Desingularization of Ore Operators
aa r X i v : . [ c s . S C ] A ug DESINGULARIZATION OF ORE OPERATORS
SHAOSHI CHEN, MANUEL KAUERS, AND MICHAEL F. SINGER
Abstract.
We show that Ore operators can be desingularized by calculatinga least common left multiple with a random operator of appropriate order. Ourresult generalizes a classical result about apparent singularities of linear dif-ferential equations, and it gives rise to a surprisingly simple desingularizationalgorithm. Introduction
Consider a linear ordinary differential equation, like for example x (1 − x ) f ′ ( x ) − f ( x ) = 0 . The leading coefficient polynomial x (1 − x ) of the equation is of special interestbecause every point ξ which is a singularity of some solution of the differentialequation is also a root of this polynomial. However, the converse is in general nottrue. In the example above, the root ξ = 1 indicates the singularity of the solution x/ (1 − x ), but there is no solution which has a singularity at the other root ξ = 0. Tosee this, observe that after differentiating the equation, we can cancel (“remove”)the factor x from it. The result is the higher order equation(1 − x ) f ′′ ( x ) − f ′ ( x ) = 0 , whose solution space contains the solution space of the original equation. Such acalculation is called desingularization. The factor x is said to be removable. Given a differential equation, it is of interest to decide which factors of its leadingcoefficient polynomial are removable, and to construct a higher order equationin which all the removable factors are removed. A classical algorithm, which isknown since the end of the 19th century [14, 11], proceeds by taking the leastcommon left multiple of the given differential operator with a suitably constructedauxiliary operator. This algorithm is summarized in Section 2 below. At the endof the 20th century, the corresponding problem for linear recurrence equations wasstudied and algorithms for identifying removable factors have been found and theirrelations to “singularities” of solutions have been investigated [3, 4, 1]. Also somesteps towards a unified theory for desingularization of Ore operators have beenmade [10, 9]. Possible connections to Ore closures of an operator ideal have beennoted in [10] and within the context of order-degree curves [9, 7, 8]. These will befurther developed in a future paper.
Mathematics Subject Classification.
Key words and phrases.
D-finite functions, Apparent Singularities, Computer Algebra.S.C. was supported by the NSFC grant 11371143 and a 973 project (2011CB302401), M.K.was supported by FWF grant Y464-N18, and M.F.S. was supported by NSF grant CCF-1017217.
Our contribution in the present article is a three-fold generalization of the clas-sical desingularization algorithm for differential equations. Our main result (Theo-rem 6 below) says that (a) instead of the particular auxiliary operator traditionallyused, almost every other operator of appropriate order also does the job, (b) alsothe case when a multiple root of the leading coefficient can’t be removed completelybut only its multiplicity can be reduced is covered, and (c) the technique works notonly for differential operators but for every Ore algebra.For every removable factor p there is a smallest n ∈ N such that removing p fromthe operator requires increasing the order of the operator by at least n . Classicaldesingularization algorithms compute for each factor p an upper bound for this n ,and then determine whether or not it is possible to remove p at the cost of increasingthe order of the operator by at most n . In the present paper, we do not addressthe question of finding bounds on n but only discuss the second part: assumingsome n ∈ N is given as part of the input, we consider the task of removing as manyfactors as possible without increasing the order of the operator by more than n .Of course, for Ore algebras where it is known how to obtain bounds on n , thesebounds can be combined with our result.Recall the notion of Ore algebras [13]. Let K be a field of characteristic zero. Let σ : K [ x ] → K [ x ] be a ring automorphism that leaves the elements of K fixed, andlet δ : K [ x ] → K [ x ] be a K -linear map satisfying the law δ ( uv ) = δ ( u ) v + σ ( u ) δ ( v )for all u, v ∈ K [ x ]. The algebra K [ x ][ ∂ ] consists of all polynomials in ∂ withcoefficients in K [ x ] together with the usual addition and the unique (in generalnoncommutative) multiplication satisfying ∂u = σ ( u ) ∂ + δ ( u ) for all u ∈ K [ x ] iscalled an Ore algebra.
