aa r X i v : . [ c s . S C ] J a n Contraction of Ore Ideals with Applications
Yi Zhang ∗ Institute for Algebra, Johannes Kepler University, Linz A-4040, AustriaKLMM, AMSS, Chinese Academy of Sciences, Beijing 100190, [email protected]
ABSTRACT
Ore operators form a common algebraic abstraction of lin-ear ordinary differential and recurrence equations. Given anOre operator L with polynomial coefficients in x , it gener-ates a left ideal I in the Ore algebra over the field k ( x ) ofrational functions. We present an algorithm for computinga basis of the contraction ideal of I in the Ore algebra overthe ring R [ x ] of polynomials, where R may be either k or adomain with k as its fraction field. This algorithm is basedon recent work on desingularization for Ore operators byChen, Jaroschek, Kauers and Singer. Using a basis of thecontraction ideal, we compute a completely desingularizedoperator for L whose leading coefficient not only has mini-mal degree in x but also has minimal content. Completelydesingularized operators have interesting applications suchas certifying integer sequences and checking special cases ofa conjecture of Krattenthaler. Categories and Subject Descriptors
I.1.2 [
Computing Methodologies ]: Symbolic and Alge-braic Manipulation—
Algorithms
General Terms
Algorithms, Theory
Keywords
Ore Algebra, Desingularization, Contraction, Syzygy
1. INTRODUCTION
There are various reasons why linear differential equationsare easier than non-linear ones. One is of course that thesolutions of linear differential equations form a vector spaceover the underlying field of constants. Another important ∗ Supported by the Austrian Science Fund (FWF) grantsY464-N18, NSFC grants (91118001, 60821002/F02) and a973 project (2011CB302401).
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ISSAC’16,
July 20–22, 2016, Waterloo, CanadaCopyright is held by the owner/author(s). Publication rights licensed to ACM.. feature concerns the singularities. While for a nonlinear dif-ferential equation the location of the singularity may dependcontinuously on the initial value, this is not possible for lin-ear equations. Instead, a solution f of a differential equation a ( x ) f ( x ) + · · · + a r ( x ) f ( r ) ( x ) = 0 , where a , . . . , a r are some analytic functions, can only havesingularities at points ξ ∈ C with a r ( ξ ) = 0.In this article, we consider the case where a , . . . , a r arepolynomials. In this case, a r can have only finitely manyroots. We shall also consider the case of recurrence equations a ( n ) f ( n ) + · · · + a r ( n ) f ( n + r ) = 0 , where again there is a strong connection between the rootsof a r and the singularities of a solution.While every singularity of a solution leaves a trace in theleading coefficient of an equation, the converse is not true. Ingeneral, the leading coefficient a r may have roots at a pointwhere no solution is singular. Such points are called appar-ent singularities, and it is sometimes of interest to identifythem. One technique for doing so is called desingularization.As an example, consider the recurrence operator L = (1 + 16 n ) ∂ − (224 + 512 n ) ∂ − (1 + n )(17 + 16 n ) , which is taken from [1, Section 4.1]. Here, ∂ denotes theshift operator f ( n ) f ( n + 1). For any choice of two initialvalues u , u ∈ Q , there is a unique sequence u : N → Q with u (0) = u , u (1) = u and L applied to u gives thezero sequence. A priori, it is not obvious whether or not u is actually an integer sequence, if we choose u , u from Z ,because the calculation of the ( n +2)nd term from the earlierterms via the recurrence encoded by L requires a divisionby (1 + 16 n ) , which could introduce fractions. In order toshow that this division never introduces a denominator, theauthors of [1] note that every solution of L is also a solutionof its left multiple T = (cid:18) n ) ∂ + (23 + 16 n )(25 + 16 n )(17 + 16 n ) (cid:19) L = 64 ∂ + (16 n + 23)(16 n − ∂ − (576 n + 928) ∂ − (16 n + 23)(16 n + 25)( n + 1) . The operator T has the interesting property that the factor(1 + 16 n ) has been “removed” from the leading coefficient.This is, however, not quite enough to complete the proof,because now a denominator could still arise from the divisionby 64 at each calculation of a new term via T . To completethe proof, the authors show that the potential denominatorsntroduced by (1 + 16 n ) and by 64, respectively, are inconflict with each other, and therefore no such denominatorscan occur at all.The process of obtaining the operator T from L is calleddesingularization, because there is a polynomial factor in theleading coefficient of L which does not appear in the leadingcoefficient of T . In the example above, the price to be paidfor the desingularization was a new constant factor 64 whichappears in the leading coefficient of T but not in the orig-inal leading coefficient of L . Desingularization algorithmsin the literature [2, 1, 3, 6, 7] care only about the removalof polynomial factors without introducing new polynomialfactors, but they do not consider the possible introductionof new constant factors. A contribution of the present pa-per is a desingularization algorithm which minimizes, in asense, also any constant factors introduced during the desin-gularization. For example, for the operator L above, ouralgorithm finds the alternative desingularization˜ T = 1 ∂ − (cid:0) n + 3976 n + 420 n + 15 (cid:1) ∂ + (cid:0) n + 11871 n + 2782 (cid:1) ∂ + 6272 n +22792 n + 30380 n + 17459 n + 3599 , (1)which immediately certifies the integrality of its solutions.In more algebraic terms, we consider the following prob-lem. Given an operator L ∈ Z [ x ][ ∂ ], where Z [ x ][ ∂ ] is anOre algebra (see Section 2 for definitions), we consider theleft ideal h L i = Q ( x )[ ∂ ] L generated by L in the extendedalgebra Q ( x )[ ∂ ]. The contraction of h L i to Z [ x ][ ∂ ] is de-fined as Cont( L ) := h L i ∩ Z [ x ][ ∂ ]. This is a left ideal of Z [ x ][ ∂ ] which contains Z [ x ][ ∂ ] L , but in general more oper-ators. Our goal is to compute a Z [ x ][ ∂ ]-basis of Cont( L ).In the example above, such a basis is given by { L, ˜ T } (seeExample 4.8). The traditional desingularization problemcorresponds to computing a basis of the Q [ x ][ ∂ ]-left ideal h L i ∩ Q [ x ][ ∂ ].The contraction problem for Ore algebras Q [ x ][ ∂ ] was pro-posed by Chyzak and Salvy [9, Section 4.3]. For the anal-ogous problem in commutative polynomial rings, there isa standard solution via Gr¨obner bases [4, Section 8.7]. Itreduces the contraction problem to a saturation problem.This reduction also works for the differential case, but inthat case it is not so helpful because it is less obvious howto solve the saturation problem. A solution was proposed byTsai [21], which involves homological algebra and D-modulestheory. Our work is based on desingularization for Ore oper-ators in [6, 7]. In particular, the p -removing operator in [7,Lemma 4] provides us with a key to determine contractionideals. The algorithm developed in this paper is consider-ably simpler than Tsai’s and at the same time it applies toarbitrary Ore algebras rather than only the differential case.Moreover, we compute a completely desingularized operatorin a contraction ideal, which has minimal leading coefficientin terms of both degree and content.The rest of this paper is organized as follows. In Section 2,we describe Ore polynomial rings over a principal ideal do-main, and extend the notion of p -removed operators to them.The notion of desingularized operators is defined and con-nected with contraction ideals in Section 3. We determinea contraction ideal in Section 4, and compute completelydesingularized operators in Section 5.
2. PRELIMINARIES
This section is divided into three parts. First, we describeOre algebras that are used in the paper. Second, we extendthe notion of p -removed operators in [6, 7]. At last, we makesome remarks on Gr¨obner basis computation over a principalideal domain. Throughout the paper, we let R be a principal ideal do-main. For instance, R can be the ring of integers or that ofunivariate polynomials over a field. We consider the Ore al-gebra R [ x ][ ∂ ; σ, δ ], where σ : R [ x ] → R [ x ] is a ring automor-phism that leaves the elements of R fixed, and δ : R [ x ] → R [ x ] is a σ -derivation, i.e. an R -linear map satisfying theskew Leibniz rule δ ( fg ) = σ ( f ) δ ( g ) + δ ( f ) g for f, g ∈ R [ x ] . The addition in R [ x ][ ∂ ] is coefficient-wise and the multi-plication is defined by associativity via the commutationrule ∂p = σ ( p ) ∂ + δ ( p ) for p ∈ R [ x ] . Given L ∈ R [ x ][ ∂ ], we can uniquely write it as L = ℓ r ∂ r + ℓ r − ∂ r − + · · · + ℓ with ℓ , . . . , ℓ r ∈ R [ x ] and ℓ r = 0. We call r the order and ℓ r the leading coefficient of L . They are denoted by deg ∂ ( L )and lc ∂ ( L ), respectively. The ring R [ x ][ ∂ ; σ, δ ] is abbreviatedas R [ x ][ ∂ ] when σ and δ are clear from the context. For asubset S of R [ x ][ ∂ ], the left ideal generated by S is denotedby R [ x ][ ∂ ] · S .Let Q R be the quotient field of R . Then Q R ( x )[ ∂ ] is anOre algebra containing R [ x ][ ∂ ]. For L ∈ R [ x ][ ∂ ], we definethe contraction ideal of L to be Q R ( x )[ ∂ ] L ∩ R [ x ][ ∂ ] anddenote it by Cont( L ). We generalize some terminologies given in [6, 7] by replac-ing the coefficient ring K [ x ] with R [ x ], where K is a field. Definition 2.1.
Let L ∈ R [ x ][ ∂ ] with positive order, and p be a divisor of lc ∂ ( L ) in R [ x ] .(i) We say that p is removable from L at order n if thereexist P ∈ Q R ( x )[ ∂ ] with order k , and w, v ∈ R [ x ] with gcd( p, w ) = 1 in R [ x ] such that P L ∈ R [ x ][ ∂ ] and σ − k (lc ∂ ( P L )) = wvp lc ∂ ( L ) . We call P a p -removing operator for L over R [ x ] , and P L the corresponding p -removed operator .(ii) p is simply called removable from L if it is removableat order k for some k ∈ N . Otherwise, p is called non-removable from L . Note that every p -removed operator lies in Cont( L ). Example 2.2.
In the example of Section 1, (1 + 16 n ) isremovable from L at order 1. And T is the corresponding (1 + 16 n ) -removed operator for L . Example 2.3.
In the differential Ore algebra Z [ x ][ ∂ ] , where ∂x = x∂ +1 , let L = x ( x − ∂ − Then (1 − x ) ∂ − ∂ = (cid:0) x ∂ (cid:1) L is an x -removed operator for L (see [6, Example 3]). The authors of [6] provide a convenient form of p -removingoperators over K [ x ] in order to get the order bound. Wederive a similar form over R [ x ] and use it in Section 5. emma 2.4. Let L ∈ R [ x ][ ∂ ] with positive order. Assumethat p ∈ R [ x ] is removable from L at order k . Then thereexists a p -removing operator for L over R [ x ] in the form p σ k ( p ) d + p σ k ( p ) d ∂ + · · · + p k σ k ( p ) d k ∂ k , where p i belongs to R [ x ] , gcd( p i , σ k ( p )) = 1 in R [ x ] , i = 0 , , . . . , k , and d k ≥ .Proof. By Definition 2.1, lc ∂ ( P ) = σ k ( w/ ( vp )) for some w, v in R [ x ] with gcd( w, p ) = 1. Then we can write a p -removingoperator for L over R [ x ] in the form P = p q σ k ( p ) d + p q σ k ( p ) d ∂ + · · · + p k q k σ k ( p ) d k ∂ k , where p i , q i ∈ R [ x ], gcd( p i q i , σ k ( p )) = 1 in R [ x ], i = 0 , . . . , k , d k ≥
1. Let ˜ P = (cid:16)Q ki =0 q i (cid:17) P , ˜ p i = p i (cid:16)Q ki =0 q i (cid:17) /q i , i =0 , . . . , k . Then˜ P = ˜ p σ k ( p ) d + ˜ p σ k ( p ) d ∂ + · · · + ˜ p k σ k ( p ) d k ∂ k , where gcd(˜ p i , σ k ( p )) = 1 in R [ x ], i = 0 , . . . , k . Moreover, σ − k (lc ∂ ( ˜ P L )) = σ − k (˜ p k ) p d k lc ∂ ( L ) . By Definition 2.1, ˜ P is a p -removing operator for L over R [ x ]with the required form. In Sections 4 and 5, we will make essential use of Gr¨obnerbases in R [ x ][ ∂ ]. When R = k [ t ] with k being a field, the no-tion of Gr¨obner bases and Buchberger’s algorithm are avail-able [14]. In our case, σ is an R -automorphism of R [ x ], whichimplies that σ ( x ) = ax + b where a, b are in R and a is a unit.Assume that ≺ is a term order on (cid:8) x i ∂ j | i, j ∈ N (cid:9) . Let P be a nonzero operator in R [ x ][ ∂ ], and c be the head coeffi-cient of P with respect to ≺ . By the commutation rule, ∂ i P has head coefficient ca i , which is associated to c , because a i is a unit. This observation enables us to extend the notionof Gr¨obner bases and Buchberger’s algorithm in [4, 19] toOre case in a straightforward way.
3. DESINGULARIZATION ANDCONTRACTION
In this section, we define the notion of desingularized op-erators, and connect it with contraction ideals. As a matterof notation, for an operator L ∈ R [ x ][ ∂ ], we set M k ( L ) = { P ∈ Cont( L ) | deg ∂ ( P ) ≤ k } . Note that M k ( L ) is a left submodule of Cont( L ) over R [ x ].We call it the k th submodule of Cont( L ). When the opera-tor L is clear from context, M k ( L ) is simply denoted by M k . Definition 3.1.
Let L ∈ R [ x ][ ∂ ] with order r > , and lc ∂ ( L ) = cp e · · · p e m m , (2) where c ∈ R and p , . . . , p m ∈ R [ x ] \ R are irreducible andpairwise coprime. An operator T ∈ R [ x ][ ∂ ] of order k iscalled a desingularized operator for L if T ∈ Cont( L ) and σ r − k (lc ∂ ( T )) = abp k · · · p k m m lc ∂ ( L ) , (3) where a, b ∈ R , and p d i i is non-removable from L for each d i > k i , i = 1 . . . m . Desingularized operators always exist by [7, Lemma 4].
Lemma 3.2.
Let L ∈ R [ x ][ ∂ ] be of order r > , and k ∈ N with k ≥ r . Assume that T is a desingularized operator for L and deg ∂ ( T ) = k .(i) deg x (lc ∂ ( T )) = min { deg x (lc ∂ ( Q )) | Q ∈ M k ( L ) \ { }} . (ii) ∂ i T is a desingularized operator for L for each i ∈ N .(iii) Set lc ∂ ( T ) = ag , where a ∈ R and g ∈ R [ x ] is primi-tive. Then, for all F ∈ Cont( L ) of order j with j ≥ k , σ j − k ( g ) divides lc ∂ ( F ) in R [ x ] .Proof. (i) Let t = lc ∂ ( T ) and d = min { deg x (lc ∂ ( Q )) | Q ∈ M k ( L ) \ { }} . Suppose that d < deg x ( t ). Then there exists Q ∈ Cont( L )with deg x (lc ∂ ( Q )) = d . Without loss of generality, we canassume that deg ∂ ( Q ) = k , because the leading coefficientsof Q and ∂ i Q are of the same degree for all i ∈ N .By pseudo-division in R [ x ], we have that st = q lc ∂ ( Q ) + h for some s ∈ R \{ } , q, h ∈ R [ x ], and h = 0 or deg x ( h ) < d .If h were nonzero, then sT − qQ would be a nonzero op-erator of order k in Cont( L ) whose leading coefficient is ofdegree less than d , a contradiction. Thus, st = q lc ∂ ( Q ). Inparticular, deg x ( q ) is positive, as d < deg x ( t ). It followsfrom (3) that σ r − k (lc ∂ ( Q )) = σ r − k (cid:18) stq (cid:19) = saσ r − k ( q ) bp k · · · p k m m lc ∂ ( L ) , which belongs to R [ x ]. Hence, σ r − k ( q ) divides lc ∂ ( L ) in R [ x ].Consequently, there exists i ∈ { . . . m } such that p i di-vides σ r − k ( q ) in R [ x ]. This implies that p k i +1 is removablefrom L , a contradiction.(ii) It is immediate from Definition 3.1.(iii) Let lc ∂ ( F ) = uf , where u ∈ R and f is primitivein R [ x ]. By (ii), ∂ j − k T is a desingularized operator whoseleading coefficient equals aσ j − k ( g ). A similar argument usedin the proof of the first assertion implies that vf = pσ j − k ( g ) for some v ∈ R \ { } and p ∈ R [ x ].By Gauss’s Lemma in R [ x ] , σ j − k ( g ) | f .We describe a relation between desingularized operatorsand contraction ideals. Let I be a left ideal in R [ x ][ ∂ ],and a ∈ R . The saturation of I with respect to a is de-fined to be I : a ∞ = n P ∈ R [ x ][ ∂ ] | a i P ∈ I for some i ∈ N o . Since a is a constant with respect to σ and δ , the satura-tion I : a ∞ is a left ideal. Theorem 3.3.
Let L ∈ R [ x ][ ∂ ] with order r > . Assumethat T is a desingularized operator for L . Let lc ∂ ( T )= ag ,where a ∈ R and g is primitive in R [ x ] . If T belongs to M k for some k ∈ N , then Cont( L ) = ( R [ x ][ ∂ ] · M k ) : a ∞ . roof. By Lemma 3.2 (ii), we may assume that the orderof T is equal to k . Set J = ( R [ x ][ ∂ ] · M k ) : a ∞ .First, assume that F ∈ J . Then there exists j ∈ N suchthat a j F ∈ R [ x ][ ∂ ] · M k . It follows that F ∈ Q R ( x )[ ∂ ] L .Thus, F ∈ Cont( L ) by definition.Next, note that Cont( L ) = ∪ ∞ i = r M i and that M i ⊆ M i +1 .It suffices to show M i ⊆ J for all i ≥ k . We proceed byinduction on i .For i = k . M k ⊆ J by definition.Suppose that the claim holds for i . For any F ∈ M i +1 \ M i ,deg ∂ ( F ) = i + 1. By Lemma 3.2 (iii), lc ∂ ( F ) = pσ i +1 − k ( g )for some p ∈ R [ x ]. Then lc ∂ ( aF ) = lc ∂ ( p∂ i +1 − k T ). Itfollows that aF − p∂ i +1 − k T ∈ M i . Since p∂ i +1 − k T ∈ R [ x ][ ∂ ] · M k ⊆ R [ x ][ ∂ ] · M i , we have that aF ∈ R [ x ][ ∂ ] · M i . On the other hand, M i ⊂ J by the induction hypothesis. Thus, aF ∈ R [ x ][ ∂ ] · J , whichis J . Accordingly, F ∈ J by the definition of saturation.
4. AN ALGORITHM FOR COMPUTINGCONTRACTION IDEALS
First, we translate an upper bound for the order of a desin-gularized operator over Q R [ x ] to R [ x ]. Lemma 4.1.
Let L ∈ R [ x ][ ∂ ] with order r > , and p ∈ R [ x ] be a primitive polynomial and a divisor of lc ∂ ( L ) . Assumethat there exists a p -removing operator for L over Q R [ x ] .Then there exists p -removing operator for L over R [ x ] withorder r .Proof. Assume that P ∈ Q R ( x )[ ∂ ] is a p -removing operatorfor L over Q R [ x ]. Let P be of order k . Then P L is of theform
P L = a k + r b k + r ∂ k + r + · · · + a b ∂ + a b for some a i ∈ R [ x ] , b i ∈ R , i = 0 , . . . , k + r . Moreover, σ − k (lc ∂ ( P L )) = wvp lc ∂ ( L ) , where w, v ∈ R [ x ] with gcd( w, p ) = 1.Let b = lcm( b , b , . . . , b k + r ) in R and P ′ = bP . Then P ′ L ∈ R [ x ][ ∂ ] and σ − k (lc ∂ ( P L )) = bwvp lc ∂ ( L ) . Since p is primitive, we have that gcd( bw, p ) = 1 in R [ x ].Thus, P ′ is a p -removing operator of order k .By the above lemma, an order bound for a p -removingoperator over Q R [ x ] is also an order bound for a p -removingoperator over R [ x ]. The former has been well-studied inthe literature. Order bounds for differential operators aregiven in [21, Algorithm 3.4] and [12, Lemma 4.3.12]. Thosefor recurrence operators are given in [6, Lemma 4] and [12,Lemma 4.3.3]. Desingularized operators are p -removing op-erators. So we can find order bounds for them.By Theorem 3.3, determining a contraction ideal amountsto finding a desingularized operator T and an R [ x ]-basisof M k , where k is an upper bound for the order of T .Next, we present an algorithm for constructing a basisfor M k ( L ), where L is a nonzero operator in R [ x ][ ∂ ] and k isa positive integer. To this end, we embed M k into the freemodule R [ x ] k +1 over R [ x ]. Let us recall the right division in Q R ( x )[ ∂ ] (see [5, Section3]). For F, G ∈ Q R ( x )[ ∂ ] with G = 0, there exist uniqueelements Q, R ∈ Q R ( x )[ ∂ ] with deg ∂ ( R ) < deg ∂ ( G ) suchthat F = QG + R . We call R the right-remainder of F by G and denote it by rrem( F, G ).Let F ∈ R [ x ][ ∂ ] with order k . Then F ∈ M k if and onlyif F ∈ Q R [ x ][ ∂ ] L , which is equivalent to rrem( F, L ) = 0.The latter gives rise to a linear system( z k , . . . , z ) A = , (4)where A is a ( k +1) × r matrix over Q R ( x ). Clearing denom-inators of the elements in A , we may further assume that A is a matrix over R [ x ]. We are concerned with the solutionsof (4) over R [ x ]. Set N k = n ( f k , . . . , f ) ∈ R [ x ] k +1 | ( f k , . . . , f ) A = o . We call N k the module of syzygies defined by (4). Theorem 4.2.
With the notation just specified, we have φ : M k −→ N k P ki =0 f i ∂ i ( f k , . . . , f ) is a module isomorphism over R [ x ] .Proof. Let F = P ki =0 f i ∂ i ∈ R [ x ]. If F belongs to M k ,then rrem( F, L ) = 0, that is, ( f k , . . . , f ) belongs to N k .Hence, φ is a well-defined map.Clearly, φ is injective. For ( f k , . . . , f ) ∈ N k , we have( f k , . . . , f ) A = . As (4) is induced by right division rrem (
F, L ) = 0, F belongsto M k . So φ is surjective. It is straightforward to see that φ is an R [ x ]-module homomorphism.By Theorem 4.2, M k is finitely generated over R [ x ]. Tofind an R [ x ]-basis of M k , it suffices to compute a basis of themodule of syzygies defined by (4). When R is a field, we justneed to solve (4) over a principal ideal domain [20, Chapter5]. When R is the ring of integers or the ring of univari-ate polynomials over a field, we can use Gr¨obner bases ofpolynomials over a principal domain [13, 11]. Their imple-mentations are available in computer algebra systems suchas Macaulay2 and
Singular .We now consider how to construct a desingularized oper-ator for L . For k ∈ Z + , we define I k = n [ ∂ k ] P | P ∈ M k o ∪ { } , where [ ∂ k ] P stands for the coefficient of ∂ k in P . It is clearthat I k is an ideal of R [ x ]. We call I k the k th coefficientideal of Cont( L ). By the commutation rule, σ ( I k ) ⊂ I k +1 . Lemma 4.3.
Let L ∈ R [ x ][ ∂ ] be of positive order. If the k thsubmodule M k of Cont( L ) has a basis { B , . . . , B ℓ } over R [ x ] ,then the k th coefficient ideal I k = D [ ∂ k ] B , . . . , [ ∂ k ] B ℓ E . Proof.
Obviously, h [ ∂ k ] B , . . . , [ ∂ k ] B ℓ i ⊆ I k . Let f ∈ I k .Then f = lc ∂ ( F ) for some F ∈ M k with deg ∂ ( F ) = k .Since M k is generated by { B , . . . , B ℓ } over R [ x ], F = h B + · · · + h ℓ B ℓ , where h , . . . , h ℓ ∈ R [ x ].hus, f = h (cid:0) [ ∂ k ] B (cid:1) + · · · + h ℓ (cid:0) [ ∂ k ] B ℓ (cid:1) . Consequently, f ∈ h [ ∂ k ] B , . . . , [ ∂ k ] B ℓ i . Theorem 4.4.
Let L ∈ R [ x ][ ∂ ] be of positive order. Assumethat the k th submodule M k of Cont( L ) contains a desingu-larized operator for L . Let s be a nonzero element in the k thcoefficient ideal with minimal degree. Then an operator S in M k with leading coefficient s is a desingularized operator.Proof. Let T be a desingularized operator in M k . By Lemma 3.2 (ii),we may assume that the order of T is equal to k . Let t =lc ∂ ( T ). Then deg( t ) = deg( s ) by Lemma 3.2 (i). Let u be the leading coefficient of s with respect to x and v bethat of t . Then ut − vs is zero. Otherwise, uT − vS wouldbe an operator of order k whose leading coefficient with re-spect to ∂ has degree lower than deg x ( t ), a contradiction toLemma 3.2 (i). It follows from ut = vs and Definition 3.1that S is also a desingularized operator.Let L be an operator in R [ x ][ ∂ ] of positive order. Wecan compute a basis { B , . . . , B ℓ } for the k th submoduleof Cont( L ) by Theorem 4.2, where k is an upper bound onthe order of a desingularized operator for L . By Lemma 4.3,we can obtain a basis { b , . . . , b ℓ } for the k th coefficientideal I k of Cont( L ). Let ¯ I k be the extension ideal of I k in Q R [ x ]. Using the extended Euclidean algorithm in Q R [ x ]and clearing denominators, we find cofactors c , . . . , c ℓ ∈ R [ x ] and s ∈ R [ x ] such that¯ I k = h s i and c b + · · · + c ℓ b ℓ = s. It follows from Theorem 4.4 that T = c B + · · · + c ℓ B ℓ is adesingularized operator for L with lc ∂ ( T ) = s . Let a be thecontent of s with respect to x . By Theorem 3.3, Cont( L ) isthe saturation of R [ x ][ ∂ ] · M k with respect to a . Note that a belongs to R , which is contained in the center of R [ x ][ ∂ ]. Soa basis of the saturation ideal can be computed in the sameway as in the commutative case. Proposition 4.5.
Let I be a left ideal of R [ x ][ ∂ ] and c benon-zero element in R . Assume that J is a left ideal R [ x, y ][ ∂ ] · ( I ∪ { − cy } ) , where y is a new indeterminate and commutes with everyelement in R [ x ][ ∂ ] . Then I : c ∞ = J ∩ R [ x ][ ∂ ] .Proof. Since both y and c commute with ∂ , the argumentin [4, page 266, Proposition 6.37] carries over.We outline our method for determining contraction ideals. Algorithm 4.6.
Given L ∈ R [ x ][ ∂ ] , where ∂x = ( x + 1) ∂ or ∂x = x∂ + 1 , compute a basis of Cont( L ) .(1) Derive an upper bound k on the order of a desingular-ized operator for L .(2) Compute an R [ x ] -basis of M k .(3) Compute a desingularized operator T , and set a to bethe content of lc ∂ ( T ) with respect to x .(4) Compute a basis of ( R [ x ][ ∂ ] · M k ) : a ∞ . The termination of this algorithm is evident. Its correct-ness follows from Theorem 3.3. We assume that the commu-tation rule in R [ x ][ ∂ ] is either ∂x = ( x + 1) ∂ or ∂x = x∂ + 1in R [ x ][ ∂ ], because we only know order bounds for thosecases. In step 1, the order bound is derived from [6, Lemma4] and [21, Algorithm 3.4]. In step 2, we need to solve linearsystems over R [ x ] as stated in Theorem 4.2. This can bedone by Gr¨obner basis computation in a free R [ x ]-moduleof finite rank. In step 3, T is computed according to The-orem 4.4 and the extended Euclidean algorithm in Q R [ x ].The last step is carried out according to Proposition 4.5. Example 4.7.
In the shift Ore algebra Q [ t ][ n ][ ∂ ] , in whichthe commutation rule is ∂n = ( n + 1) ∂ . Consider L = ( n − n + t ) ∂ + n + t + 1 . By [6, Lemma 4], we obtain an order bound for a desin-gularized operator. Thus, M contains a desingularized op-erator for L . In step 2 of Algorithm 4.6, we find that M isgenerated by T = (2 + t ) n∂ + (4 − n + t ) ∂ − ,T = ( n − n∂ + 2( n − ∂ + 1 , where T is a desingularized operator, lc ∂ ( T ) = (2 + t ) n .Using Gr¨obner bases, Cont( L ) = ( Q [ t ][ n ][ ∂ ] · M ) : (2 + t ) ∞ is generated by { L, T } . Let us consider the example in Section 1.
Example 4.8.
In the shift Ore algebra Z [ n ][ ∂ ] , let L = (1 + 16 n ) ∂ − (224 + 512 n ) ∂ − (1 + n )(17 + 16 n ) . By [6, Lemma 4], we obtain an order bound for a desingu-larized operator. Thus, M contains a desingularized opera-tor for L . In step 2 of Algorithm 4.6, we find that M is gen-erated by { L, ˜ T } , where ˜ T is given in (1) . Note that lc ∂ ( ˜ T )=1 .Thus, ˜ T is a desingularized operator. Consequently, Cont( L ) = ( Z [ n ][ ∂ ] · { L, ˜ T } ) : 1 ∞ = Z [ n ][ ∂ ] · { L, ˜ T } . Example 4.9.
In the differential Ore algebra Z [ x ][ ∂ ] , inwhich the commutation rule is ∂x = x∂ + 1 . Consider theoperator L = x∂ − ( x + 2) ∂ + 2 ∈ Z [ x ][ ∂ ] in [3]. By [21,Algorithm 3.4], we obtain an order bound for a desingular-ized operator. Thus, M contains a desingularized operatorfor L . In step 2 of Algorithm 4.6, we find that M is gener-ated by { L, ∂L, T } , where T = ∂ − ∂ . Note that lc ∂ ( T ) = 1 .Thus, T is a desingularized operator. Consequently, Cont( L ) = ( Z [ x ][ ∂ ] · { L, ∂L, T } ) : 1 ∞ = Z [ x ][ ∂ ] · { L, T } .
5. COMPLETE DESINGULARIZATION
As seen in Section 1, the shift operator L = (1 + 16 n ) ∂ − (224 + 512 n ) ∂ − (1 + n )(17 + 16 n ) has a desingularized operator T with leading coefficient 64.But the content of lc ∂ ( L ) is 1. The redundant content 64has been removed by computing another desingularized op-erator ˜ T in (1). This enables us to see immediately that thesequence annihilated by L is an integer sequence when itsinitial values are integers.Krattenthaler proposes a conjecture in [10]: Let ( a n ) n ≥ and ( b n ) n ≥ be two P-recursive sequences over Z with lead-ing coefficients n . Then ( n ! a n b n ) n ≥ is also a P-recursiveequence over Z with leading coefficient n . To test the con-jecture for the two particular sequences, one may first com-pute an annihilator L of ( n ! a n b n ) n ≥ , and then look for anonzero operator in Cont( L ) whose leading coefficient has“minimal” content with respect to n . When the content isequal to 1, the conjecture is true for these sequences.These two observations motivate us to define the notionof completely desingularized operators. Definition 5.1.
Let L ∈ R [ x ][ ∂ ] with positive order, and Q a desingularized operator for L . Set lc ∂ ( Q ) = c g , where c isthe content of lc ∂ ( Q ) with respect to x and g the correspond-ing primitive part. We call Q a completely designularizedoperator for L if c is a divisor of the content of the leadingcoefficient of every desingularized operator for L . To see the existence of completely designularized opera-tors, we assume that L is of order r . For a desingularizedoperator T of order k , equations (2) and (3) in Definition 3.1enable us to write σ r − k (lc ∂ ( T )) = c T g, (5)where c T ∈ R and g = p e − k · · · p e m − k m s . Note that g isprimitive and independent of the choice of desingularizedoperators. Lemma 5.2.
Let L ∈ R [ x ][ ∂ ] with order r > . Set I to be the set consisting of zero and c T given in (5) for alldesingularized operators for L . Then I is an ideal of R .Proof. By Definition 3.1, the product of a nonzero elementof R and a desingularized operator for L is also a desingu-larized one. So it suffices to show that I is closed underaddition. Let T and T be two desingularized operators oforders k and k , respectively. Assume that k ≥ k . By (5), σ r − k (lc ∂ ( T )) = c g and σ r − k (lc ∂ ( T )) = c g. If c + c = 0, then there is nothing to prove. Otherwise, adirect calculation implies thatlc ∂ ( T ) = c σ k − r ( g ) and lc ∂ (cid:16) ∂ k − k T (cid:17) = c σ k − r ( g ) . Thus, T + ∂ k − k T has leading coefficient ( c + c ) σ k − r ( g ) . