , ∂ , . . . , ∂ n | for ( i, j ) ∈ n × n : ∂ i x j = ( x j ∂ i , if i = jq i x j ∂ i + 1 , if i = j , ∂ i ∂ j − ∂ j ∂ i = x i x j − x j x i = 0 (cid:29) , where q , . . . , q n are units in K . For the special case where q = · · · = q n = 1 we have the n th Weyl algebra , whichis denoted by A n . For notational convenience, we write X e D w := x e · · · x e n n ∂ w · · · ∂ w n n for every monomial, where e, w ∈ N n . Definition The n th shift algebra S n is defined as S n := K (cid:28) x , . . . , x n , s , . . . , s n | for ( i, j ) ∈ n × n : s i x j = ( x j s i , if i = j ( x j + 1) s i , if i = j , s i s j − s j s i = x i x j − x j x i = 0 (cid:29) . For notational convenience, we write as above X e S w := x e · · · x e n n s w · · · s w n n , where e, w ∈ N n . Remark Throughout this paper we view Z n , equippedwith the coordinate-wise addition, as an ordered monoid withrespect to a total ordering < , compatible with addition andsatisfying the following property: for any z , z ∈ Z n , suchthat z < z , the set { w ∈ Z n , z < w < z } is finite. The n th ( q -)Weyl algebra possesses a nontrivial Z n -gradingusing the weight vector [ − w, w ] for a 0 = w ∈ Z n on theelements x , . . . , x n , ∂ , . . . , ∂ n . For simplicity, we choose w := [1 , . . . , X a D b ) := [ b − a , . . . , b n − a n ] for a, b ∈ N n . Note, that a Z -grading, aris-ing from V -filtration [16, 21] prescribes to X a D b the grade P ni =1 ( b i − a i ) ∈ Z .We call a polynomial homogeneous or graded , if ev-ery summand is weighted homogeneous with respect to theweight vector [ − w, w ] as above. Example In the second Weyl algebra A one has deg( x x ∂ ∂ ) = deg(( ∂ x + 1) x ∂ ) = [0 , . The polynomial x ∂ x + x ∂ x + ∂ x is homogeneous ofdegree [1 , − . The monomials x ∂ , resp. x ∂ , have de-grees [ − , , resp. [1 , − , hence their sum is not homoge-neous. ote that the so-called Euler operators θ i := x i ∂ i , i ∈ n have degree 0 for all i , and thus play an important role, aswe shall soon see.First, we study some commutation rules the Euler opera-tor θ i has with x i and ∂ i . For Q n , in order abbreviate thesize of our formulae, we introduce the so called q -bracket. Definition For n ∈ N and q ∈ K \{ } , the q -bracketof n is defined to be [ n ] q := − q n − q = P n − i =0 q i . Lemma 1 (Compare with [24]). In A n , the followingcommutation rules hold for m ∈ N and i ∈ n : θ i x mi = x mi ( θ i + m ) , θ i ∂ mi = ∂ mi ( θ i − m ) . More generally, in Q n , the following commutation rules holdfor m ∈ N and i ∈ n : θ i x mi = x mi ( q mi θ i +[ m ] q i ) , θ i ∂ mi = ∂ mi q i (cid:18) θ i − q m − i − q − m +2 i − q i − q i (cid:19) . The commutation rules described in Lemma 1 can, ofcourse, be extended to arbitrary polynomials in the θ i , i ∈ n . Corollary Consider f ( θ ) ∈ K [ θ ] . Then, in A n wehave f ( θ ) X e = X e f ( θ + e , . . . , θ n + e n ) , and f ( θ ) D e = D e f ( θ − e , . . . , θ n − e n ) . Analogous identities with the re-spective commutation rules as given in Lemma 1 hold for Q n .
2. FACTORING GRADED POLYNOMIALS
For graded polynomials, the main idea of our factorizationtechnique lies in the reduction to a commutative univariatepolynomial subring of A n , respectively Q n , namely K [ θ ].Actually, it appears that this subring is quite large in thesense of reducibility of its elements in A n (resp. Q n ).Due to the commutativity of x i with ∂ j , for i = j , we canwrite X a D b = x a · · · x a n n · ∂ b · · · ∂ b n n = x a · ∂ b · · · x a n n · ∂ b n n for any a, b ∈ N n . By definition, a monomial X a D b has de-gree 0 := [0 , . . . ,
0] if and only if a = b . The following lemmashows, how we can rewrite every homogeneous polynomialof degree 0 in A n (resp. Q n ) as a polynomial in K [ θ ]. Lemma 2 (Compare with [24], Lemma 1.3.1). In A n ,we have the identity x mi ∂ mi = Q m − j =0 ( θ i − j ) . for m ∈ N and i ∈ n . In Q n , one can rewrite x mi ∂ mi as element in K [ θ ] andit is equal to q Tm − i Q m − j =0 ( θ i − [ j ] q i ) , where T j denotes the j th triangular number, i.e., T j := j ( j + 1) / for all j ∈ N . Corollary The th graded part of Q n , respectively A n , is K [ θ , . . . , θ n ] . Recall, that the z th graded part for z ∈ Z n of Q n , resp. A n , is defined to be the K -vector space: Q ( z ) n := K { X n D n : n , n ∈ N n , n − n = z } , i.e., the degree of a monomial is determined by the differenceof its powers in the x i and the ∂ i . Moreover, since in agrading Q ( z ) n · Q ( z ) n ⊆ Q ( z + z ) n holds for all z , z ∈ Z n , Q ( z ) n is naturally a Q (0) n -module. Proposition For z ∈ Z n \ { } , the z th graded part Q ( z ) n , resp. A ( z ) n , is a cyclic K [ θ ] -bimodule, generated by theelement X e ( z ) D w ( z ) , exponent vectors of which read for i ∈ n as follows: e i ( z ) := ( − z i , if z i < , , otherwise , , w i ( z ) := ( z i , if z i > , , otherwise . Proof.
A polynomial p ∈ Q ( z ) n resp. p ∈ A ( z ) n is ho-mogeneous of degree z ∈ Z n if and only if every mono-mial of p is of the form X k + e ( z ) D k + w ( z ) , where k ∈ N and k := [ k, . . . , k ]. By doing a rewriting, similar to theabove, we obtain X k + e ( z ) D k + w ( z ) = X e ( z ) X k D k D w ( z ) = X e ( z ) f k ( θ ) D w ( z ) , where f k ( θ ) is computed utilizing Lemma2. Moreover, by Corollary 1, we conclude that X e ( z ) f k ( θ ) D w ( z ) = f k ( θ − e ( z ) , . . . , θ n − e n ( z )) X e ( z ) D w ( z ) or, equivalently, X e ( z ) D w ( z ) f k ( θ + w ( z ) , . . . , θ n + w n ( z )),showing the cyclic bimodule property.Therefore, the factorization of a homogeneous polynomialof degree zero can be done by rewriting the polynomial as anelement in K [ θ ] and applying a commutative factorization onthe polynomial, a much-better-understood problem which isalso well implemented in every computer algebra system.Of course, this would not be a complete factorization,as there are still elements irreducible in K [ θ ] which are re-ducible in Q n , resp. A n . An obvious example are the θ i themselves. Fortunately, there are only 2 n monic polynomi-als irreducible in K [ θ ] that are reducible in A n , resp. Q n ,and these are of quite a special form. This extends the prooffor A and Q presented in [14]. Lemma Let i ∈ n . The polynomials θ i and θ i + q i arethe only irreducible monic elements in K [ θ ] that are reduciblein Q n . Respectively, θ i and θ i + 1 are the only irreduciblemonic polynomials in K [ θ ] that are reducible in A n . Proof.
We only consider the proof for A n , as the prooffor Q n is done in an analogous way. Let f ∈ K [ θ ] be amonic polynomial. Assume that it is irreducible in K [ θ ], butreducible in A n . Let ϕ, ψ be elements in A n with ϕψ = f .Then ϕ and ψ are homogeneous and ϕ ∈ A ( − z ) n , ψ ∈ A ( z ) n fora z ∈ Z n . Let [ e, w ] := [ e ( z ) , w ( z )] be as in Proposition 1.Note, that then w ( − z ) = e ( z ) = e and e ( − z ) = w ( z ) = w holds. That is, A ( z ) n = K [ θ ] X e D w whereas A ( − z ) n = K [ θ ] X w D e . Then for ˜ ϕ, ˜ ψ ∈ K [ θ ], we have ϕ = ˜ ϕ ( θ ) X e D w and ψ = ˜ ψ ( θ ) X w D e . Using Corollary 1, we obtain f = ˜ ϕ ( θ ) X e D w ˜ ψ ( θ ) X w D e = ˜ ϕ ( θ ) X e D w X w D e ˜ ψ ( θ + w − e ) , where, by Lemma 2, X e D w X w D e = g ( θ ) ∈ K [ θ ]. Since vec-tors e and w have disjoint support and e + w = [ | z | , . . . , | z n | ], g is irreducible by Lemma 2 only if there is at most onenonzero z i . If z = 0, then e = w = 0, hence g = 1 and φ, ψ ∈ K [ θ ]. Since f has been assumed to be monic irre-ducible in K [ θ ], one φ and ψ give us a trivial factorization.Now, suppose that there exists exactly one i such that z i >
0. Then e ( z ) = 0 and w ( z ) = z is zero on all but i thplace. By the irreducibility assumption on f ∈ K [ θ ] we musthave ˜ ϕ, ˜ ψ ∈ K ; since f is monic, we must also have ˜ ϕ = ˜ ψ − .By Lemma 2 we obtain z i = 1. As a result, the only possible f in this case is f = θ i + 1. For analogous reasons for thecase when z i <
0, we conclude, that the only possible f inthat case is f = θ i .The result in Lemma 3 provides us with an easy way tofactor a homogeneous polynomial p ∈ A n , resp. p ∈ Q n ,of degree 0. Obtaining one possible factorization into irre-ducible polynomials can be done using the following steps:1. Rewrite p as an element in K [ θ ];2. Factorize this resulting element in K [ θ ] with commu-tative methods;3. If there is θ i or θ i + 1 for i ∈ n among the factors,rewrite it as x i · ∂ i resp. ∂ i · x i .s mentioned earlier, the factorization of a polynomial ina noncommutative ring is unique up to a weak similarity [8].This notion is much more involved than the similarity upto multiplication by units or up to interchanging factors,as in the commutative case. Indeed, several different non-trivial factorizations can occur. Fortunately, in the case ofthe polynomial first ( q -)Weyl algebra, there are only finitelymany different nontrivial factorizations possible due to [27].In order to obtain all these different factorizations, one canapply the commutation rules for x i and ∂ i with θ i for i ∈ n .That these are all possible factorizations up to multiplica-tion by units can be seen using an analogue approach as inthe proof of Lemma 3. Consider the following example. Example Let p := x x ∂ ∂ + 2 x x ∂ ∂ + x ∂ +1 ∈ A . The polynomial p is homogeneous of degree , andhence belongs to K [ θ ] as θ ( θ − θ + 2 θ θ + θ + 1 . Thispolynomial factorizes in K [ θ ] into ( θ θ + 1)( θ + 1) . Since θ + 1 factorizes as ∂ · x , we obtain the following possibledifferent nontrivial factorizations: ( θ θ +1) · ∂ · x = ∂ · (( θ − θ +1) · x = ∂ · x · ( θ θ +1) . Note that x ∂ + 1 is not irreducible, since it factorizes non-trivially as ∂ · x . Now we consider the factorization of homogeneous poly-nomials of arbitrary degree z ∈ Z n . Fortunately, the hardwork is already done in Proposition 1. Indeed, one factoriza-tion of a homogeneous polynomial p ∈ Q ( z ) n , resp. p ∈ A ( z ) n ,of degree z ∈ Z n can be obtained using the following steps.1. Write p ( X, D ) as ˜ p ( θ ) X e D w , where ˜ p ∈ A (0) n = K [ θ ]and e, w are constructed according to Proposition 1.2. Factorize ˜ p using the technique described for polyno-mials of degree 0. Append to such a factorizationthe natural expansion of the monomial X e D w into theproduct of occuring single variables.This leads to one nontrivial factorization. A characteriza-tion of how to obtain all factorizations is given provided bythe following lemma. Lemma Let z ∈ Z n and p ∈ A ( z ) n , resp. p ∈ Q ( z ) n , ismonic. Suppose, that one factorization has been constructedas above and has the form Q ( θ ) · T ( θ ) · X e D w , where T ( θ ) = Q ni =1 ( x i ∂ i ) t i ( ∂ j x j ) s i is a product of irreducible factors in K [ θ ] , which are reducible in A n , resp. Q n , and Q ( θ ) is theproduct of irreducible factors in both K [ θ ] and A n , resp. Q n ) .Let p · · · p m for m ∈ N be another nontrivial factorizationof p . Then this factorization can be derived from Q ( θ ) · T ( θ ) · X e D w by using two operations, namely (i) “swapping”,that is interchanging two adjacent factors according to thecommutation rules and (ii) “rewriting” of occurring θ i resp. θ i + 1 ( θ i + q in the q -Weyl case) by x i · ∂ i resp. ∂ i · x i . Proof.
Since p is homogeneous, all p i for i ∈ m are ho-mogeneous, thus each of them can be written in the form p i = ˜ p i ( θ ) · X e ( i ) D w ( i ) , where e ( i ) , w ( i ) ∈ N n . With respectto the commutation rules as stated in Corollary 1, we canswap the ˜ p i ( θ ) to the left for any 2 ≤ i ≤ m . Note that itis possible for them to be transformed to the form θ j resp. θ j + 1 ( θ j + q in the q -Weyl case), j ∈ n , after perform-ing these swapping steps. I.e., we have commuting factors,both belonging to Q ( θ ), as well as to T ( θ ) at the left. Ourresulting product is thus ˜ Q ( θ ) ˜ T ( θ ) Q mj =1 X e ( j ) D w ( j ) , wherethe factors in ˜ Q ( θ ), resp. ˜ T ( θ ), contain a subset of thefactors of Q ( θ ) resp. T ( θ ). By our assumption of p hav-ing degree z , we are able to swap X e D w to the right in F := Q mj =1 X e ( j ) D w ( j ) , i.e., F = ˜ F X e D w for ˜ F ∈ A (0) n .This step may involve combining some x j and ∂ j to θ j resp. θ j + 1, j ∈ n . Afterwards, this is also done to the remainingfactors in ˜ F that are not yet polynomials in K [ θ ] using theswapping operation. These polynomials are the remainingfactors that belong to Q ( θ ), resp. T ( θ ), and can be swappedcommutatively to their respective positions. Since reverseengineering of those steps is possible, we can derive the fac-torization p · · · p m from Q ( θ ) · T ( θ ) · X e D w as claimed.Summarizing, we are now able to effectively factor gradedpolynomials in the n th Weyl and q -Weyl algebra. All differ-ent factorizations are obtainable using our technique. Remark A reader might ask what are the merits ofour “graded-driven” approach as opposed to a somewhat moredirect approach to factorization using leading monomials.Since, for monomials m, m ′ ∈ A n , one has lexp( m · m ′ ) =lexp( m )+lexp( m ′ ) , indeed h = p · q implies lexp( p )+lexp( q ) =lexp( h ) . Thus by considering, say, degree lexicographic or-dering on A n , the set C h := { ( a, b ) ∈ N n × N n : a, b =0 , a + b = lexp( h ) } contains all possible pairs of leadingmonomials of p and q . Then, since with respect to the cho-sen ordering, for any monomials there are only finitely manysmaller monomials, one can make an ansatz with unknowncoefficients for p and q . Each ( a, b ) ∈ C h leads to a systemof nonlinear polynomial equations in finitely many variables.We compare this “leading monomial” approach with our“graded-driven” one. At first, the factorization of a Z n -graded polynomial, which is very easy to accomplish with ourapproach, requires solving of several systems within the lead-ing monomial approach. Second, the number of all elementsin the set C h above is significantly bigger than the number offactorizations of the highest graded part of a polynomial, say ˜ p ( θ ) X e D w : suppose that ˜ p ( θ ) is irreducible over K [ θ ] . Thenfactorization with the “graded-driven” approach are obtainedvia moving x , resp. ∂ , past ˜ p ( θ ) to the left. Thus the num-ber of such factorizations is much smaller than the numberof ways of writing the exponent vector of lm(˜ p ( θ ) X e D w ) = θ α X e D w as a sum of two exponent vectors. In the next section, we show how the developed techniquehelps us to tackle the factorization problem for arbitrarypolynomials in A n .
3. FACTORING ARBITRARYPOLYNOMIALS3.1 Preliminaries
The techniques described in this section solve the factor-ization problem in A n . A generalization for Q n is moreinvolved and the subject of ongoing research.We begin by fixing some notation used throughout thissection. From now on, let “ < ” be an ordering on Z n sat-isfying the conditions of Remark 1. Let h ∈ A n be thepolynomial we want to factorize. As we are deducing in-formation from the graded summands of h , let furthermore M := { z (1) , . . . , z ( m ) } , where m ∈ N and z (1) > . . . > z ( m ) ,be a finite subset of Z n containing the degrees of thosegraded summands. Hence, h can be written in the form h = P z ∈ M h z ∈ A n , where h z ∈ A ( z ) n for z ∈ M . Let usassume that h possesses a nontrivial factorization of at leasttwo factors, which are not necessary irreducible. Moreover,e assume that m >
1, which means that h is not graded,since we have dealt with graded polynomials in A n already.Let us denote the factors by h = X z ∈ M h z := ( p η + . . . + p η k ) | {z } := p ( q µ + . . . + q µ l ) | {z } := q , (1)where η > η > . . . > η k and µ > µ > . . . > µ l ∈ Z n , p η i ∈ A ( η i ) n for all i ∈ k , q µ j ∈ A ( µ j ) n for all j ∈ l . Weassume that p and q are not graded, since we could easilyobtain those factors by simply comparing all factorizationsof the graded summands in h . In general, while trying tofind a factorization of h , we assume that the values of k and l are not known to us beforehand. We will soon see howwe can obtain them. One can easily see that h z (1) = p η q µ and h z ( m ) = p η k q µ l , as the degree-wise biggest summandof h can only be combined by multiplication of the highestsummands of p and q ; analogously this holds for the degree-wise lowest summand.A finite set of candidates for p η , q µ , p η k and q µ l can beobtained by factoring h z (1) and h z ( m ) using the technique de-scribed in the previous section. Since the set of candidatesis finite, we can assume that the correct representatives for p η , q µ , p η k and q µ l are known to us. In practice, we wouldapply our method to all candidates and would succeed inat least one case to factorize the polynomial due to our as-sumption of h being reducible.One may ask now how many valid degrees could occur insummands of such factors p and q , i.e., what are the valuesof l and k . Theoretically, there exist choices for η and η k (resp. µ and µ l ) where there are infinitely many z ∈ Z n such that η > z > η k (resp. µ > z > µ l ). Fortunately,only finitely many are valid degrees that can appear in afactorization, as the next lemma shows. Lemma For fixed h, p η , q µ , p η k and q µ l ∈ A n ful-filling the assumptions stated above, there are only finitelymany possible η i resp. µ j ∈ Z n , i, j ∈ N , that can appear asdegrees for graded summands in p and q . Proof.
For every variable v ∈ { x , . . . , x n , ∂ , . . . , ∂ n } ⊂ A n , there exists a j ∈ N that represents the maximal degreeof v that occurs among the monomials in h . The number j can be seen as a lower bound of the associated position of v in η i , resp. µ i , if v is one of the x s, or as an upper bound if v is one of the ∂ s. If the degree of one of the homogeneoussummands of p or q would go higher, resp. lower, than thisdegree-bound, v would appear in h in a higher degree than j , which contradicts our choice of j . Example Let us consider h = x ∂ ∂ + ∂ | {z } degree: [1 , + x x ∂ | {z } degree: [1 , − + 4 ∂ |{z} degree: [0 , + 4 x ∂ | {z } degree: [0 , ∈ A . One possible factorization of x ∂ ∂ + ∂ is ∂ · x ∂ =: p η · q µ and, on the other end, one possible factorization of x ∂ is x ∂ · p η k · q µ l . Concerning p , there are noelements in Z n that can occur between deg( p η ) = [0 , and deg( p η k ) = [0 , ; therefore we can set k = 2 . For q , theonly degree that can occur between deg( q µ ) = [1 , − and deg( q µ l ) = [0 , is [0 , , as every variable except ∂ appearswith maximal degree in h . We have l = 3 in this case. In the previous subsection we saw that, given h ∈ A n that possesses a factorization as in (1), we are able to ob-tain the elements p η , q µ , p η k and q µ l . Moreover, we can compute the numbers k and l of homogeneous summands inthe factors. Now our goal is to find values for the unknownhomogeneous summands, i.e. p η , . . . , p η k − , q µ , . . . , q l − .Our goal is to reduce this to a commutative problem to thegreatest extent we can.For this, we use Proposition 1 and define for all i ∈ k thepolynomial ˜ p η i ∈ A (0) n by ˜ p η i X e D w = p η i . In the same waywe define ˜ q µ j for all j ∈ l and ˜ h z for z ∈ M . The latter areknown to us since ˜ h z can easily be obtained from the inputpolynomial h . We can refer to the ˜ h z , ˜ p η i , ˜ q µ j as elementsin the commutative ring K [ θ ] using Lemma 2.The next fact about the degree of the remaining unknownscan be easily proven and is useful for our further steps. Lemma The degree of the ˜ p η i and the ˜ q µ j , ( i, j ) ∈ k × l ,in θ t , t ∈ n , is bounded by min { deg x t ( h ) , deg ∂ t ( h ) } , where deg v ( f ) denotes the degree of f ∈ A n in the variable v . There are certain equations that the ˜ p η i and the ˜ q µ j mustfulfil in order for p and q to be factors of h . Definition For α, β ∈ Z n we define γ α,β = Q nκ =1 ˜ γ ( κ ) α κ ,β κ ;in the latter expression we define for a, b ∈ Z and κ ∈ n ˜ γ ( κ ) a,b := , if a, b ≥ ∨ a, b ≤ , Q | a |− τ =0 ( θ κ − τ ) , if a < , b > , | a | ≤ | b | , Q | b |− τ =0 ( θ κ − τ − | a | + | b | ) , if a < , b > , | a | > | b | , Q aτ =1 ( θ κ + τ ) , if a > , b < , | a | ≤ | b | , Q | b | τ =1 ( θ κ + τ + | a | − | b | ) , if a > , b < , | a | > | b | . Theorem Suppose that, with the notation as above,we have h = pq and ˜ p η , ˜ q µ , ˜ p η k , ˜ h z (1) , . . . , ˜ h z ( m ) are known.Define ˜ h z := 0 for z (1) > z > z ( m ) and z M . Then theremaining unknown ˜ p η , . . . , ˜ p η k − , ˜ q µ , . . . , ˜ q µ l − are solu-tions of the following finite set of equations: ( X λ,̺ ∈ k × lηλ + µ̺ = z ˜ p η λ ( θ )˜ q µ ̺ ( θ + ( η λ ) , . . . , θ n + ( η λ ) n ) γ η λ ,µ ̺ = ˜ h z | z ∈ Z n , z (1) ≥ z ≥ z ( m ) ) . (2) Moreover, a factorization of h in A corresponds to ˜ q µ i and ˜ p η j for ( i, j ) ∈ k × l being polynomial solutions with boundsas stated in Lemma 6. Proof.
We only sketch this technical proof. Inspectingthe product in (2), we split it into its graded summands.By repeated application of Lemma 1, we arrive at the de-scribed set of equations via coefficient comparison. The de-gree bound has been established in Lemma 6 above.
Corollary The problem of factorizing a polynomialin the n th Weyl algebra can be solved via finding polynomialunivariate solutions of degree at most · P ni =0 | deg( h ) i | fora system of difference equations with polynomial coefficients,involving linear and quadratically nonlinear inhomogeneousequations. As this part of the method is rather technical, let us illus-trate it via an example.
Example Let p := θ ∂ |{z} = p [0 , + ( θ + 3) θ | {z } = p [0 , + x |{z} = p [0 , − ,q := ( θ + 4) x ∂ | {z } = q [ − , + x |{z} = q [ − , + ( θ + 1) x x | {z } = q [ − , − ∈ A and := pq = θ ( θ + 4) x ∂ (3)+ ( θ ( θ − θ + 8 θ θ + θ + 12 θ ) x ∂ (4)+ ( θ ( θ − θ + θ − θ + 4 θ θ + 2 θ + 7 θ ) x (5)+ ( θ ( θ − θ + 5 θ θ + 3 θ + 1) x x (6)+ ( θ + 1) x x . (7) We have written every coefficient in terms of the θ i alreadyfor better readability.By assumption, the only information we have about p and q are the values of p [0 , =: p η , p [0 , − =: p η , q [ − , =: q µ and q [ − , − =: q µ l . Thus we have, using the above notation, ˜ p η = θ , ˜ p η k = 1 , ˜ q µ = ( θ + 4) and ˜ q µ l = ( θ + 1) . Weset k := l := 3 , and it remains to solve for ˜ q [ − , and ˜ p [0 , .In h , every variable appears in degree 2, except from x ,which appears in degree 3. That means that the degree boundsfor θ and θ in ˜ q µ i can be set to be two.The product of ( p η + p η + p η )( q µ + q µ + q µ ) withknown values inserted is θ ( θ + 4) x ∂ (8)+ ( θ ˜ q µ ( θ , θ + 1) + ˜ p η ( θ + 4)) x ∂ (9)+ ( θ ( θ + 1)( θ + 1) + ( θ + 4) θ + ˜ p η ˜ q µ ) x (10)+ (˜ q µ ( θ , θ −
1) + ˜ p η ( θ + 1)) x x (11)+ ( θ + 1) x x . (12) The coefficients in K [ θ ] in the terms (8)-(12) have to coin-cide with the respective coefficients in the terms (3)-(7) forthe factorization to be correct. The equations with respect tothose coefficents are exactly the ones given in (2). There are many ways of dealing with finding solutions forthe system as described by the set (2). The first way wouldbe to solve the appearing partial difference equations andderive polynomial solutions. To the best of our knowledge,there is no general algorithm for finding polynomial solutionsof a system of nonlinear difference equations ([1], [2] and [3]).However, by Theorem 1 and Lemma 6, we are looking forbounded solutions, where explicit bounds are given. Thisproblem is clearly algorithmically solvable.Here, we present one of the possible approaches to solvethe commutative system of equations, which we also chosefor the implementation. We give an outline of the basic ideashere. A detailed description and discussion will become sub-ject of a journal version of this paper.We begin by studying the equations as given in Theorem 1.
Lemma Let us sort the equations as given in the setstated in (2) by the degree of the graded part they represent,from highest to lowest. Let moreover ν ∈ N be the numberof those equations, and κ be the number of all unknowns.We define χ i for i ∈ ν to be the number of ˜ p η κ and ˜ q µ ι , ( κ, ι ) ∈ l × k , appearing in equations , . . . , i . Then we have,for i ≤ ⌈ κ/ ⌉ , χ i = 2 · ( i − . The same holds if we sort theequations from lowest to highest. Proof.
The proof of this statement can be obtained us-ing induction on i . We outline the main idea here. For i = 1,we have the known equation ˜ h z (1) = p η q µ = ˜ p η ˜ q µ ( θ +( η ) , . . . , θ n + ( η ) n ) γ η ,µ , i.e. χ = 0. For the next equa-tion, as we regard the directly next lower homogeneous sum-mand, only the directly next lower unknowns ˜ p η and ˜ q µ appear, multiplied by ˜ q µ resp. ˜ p η . Hence, we get χ = 2.This process can be iterated until χ ⌈ κ/ ⌉ = κ . An analogousargument can be used when the equations are sorted fromlowest to highest. Using Lemma 7, we can reduce the unknowns we need tosolve for to the ˜ q µ i . Sorting the equations in the set (2) fromhighest to lowest, we can rearrange them by putting the ˜ p η i on the left hand side and backwards substituting the appear-ing ˜ p η j on the respective right hand side by the formulae inthe former equations. The same can be done when sortingthe equations from lowest to highest, which lead to a second– different – set of equations for the ˜ p η i . The remainingstep is then to concatenate the two respective descriptionsfor the ˜ p η i and then solve the resulting nonlinear system ofequations in the coefficients of the ˜ q µ j using e.g. Gr¨obnerbases [7]. We depict this process in the next example. Example Let us consider h = pq from Example 4,using all notations that were introduced there.We assume that the given form of ˜ p η is ˜ p η = ˜ p (0) η +˜ p (1) η θ +˜ p (2) η θ +˜ p (3) η θ +˜ p (4) η θ θ +˜ p (5) η θ θ +˜ p (6) η θ +˜ p (7) η θ θ +˜ p (8) η θ θ , and that ˜ q µ has an analogous shape with coeffi-cients q ( i ) µ , where ˜ p ( i ) η , ˜ q ( i ) µ ∈ K for i ∈ ∪ { } .We use our knowledge of the form of h and the product of pq with unknowns as depicted (8)-(12). Therefore, startingfrom the top and starting from the bottom, we obtain twoexpressions of ˜ p η , namely ˜ p η = θ ( θ − θ + 8 θ θ + θ + 12 θ − θ ˜ q µ ( θ , θ + 1) θ + 4= θ ( θ − θ + 5 θ θ + 3 θ + 1 − ˜ q µ ( θ , θ − θ + 1 . Thus, ˜ q µ has to fulfil the equation ( θ ( θ − θ + 8 θ θ + θ + 12 θ − θ ˜ q µ ( θ , θ + 1))( θ + 1)= ( θ ( θ − θ + 5 θ θ + 3 θ + 1 − ˜ q µ ( θ , θ − θ + 4) . Note here, that we could consider more equations that ˜ q µ must fulfill, but we refrained from it in this example for thesake of brevity.Using coefficient comparison, one can form from this equa-tion a nonlinear system of equations with the ˜ q ( i ) µ , i ∈ ∪{ } ,as indeterminates. The reduced Gr¨obner basis of this systemis { ˜ q (0) µ − , ˜ q (1) µ , ˜ q (2) µ , . . . , ˜ q (8) µ } , which tells us, that ˜ q µ = 1 and hence, ˜ p η = ( θ +3) θ . Thus, we have exactly recoveredboth p and q in the factorization of h . The concrete originalsystem is stated in Appendix A. This approach of course raises the question, if those sys-tems of equations that we construct are over- resp. under-determined. In the latter case, we might end up with someambiguity regarding the solutions of the systems. The nextlemma will show that our construction in fact leads to anoverdetermined system.
Lemma Let ν denote amount of the vectors in N n , thatare in each component t smaller or equal to min { deg x t ( h ) , deg ∂ t ( h ) } . After the reduction of the unknownsto the ˜ q µ i for i ∈ { , . . . , l − } , the amount of equations sat-isfied by the ˜ q µ i will be between · ( l − · ν and ( l − · ν , andthe amount of variables that we have to solve for is ( l − · ν . Proof.
The number ( l − · ν is obvious for the numberof unknowns, as we have for every polynomial ˜ q µ i for i ∈{ , . . . , l − } exactly ν unknown coefficients.In order to obtain expressions for our unknowns, we areconsidering two times l − l − l − q µ l when starting from the top, and the equa-tion for q µ when starting from the bottom, as we obtainore equations fulfilled by the unknown variables in thisway, where part of it is known to us. In the backwardssubstitution phase, we obtain different products of the poly-nomials ˜ q µ i . The amount of terms in the θ j for j ∈ n of thoseproducts is greater or equal to 2 · ν and at most ( l − · ν .This leads to the claimed bounds. In practice, one is often interested in differential equationsover the field of rational functions in the indeterminates x i .We refer to the corresponding operator algebras as the ra-tional Weyl algebras . We have the same commutationrules there, but with extension to the case where x i appearsin the denominator. These algebras can be recognized as Orelocalization of polynomial Weyl algebras with respect to themultiplicatively closed Ore set S = K [ x , . . . , x n ] \ { } .Unlike in the polynomial Weyl algebra, an infinite num-ber of nontrivial factorizations of an element is possible. Theeasiest example is the polynomial ∂ ∈ A , having nontriv-ial factorizations ( ∂ + x + c )( ∂ − x + c ) for all c ∈ K ; theonly polynomial factorization is ∂ · ∂ . Thus, at first glance,the factorization problem in both the rational and the poly-nomial Weyl algebras seems to be distinct in general. Butthere are still many things in common.Consider the more general case of localization of Ore alge-bras. In what follows, we denote by S ⊂ R the denomina-tor set of an arbitrary localization of a Noetherian integraldomain R . For properties that S has to fulfil and calcula-tion rules of elements in S − R please consider [8], Chapter8. Let us clarify the connection between factorizations in S − R and factorizations in R . Theorem Let h be an element in S − R \{ } . Suppose,that h = h · · · h m , m ∈ N , h i ∈ S − R for i ∈ m . Then thereexists q ∈ S and ˜ h , . . . , ˜ h m ∈ R , such that qh = ˜ h · · · ˜ h m . Thus, by clearing denominators in an irreducible elementin S − R one obtains an irreducible element in R . The otherdirection does not hold in general. However, one can useour algorithms in a pre-processing step of finding factoriza-tion over S − R . In particular, a reducible element of R isnecessarily reducible over S − R .The theorem says that we can lift any factorization fromthe ring S − R to a factorization in R by a left multiplicationwith an element of S . This means that in our case, where S = K [ x , . . . , x n ] \ { } , it suffices to multiply a polynomial h by a suitable element in K [ x , . . . , x n ] in order to obtaina representative of a rational factorization. Finding thiselement is subject of future research. As we already haveshown in [14], a polynomial factorization of an element in A n is often more readable than the factorization producedby rational factorization methods. Thus a pre-computationthat finds such a premultiplier so that we can just performpolynomial factorization would be a beneficial ansatz in therational factorization. Example Consider the polynomial h := ∂ − x ∂ − ∈ A . h is irreducible in A , but in the first rational Weylalgebra, we obtain a factorization given by ( ∂ + x )( ∂ − x ∂ − x ) . If we multiply h by x from the left, our fac-torization method reveals two different factorizations. Thefirst one is x · h itself, and the second one is given by ∂ · ( x ∂ − x − ∂ ) , which represents the rational factoriza-tion in the sense of Theorem 2. With the help of the Lemma 1 one can see that S n is a sub-algebra of the n th Weyl algebra A n via the following homo-morphism of K -algebras: ι : S n → A n , x i θ i , s j ∂ j . One can easily prove that ι is, in fact, a monomorphism.This observation leads to the following result, which tells usthat we do not have to consider the algebra S n separatelywhen dealing with factorization of its elements. Corollary The factorization problem for a polyno-mial p ∈ S n can be obtained from the solution of a factor-ization problem of ι ( p ) ∈ A n by refining. Theorem 2 also applies to the rational shift algebra. Thus,the approach to lift factorizations in the shift algebras withrational coefficients can also be applied here. The remainingresearch is also here to find suitable elements in K [ x , . . . , x n ]for pre-multiplication.
4. IMPLEMENTATION AND TIMINGS
We have implemented the described method for A n in thecomputer algebra system Singular . Our goal was to testthe performance of our approach and the versatility of theresults in practice and compare it to given implementations.Our implementation is in a complete but experimental stage,and we see potential for optimization in several areas.The implementation extends the library ncfactor.lib ,which contains the functionality to factorize polynomials inthe first Weyl algebra, the first shift algebra and gradedpolynomials in the first q -Weyl algebra. The actual libraryis distributed with Singular since version 3-1-3.In the following examples, we consider different polynomi-als and present the resulting factorizations and timings. Ourfunction to factorize polynomials in the n th Weyl algebra iswritten to solve problem (ii) as given in the introduction,i.e. finding all possible factorizations of a given polynomial.All computations were done using Singular version 3-1-6.We compare our performance and our outputs to
REDUCE version 3.8. There, we use the function nc_factorize_all in the library
NCPOLY . The calculations were run on a on acomputer with a 4-core Intel CPU (Intel R (cid:13) Core TM i7-3520MCPU with 2.90GHz, 2 physical cores, 2 hardware threads,32K L1[i,d], 256K L2, 4MB L3 cache) and 16GB RAM.In order to make the tests reproducible, we used the SDE-val [15] framework, created for the
Symbolic Data project[5], for our benchmarking. The functions of
Symbolic Data as well as the data are free to use. In such a way our com-parison is easily reproducible by any other person.Our set of examples is given by h := ( ∂ + 1) ( ∂ + x ∂ ) ∈ A ,h := ( θ ∂ + ( θ + 3) θ + x ) · (( θ + 4) x ∂ + x + ( θ + 1) x x ) ∈ A ,h := x x x ∂ ∂ + x x ∂ ∈ A ,h := ( x ∂ + x x ∂ )( ∂ ∂ + ∂ ∂ x x ) ∈ A . The polynomial h can be found in [18], the polynomial h isthe polynomial from Example 5 and the last two polynomialsare graded polynomials.Our implementation in Singular managed to factor allthe polynomials that are listed above. For h , it took 2.83sto find two distinct factorizations. Besides the given oneabove, we have h = ( x ∂ ∂ + ∂ + x ∂ + ∂ +2 ∂ )( ∂ +1) . Inorder to factorize h , Singular took 23.48s to find three fac-orizations. For the graded polynomials h and h , our im-plementation finished its computations as expected quickly(0.46s and 0.32s) and returned 60 distinct factorizations foreach h and h . REDUCE only terminated for h (within two hours). For h it returned 3 different factorizations (within 0.1s), andone of the factorizations contained a reducible factor. For h , h and h , we cancelled the process after two hours.Factoring Z -graded polynomials in the first Weyl algebrawas already timed and compared with several implementa-tions on various examples in [14]. The comparison there alsoincluded the functionality in the computer algebra system Maple for factoring polynomials in the first Weyl algebrawith rational coefficients.The next example shows the performance of our imple-mentation for the first Weyl algebra.
Example This example is taken from [17], page 200.We consider h := ( x − x ∂ + (1 + 7 x ) ∂ + 8 x . Ourimplementation takes 0.75 seconds to find 12 distinct fac-torizations in the algebra A . Maple
17, using
DFactor from the
DETools package, takes the same amount of timeand reveals one factorization in the first Weyl algebra withrational coefficients.
REDUCE outputs 60 factorizations in A after 3.27s. However, these factorizations contain fac-torizations with reducible factors. After factoring such casesand removing duplicates from the list, the number of differ-ent factorizations reduced to 12.
5. CONCLUSIONS
An approach to factoring polynomials in the operator al-gebras A n , Q n and S n based on nontrivial Z n -gradings hasbeen presented, and an experimental implementation hasbeen evaluated. We have shown that the set of polynomi-als that we can factorize using our technique in a feasibleamount of time has been greatly extended. Especially for Z n -graded polynomials, we have shown that the problem offinding all nontrivial factorizations in A n resp. Q n can bereduced to commutative factorization in multivariate ringsand some basic combinatorics. Thus, the performance ofthe factorization algorithm regarding graded polynomials isdominated by the performance of the commutative factor-ization algorithm that is available.Our future work consists of implementing the remainingfunctionalities into ncfactor.lib . Furthermore, it would beinteresting to extend our technique to deal with the factor-ization problem in A n to polynomials in Q n . Additionally,there exist many other operator algebras, and it would beinteresting to investigate to what extent we can use the de-scribed methodology there.Applying our techniques for the factorization problem inthe case of algebras with coefficients in rational functions isalso interesting, albeit more involved. Amongst other prob-lems, in that case infinitely many different factorizations canoccur. One has to find representatives of parametrized fac-torizations, and use these to obtain a factorization in thepolynomial sense. This approach could be beneficial, and ithas been developed in [14].
6. ACKNOWLEDGMENTS
We would like to thank to Dima Grigoriev for discussionson the subject, and to Mark van Hoeij for his expert opin-ion. We are grateful to Wolfram Koepf and Martin Leefor providing us with interesting examples and to Michael Singer, Shaoshi Chen and Daniel Rettstadt for sharing withus interesting points of view on our research problems.We would like to express our gratitude to the GermanAcademic Exchange Service DAAD for funding our projectin the context of the German-Canadian PPP program.
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Proc. ISSAC 2010 , pages 37–44, 2010.
APPENDIXA. COMMUTATIVE POLYNOMIAL SYSTEMOF EQUATIONS
The commutative polynomial system of equations that isformed in Example 5 is given as follows. {− ˜ q (8) µ , − ˜ q (7) µ − q (8) µ , − ˜ q (6) µ − ˜ q (7) µ − ˜ q (8) µ , − ˜ q (5) µ , − ˜ q (4) µ − q (5) µ − q (8) µ , − ˜ q (3) µ − ˜ q (4) µ − ˜ q (5) µ − q (7) µ , − ˜ q (2) µ + 4˜ q (8) µ , − ˜ q (1) µ − q (2) µ − q (5) µ + 4˜ q (7) µ − q (8) µ , − ˜ q (0) µ − ˜ q (1) µ − ˜ q (2) µ − q (4) µ + 4˜ q (6) µ − q (7) µ + 4˜ q (8) µ + 1 , − q (2) µ + 4˜ q (4) µ − q (5) µ , − q (1) µ + 4˜ q (3) µ − q (4) µ + 4˜ q (5) µ , q (2) µ , q (1) µ − q (2) µ , q (0) µ − q (1) µ + 4˜ q (2) µ − , (˜ q (8) µ ) , q (7) µ ˜ q (8) µ + 2(˜ q (8) µ ) , q (6) µ ˜ q (8) µ + (˜ q (7) µ ) + 3˜ q (7) µ ˜ q (8) µ +(˜ q (8) µ ) , q (6) µ ˜ q (7) µ + 2˜ q (6) µ ˜ q (8) µ + (˜ q (7) µ ) + ˜ q (7) µ ˜ q (8) µ , (˜ q (6) µ ) +˜ q (6) µ ˜ q (7) µ + ˜ q (6) µ ˜ q (8) µ , q (5) µ ˜ q (8) µ , q (4) µ ˜ q (8) µ + 2˜ q (5) µ ˜ q (7) µ + 4˜ q (5) µ ˜ q (8) µ − ˜ q (8) µ , q (3) µ ˜ q (8) µ + 2˜ q (4) µ ˜ q (7) µ + 3˜ q (4) µ ˜ q (8) µ + 2˜ q (5) µ ˜ q (6) µ + 3˜ q (5) µ ˜ q (7) µ +2˜ q (5) µ ˜ q (8) µ − ˜ q (7) µ , q (3) µ ˜ q (7) µ + 2˜ q (3) µ ˜ q (8) µ + 2˜ q (4) µ ˜ q (6) µ + 2˜ q (4) µ ˜ q (7) µ +˜ q (4) µ ˜ q (8) µ + 2˜ q (5) µ ˜ q (6) µ + ˜ q (5) µ ˜ q (7) µ − ˜ q (6) µ , q (3) µ ˜ q (6) µ + ˜ q (3) µ ˜ q (7) µ +˜ q (3) µ ˜ q (8) µ + ˜ q (4) µ ˜ q (6) µ + ˜ q (5) µ ˜ q (6) µ , q (2) µ ˜ q (8) µ + (˜ q (5) µ ) , q (1) µ ˜ q (8) µ +2˜ q (2) µ ˜ q (7) µ + 4˜ q (2) µ ˜ q (8) µ + 2˜ q (4) µ ˜ q (5) µ + 2(˜ q (5) µ ) − ˜ q (5) µ − q (8) µ , q (0) µ ˜ q (8) µ + 2˜ q (1) µ ˜ q (7) µ + 3˜ q (1) µ ˜ q (8) µ + 2˜ q (2) µ ˜ q (6) µ + 3˜ q (2) µ ˜ q (7) µ +2˜ q (2) µ ˜ q (8) µ + 2˜ q (3) µ ˜ q (5) µ + (˜ q (4) µ ) + 3˜ q (4) µ ˜ q (5) µ − ˜ q (4) µ + (˜ q (5) µ ) − q (7) µ − ˜ q (8) µ , q (0) µ ˜ q (7) µ + 2˜ q (0) µ ˜ q (8) µ + 2˜ q (1) µ ˜ q (6) µ + 2˜ q (1) µ ˜ q (7) µ +˜ q (1) µ ˜ q (8) µ + 2˜ q (2) µ ˜ q (6) µ + ˜ q (2) µ ˜ q (7) µ + 2˜ q (3) µ ˜ q (4) µ + 2˜ q (3) µ ˜ q (5) µ − ˜ q (3) µ +(˜ q (4) µ ) + ˜ q (4) µ ˜ q (5) µ − q (6) µ − ˜ q (7) µ , q (0) µ ˜ q (6) µ + ˜ q (0) µ ˜ q (7) µ + ˜ q (0) µ ˜ q (8) µ +˜ q (1) µ ˜ q (6) µ + ˜ q (2) µ ˜ q (6) µ + (˜ q (3) µ ) + ˜ q (3) µ ˜ q (4) µ + ˜ q (3) µ ˜ q (5) µ − ˜ q (6) µ , q (2) µ ˜ q (5) µ , q (1) µ ˜ q (5) µ + 2˜ q (2) µ ˜ q (4) µ + 4˜ q (2) µ ˜ q (5) µ − ˜ q (2) µ − q (5) µ − q (8) µ , q (0) µ ˜ q (5) µ + 2˜ q (1) µ ˜ q (4) µ + 3˜ q (1) µ ˜ q (5) µ − ˜ q (1) µ + 2˜ q (2) µ ˜ q (3) µ + 3˜ q (2) µ ˜ q (4) µ +2˜ q (2) µ ˜ q (5) µ − q (4) µ − ˜ q (5) µ − q (7) µ , q (0) µ ˜ q (4) µ + 2˜ q (0) µ ˜ q (5) µ − ˜ q (0) µ +2˜ q (1) µ ˜ q (3) µ + 2˜ q (1) µ ˜ q (4) µ + ˜ q (1) µ ˜ q (5) µ + 2˜ q (2) µ ˜ q (3) µ + ˜ q (2) µ ˜ q (4) µ − q (3) µ − ˜ q (4) µ − q (6) µ + 1 , q (0) µ ˜ q (3) µ + ˜ q (0) µ ˜ q (4) µ + ˜ q (0) µ ˜ q (5) µ + ˜ q (1) µ ˜ q (3) µ +˜ q (2) µ ˜ q (3) µ − ˜ q (3) µ , (˜ q (2) µ ) , q (1) µ ˜ q (2) µ + 2(˜ q (2) µ ) − q (2) µ − q (5) µ , q (0) µ ˜ q (2) µ + (˜ q (1) µ ) + 3˜ q (1) µ ˜ q (2) µ − q (1) µ + (˜ q (2) µ ) − ˜ q (2) µ − q (4) µ , q (0) µ ˜ q (1) µ + 2˜ q (0) µ ˜ q (2) µ − q (0) µ + (˜ q (1) µ ) + ˜ q (1) µ ˜ q (2) µ − ˜ q (1) µ − q (3) µ +7 , (˜ q (0) µ ) + ˜ q (0) µ ˜ q (1) µ + ˜ q (0) µ ˜ q (2) µ − ˜ q (0) µ , − q (1) µ , − q (0) µ + 12 }}