The field K is called the constant field of the algebra. Everynonzero element L of an Ore algebra K [ x ][ ∂ ] can be written uniquely in the form L = ℓ + ℓ ∂ + · · · + ℓ r ∂ r with ℓ , . . . , ℓ r ∈ K [ x ] and ℓ r = 0. We call deg ∂ ( L ) := r the order of L andlc ∂ ( L ) := ℓ r the leading coefficient of L . Roots of the leading coefficient ℓ r arecalled singularities of L . Prominent examples of Ore algebras are the algebra oflinear differential operators (with σ = id and δ = ddx ; we will write D instead of ∂ in this case) and the algebra of linear recurrence operators (with σ ( x ) = x + 1 and δ = 0; we will write S instead of ∂ in this case).We shall suppose that the reader is familiar with these definitions and facts,and will make free use of well-known facts about Ore algebras, as explained, forinstance, in [13, 6, 2]. In particular, we will make use of the notion of least commonleft multiples (lclm) of elements of Ore algebras: L ∈ K ( x )[ ∂ ] is a common leftmultiple of P, Q ∈ K ( x )[ ∂ ] if we have L = U P = V Q for some
U, V ∈ K ( x )[ ∂ ],it is called a least common left multiple if there is no common left multiple oflower order. Least common left multiples are unique up to left-multiplication bynonzero elements of K ( x ). By lclm( P, Q ) we denote a least common left multiplewhose coefficients belong to K [ x ] and share no common divisors in K [ x ]. Note thatlclm( P, Q ) is unique up to (left-)multiplication by nonzero elements of K . Efficientalgorithms for computing least common left multiples are available [5]. ESINGULARIZATION OF ORE OPERATORS 3 The Differential Case
In order to motivate our result, we begin by recalling the classical results con-cerning the desingularization of linear differential operators. See the appendix of [1]for further details on this case.Let L = ℓ + ℓ D + · · · + ℓ r D r ∈ K [ x ][ D ] be a differential operator of order r .Consider the power series solutions of L . It can be shown that x ∤ ℓ r if and onlyif L admits r power series solutions of the form x α + · · · , for α = 0 , . . . , r − x | ℓ r , then this factor is removable if and only if there exists someleft multiple M of L , say with deg ∂ ( M ) = s , such that M admits a power seriessolution with minimal exponent α for every α = 0 , . . . , s −
1. This is the case if andonly if L has r linearly independent power series solutions with integer exponents0 ≤ α < α < · · · < α r , because in this case (and only in this case) we canconstruct a left multiple M of L with power series solutions x α + · · · for each α = 0 , . . . , max { α , . . . , α r } −
1, by adding power series of the missing orders to thesolution space of L .These observations suggest the following desingularization algorithm for oper-ators L ∈ K [ x ][ ∂ ] with x | lc ∂ ( L ). First find the set { α , . . . , α ℓ } ⊆ N of allexponents α i for which there exist power series solutions x α i + · · · . If ℓ < r , re-turn “not desingularizable” and stop. Otherwise, let m = max { α , . . . , α ℓ } and let e , e , . . . , e m − ℓ be those nonnegative integers which are at most m but not amongthe α i . Return the operator M = lclm( L, A ) , where A := lclm( xD − e , xD − e , . . . , xD − e m − ℓ ) . Note that among the solutions of A there are the monomials x e , x e , . . . , x e m − ℓ , andthat the solutions of M are linear combinations of solutions of A and solutions of L .Therefore, by the choice of the e j and the remarks made above, M is desingularized. Example 1.
Consider the operator L = ( x − x − x +3) xD − ( x − x − x +2) D +( x − x − x +3) ∈ K [ x ][ D ] . This operator has power series solutions with minimal exponents α = 0 and α = 3 .Their first terms are x + x − x − x − x + · · · ,x + x + x + x + · · · . The missing exponents are e = 1 and e = 2 . Therefore we take A := lclm( xD − , xD −
2) = x D − xD + 2 and calculate M = lclm( L, A ) = ( x − x + 4 x − x + 12 x − D − ( x − x + x − x + 24 x − D − (3 x + 9 x ) D + (6 x + 18 x ) D − (6 x + 18) . Note that we have x ∤ lc ∂ ( M ) , as predicted. SHAOSHI CHEN, MANUEL KAUERS, AND MICHAEL F. SINGER
In the form sketched above, the algorithm applies only to the singularity 0. Inorder to get rid of a different singularity, move this singularity to 0 by a suitablechange of variables, then proceed as described above, and after that undo the changeof variables. Note that by removing the singularity 0 we will in general introducenew singularities at other points.3.
Removable Factors
We now turn from the algebra of linear differential operators to arbitrary Orealgebras. In the general case, removability of a factor of the leading coefficient isdefined as follows.
Definition 2.
Let L ∈ K [ x ][ ∂ ] and let p ∈ K [ x ] be such that p | lc ∂ ( L ) ∈ K [ x ] . Wesay that p is removable from L at order n if there exists some P ∈ K ( x )[ ∂ ] with deg ∂ ( P ) = n and some v, w ∈ K [ x ] with gcd( p, w ) = 1 such that P L ∈ K [ x ][ ∂ ] and σ − n (lc ∂ ( P L )) = wvp lc ∂ ( L ) . We then call P a p -removing operator for L , and P L the corresponding p -removed operator. p is simply called removable from L if it isremovable at order n for some n ∈ N . Example 3. (1)
In the example from the introduction, we have L = x (1 − x ) D − ∈ K [ x ][ D ] . An x -removing operator is P = x D : we have P L =(1 − x ) D − D . Because of deg ∂ ( P ) = 1 we say that x is removable atorder 1.If P is a p -removing operator then so is QP , for every Q ∈ K [ x ][ ∂ ] with gcd(lc ∂ ( Q ) , σ deg ∂ ( P )+deg ∂ ( Q ) ( p )) = 1 . In particular, note that the definitionpermits to introduce some new factors w into the leading coefficient while p is being removed. For instance, in our example also − xx D is an x -removing operator for L . (2) The definition does not imply that the leading coefficient of a p -removedoperator is coprime with (a shifted copy of ) p . In general, it only requiresthat the multiplicity is reduced. As an example, consider the operator L = x ( x − x − D + 2 x ( x − x + 1) D − ∈ K [ x ][ D ] and p = x . The operator P = x − x − x +2 x − x − x D − ( x + 5 x + 3) ∈ K ( x )[ D ] is a p -removing operator because the leading coefficient of P L = x ( x − x − x − x + 2 x − D − ( x − x − x + 22 x − x + 18 x − D − x − x − x + 8 x − x + 6) D + 2( x + 5 x + 3) contains only one copy of p while there are two of them in L . This is called partial desingularization. Observe that the definition permits to removesome factors v from the leading coefficient in addition to p . (3) In the shift case, or more generally, in an Ore algebra where σ is not theidentity, the leading coefficient changes when an operator is multiplied by apower of ∂ from the left. The application of σ − n in the definition compen-sates this change. As an example, consider the operator L = x ( x + 1)(5 x − S − x (5 x − x − S + ( x − x + 2)(5 x + 3) ∈ K [ x ][ S ] ESINGULARIZATION OF ORE OPERATORS 5 and p = x + 1 . The operator P = x +13 x − x − x +2)(5 x +3) S − x +28 x +23 x − x +2)(5 x +3) is a p -removing operator because the leading coefficient of P L = ( x + 1)(5 x + 13 x − x − S − x + 1)(10 x + 21 x − x + 24) S + (25 x + 60 x − x − x + 288) S − x − x + 28 x + 23 x − does not contain σ ( p ) = x + 2 . It is irrelevant that it contains x + 1 . As indicated in the examples, when removing a factor p from an operator L ,Def. 2 allows that we introduce other factors w , coprime to p . We are also alwaysallowed to remove additional factors v besides p . The freedom for having v and w is convenient but not really necessary. In fact, whenever there exists an operator P ∈ K ( x )[ ∂ ] of order n such that σ − n (lc ∂ ( P L )) = wvp lc ∂ ( L ), then there also existsan operator Q ∈ K ( x )[ ∂ ] of order n such that σ − n (lc ∂ ( QL )) = p lc ∂ ( L ). To seethis, note that by the extended Euclidean algorithm there exist s, t ∈ K [ x ] suchthat sw + tp = 1. Set Q = σ n ( sv ) P + σ − n ( t ) ∂ n . Then σ − n (lc ∂ ( QL )) = sv σ − n (lc ∂ ( P L )) + t lc ∂ ( ∂ n L )= sv wvp lc ∂ ( L ) + tpp lc ∂ ( L ) = 1 p lc ∂ ( L ) , as desired. This argument is borrowed from [1]. The same argument can also beused to show the existence of operators that remove all the removable factors inone stroke: Lemma 4.
Let L ∈ K [ x ][ ∂ ] , let n ∈ N , and let lc ∂ ( L ) = p e p e · · · p e m m be afactorization of the leading coefficient into irreducible polynomials. For each i =1 , . . . , m , let k i ≤ e i be maximal such that p i is removable from L at order n .Then there exists an operator P ∈ K ( x )[ ∂ ] of order n such that σ − n (lc ∂ ( P L )) = p k p k ··· p kmm lc ∂ ( L ) .Proof. By the remark preceding the lemma, we may assume that for every i thereexists an operator P i ∈ K ( x )[ ∂ ] of order n with P i L ∈ K [ x ][ ∂ ] and σ − n (lc ∂ ( P i L )) = p − k i i lc ∂ ( L ) (i.e., w = v = 1).Next, observe that when p and q are two coprime factors of lc ∂ ( L ) which both areremovable at order n , then also their product pq is removable at order n . Indeed,if P, Q ∈ K ( x )[ ∂ ] are such that deg ∂ ( P ) = deg ∂ ( Q ) = n , P L, QL ∈ K [ x ][ ∂ ], σ − n (lc ∂ ( P L )) = p lc ∂ ( L ), and σ − n (lc ∂ ( QL )) = q lc ∂ ( L ), and if s, t ∈ K [ x ] aresuch that sq + tp = 1, then for R := σ − n ( s ) P + σ − n ( t ) Q we have σ − n (lc ∂ ( RL )) = pq lc ∂ ( L ), as desired.The claim of the lemma now follows by induction on m , taking p = p e · · · p e m m − and q = p e m m . (cid:3) Desingularization by Taking Least Common Left Multiples
As outlined in Section 2, the classical algorithm for desingularizing differentialoperators relies on taking the lclm of the operator to be desingularized with asuitably chosen auxiliary operator. Our contribution consists in a three-fold gener-alization of this approach: first, we show that it works in every Ore algebra and not
SHAOSHI CHEN, MANUEL KAUERS, AND MICHAEL F. SINGER just for differential operators, second, we show that almost every operator qualifiesas an auxiliary operator in the lclm and not just the particular operator used tradi-tionally, and third, we show that the approach also covers partial desingularization.From the second fact it follows directly that taking the lclm with a random operatorof appropriate order removes, with high probability, all the removable singularitiesof the operator under consideration and not just a given one.Consider an operator L ∈ K [ x ][ ∂ ] in an arbitrary Ore algebra, and let p | lc ∂ ( L )be a factor of its leading coefficient. Assume that this factor is removable at order n .Our goal is to show that for almost all operators A ∈ K [ ∂ ] of order n with constantcoefficients the operator lclm( L, A ) is p -removed.One way of computing the least common left multiple of two operators L, A ∈ K [ x ][ ∂ ] with deg ∂ ( L ) = r and deg ∂ ( A ) = n is as follows. Make an ansatz withundetermined coefficients u , . . . , u n , v , . . . , v r and compare coefficients of ∂ i ( i =0 , . . . , n + r ) in the equation( u + · · · + u n − ∂ n − + u n ∂ n ) L = ( v + · · · + v r − ∂ r − + v r ∂ r ) A. This leads to a system of homogeneous linear equations over K ( x ) for the unde-termined coefficients, which has more variables than equations and therefore musthave a nontrivial solution. For each solution, the operator on either side of theequation is a common left multiple of L and A .For most choices of A the solution space will have dimension 1, and in this case,for every nontrivial solution we have u n = 0. In particular the least common leftmultiple M = lclm( L, A ) has then order r + n . The singularities of M are then theroots of σ n (lc ∂ ( L )) plus the roots u n minus the common roots of u , . . . , u n , whichare cancelled out by convention. It is not obvious at this point why removablefactors should appear among the common factors of u , . . . , u n . To see that theysystematically do, consider a p -removing operator P ∈ K ( x )[ ∂ ] of order n , andobserve that the operators 1 , ∂, . . . , ∂ n − , ∂ n generate the same K ( x )-vector spaceas 1 , ∂, . . . , ∂ n − , P . If we use the latter basis in the ansatz for the lclm, i.e., docoefficient comparison in( u + · · · + u n − ∂ n − + u n P ) L = ( v + · · · + v r − ∂ r − + v r ∂ r ) A, then every nontrivial solution vector ( u , . . . , u n , v , . . . , v r ) of the resulting linearsystem gives rise to a common left multiple of L and A in K [ x ][ ∂ ] whose singularitiesare the roots of lc ∂ ( P L ) = σ n ( p lc ∂ ( L )) plus the roots of u n minus the commonroots of u , . . . , u n . This argument shows that the removable factor p will havedisappeared in the lclm unless it is reintroduced by u n . The main technical difficultyto be addressed in the following is to show that this can happen only for very specialchoices of A . For the proof of this result we need the following lemma. Lemma 5.
Let n, m ∈ N , let v , . . . , v n ∈ K n + m be linearly independent over K ,and let w , . . . , w m ∈ K [ x , . . . , x n ] n + m be defined by w = ( x , . . . , x n , , , . . . , w = (0 , x , . . . , x n , , , . . . , ... w m = (0 , . . . , , x , . . . , x n , . Then ∆ := det( w , . . . , w m , v , . . . , v n ) is a nonzero polynomial in K [ x , . . . , x n ] . ESINGULARIZATION OF ORE OPERATORS 7
Proof.
Simultaneous induction on n and m : We show that the lemma holds for( n, m ) if it holds for ( n − , m ) and for ( n, m − n = 1, m arbitrary, andalso for n arbitrary, m = 1.Now let ( n, m ) ∈ N with n ≥ , m ≥ v , . . . , v n ∈ K n + m belinearly independent. Write v i = ( v ,i , . . . , v n + m,i ) for the coefficients.Case 1. v , = v , = · · · = v ,n = 0. In this case, the vectors ¯ v i ∈ K n +( m − obtained from the v i by chopping the first coordinate must be linearly independent.By expanding along the first row, we have∆ = x det( ¯ w , . . . , ¯ w m , ¯ v , . . . , ¯ v n ) . The determinant on the right is nonzero by applying the lemma with n and m − v ,j is nonzero, then we may assume withoutloss of generality that v , = 1 and v , = v , = · · · = v ,n = 0, by perform-ing suitable column operations on ( v , . . . , v n ) ∈ K ( n + m ) × n . Then the vectors¯ v , . . . , ¯ v n ∈ K ( n − m obtained from the v i by chopping the first coordinate arelinearly independent. Expanding along the first row, we now have∆ = x [[poly]] + v , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x · · · v , · · · v ,n x x . . . . . . ... ... ...... . . . . . . . . . 0 ... ...... . . . . . . x ... ... x n . . . x ... ...1 . . . x ... ...0 . . . . . . ... ... ...... . . . . . . . . . ... ... ...... . . . . . . x n ... ...0 · · · · · · v n + m, · · · v n + m,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By setting x = 0, the first term on the right hand side disappears, and so do theentries x in the determinant of the second term. By induction hypothesis, thedeterminant on the right with x set to zero is a nonzero polynomial in x , . . . , x n .Since also v , = 0, the whole right hand side is nonzero for x = 0. Consequently,when x is not set to zero, it cannot be the zero polynomial. (cid:3) Theorem 6 (Main result) . Let K [ x ][ ∂ ] be an Ore algebra, let L ∈ K [ x ][ ∂ ] be anoperator of order r , and let n ∈ N . Let p ∈ K [ x ][ ∂ ] be an irreducible polynomialwhich appears with multiplicity e in lc ∂ ( L ) and let k ≤ e be maximal such that p k is removable from L at order n . Let A = a + a ∂ + · · · + a n − ∂ n − + ∂ n in K [ a , . . . , a n − ][ ∂ ] , where a , . . . , a n − are new constants, algebraically independentover K . Then the multiplicity of σ n ( p ) in lc ∂ (lclm( L, A )) is e − k .Proof. Let P , . . . , P n ∈ K ( x )[ ∂ ] be such that each P i has order i and removes from L all the factors of lc ∂ ( L ) that can possibly be removed by an operator of order i . SHAOSHI CHEN, MANUEL KAUERS, AND MICHAEL F. SINGER
Such operators exist by Lemma 4. Consider an ansatz u P L + u P L + · · · + u n P n L = v A + v ∂A + · · · + v r ∂ r A with unknown u i , v j ∈ K [ a , . . . , a n − ][ x ]. Compare coefficients with respect topowers of ∂ on both sides and solve the resulting linear system. This gives apolynomial solution vector with u n = det (cid:0) [ P L ] , [ P L ] , · · · [ P n − L ] , [ A ] , [ ∂A ] , · · · , [ ∂ r − A ] (cid:1) , where the notation [ U ] refers to the coefficient vector of the operator U (paddedwith zeros, if necessary, to dimension r + n ).If σ n ( p ) | u n , then the columns of the determinant are linearly dependent whenviewed as elements of F [ a , . . . , a n − ] with F = K [ x ] / h σ n ( p ) i . Then Lemma 5with F in place of K implies that already [ P L ] , . . . , [ P n − L ] are linearly dependentmodulo σ n ( p ). In other words, there are polynomials u , . . . , u n − ∈ K [ x ] of degree < deg( p ), not all zero, such that the linear combination u P L + · · · + u n − P n − L hascontent σ n ( p ). If d is maximal such that u d = 0, then this means that σ d ( p ) ( u P + · · · + u d P d ) is an operator of order d which removes from L one factor σ n − d ( p )more than P d does, in contradiction to the assumption that P d removes as muchas possible. (cid:3) The theorem continues to hold when the indeterminates a , . . . , a n − are replacedby values in K which do not form a point on the zero set of the determinant poly-nomial u n mod σ n ( p ), as discussed in the proof. As this is not the zero polynomialand we assume throughout that K has characteristic zero, it follows that almostall choices of A ∈ K [ ∂ ] will successfully remove all the factors of lc ∂ ( L ) that areremovable at order deg ∂ ( A ).The theorem thus gives rise to the following very simple probabilistic algorithmfor removing, with high probability, as many factors as possible from a given oper-ator L ∈ K [ x ][ ∂ ] at a given order n : • Pick an operator A ∈ K [ ∂ ] of order n at random. • Return lclm(
L, A ).This is a Monte Carlo algorithm: it always terminates but with low probabilitymay return an incorrect answer. For a Las Vegas algorithm (low probability of notterminating but every answer is guaranteed to be correct), inspect the multiplier u n which appears during the construction of the lclm: if it is coprime with σ n (lc ∂ ( L )),then no removed singularities get mistakenly re-introduced and the result is there-fore correct. Otherwise, try again. For a deterministic algorithm, don’t take theoperators A at random but from some enumeration of K [ ∂ ] which is chosen in sucha way that the Zariski closure of the set of the corresponding coefficient vectors isall of K n .The Monte Carlo version of the algorithm is included in the new ore_algebra package for Sage [12], and works very efficiently thanks to the efficient implemen-tation of least common left multiples also available in this package. This packagehas been used for the calculations in the following concluding examples. The com-putation time for all these examples is negligible. ESINGULARIZATION OF ORE OPERATORS 9
Example 7. (1)
For L ∈ Q [ x ][ D ] from Example 1 and the “randomly chosen”operator A = D + D + 1 we have lclm( L, A ) = ( x − x + 6 x − x + x + 6 x − D − (2 x − x + 15 x − x + 3 x − D − ( x − x + 6 x − x + x + 6 x − D + (2 x − x + 15 x − x + 3 x − . This is not the same result as in Example 1, but it does have the requiredproperty x ∤ lc ∂ (lclm( L, A )) . (2) This is an example for the recurrence case. Let L = 2( x + 3) (59 x + 94) S − (2301 x + 15171 x + 32696 x + 22876) S − x + 330 x + 600 x + 359) S − (59 x + 153)( x + 1) . Among the factors of ( x + 3) and (59 x + 94) of the leading coefficient, thelatter is removable at order 1 and the former is not removable. Accordingly,for the “randomly chosen” operator A = S − we have lclm( L, A ) = 2( x + 4) (8909 x + 57087 x + 119629 x + 81711) S + ( · · · ) S + ( · · · ) S + ( · · · ) S + ( · · · ) , where ( · · · ) stands for some other polynomials. Note that the leading coef-ficient is coprime to σ (59 x + 94) = 59 x + 153 . (3) As an example for partial desingularization, consider the operator L = x D − x D − xD + 10 ∈ Q [ x ][ D ] . Of the three copies of x in theleading coefficient, one is removable at order 2, another one at order 4, andthe third is not removable. In perfect accordance, we find for example lc ∂ (lclm( L, D + 2)) = x (4 x + 6 x − x − , lc ∂ (lclm( L, D + 1)) = x ( x + 10 x + 40 x + 80) , lc ∂ (lclm( L, D + 3 D − x ( x − x + · · · + 2160 x + 1920) , lc ∂ (lclm( L, D − D + 1)) = x ( x − x + 120 x − x − , lc ∂ (lclm( L, D + D − x ( x − x + · · · + 25600 x − . (4) There are unlucky choices for A . For example, consider L = ( x − x − x − S − (3 x − x − x + 291) S + 2( x − x − ∈ Q [ x ][ S ] . The factor x − is removable, as can be seen, for example, from the factthat lc ∂ (lclm( L, S − x − x − is coprime to σ ( x −
7) = x − .However, if we take A = S − , then lclm( L, A ) = 4( x − x − x − S − ( x − − x + 105 x ) S + ( x − − x + 175 x ) S − x − x − x − , which has x − in the leading coefficient. (It is irrelevant that also x − appears as a factor.) (5) Finally, as an example with an unusual Ore algebra, consider Q [ x ][ ∂ ] with σ : Q [ x ] → Q [ x ] defined by σ ( x ) = x and δ : Q [ x ] → Q [ x ] defined by δ ( x ) = 1 − x . Let L = (2 x + 1) ∂ + ( x + 3 x − ∂ − (2 x + 2 x + x + 1) . The factor x + 1 is removable at order . For example, for A = ∂ − wefind that lclm( L, A ) equals (2 x + 4 x + 4 x − ∂ − (2 x − x − x − x + x + 5) ∂ − (2 x + 4 x + 6 x + 4 x + 2 x + 3 x + 2 x + 3 x + 3 x − ∂ + (2 x + 4 x + 6 x + 6 x + 2 x + 2 x − x − x + 4) . As expected, the leading coefficient does not contain the factor σ (lc ∂ ( L )) =2 x + 1 . References [1] Sergei A. Abramov, Moulay A. Barkatou, and Mark van Hoeij. Apparent singularities oflinear difference equations with polynomial coefficients.
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ESINGULARIZATION OF ORE OPERATORS 11
Shaoshi Chen, KLMM, AMSS, Chinese Academy of Sciences, 100190 Beijing, China
E-mail address : [email protected] Manuel Kauers, Research Institute for Symbolic Computation, J. Kepler UniversityLinz, Austria
E-mail address : [email protected] Michael F. Singer, Department of Mathematics, North Carolina State University,Raleigh, NC, USA
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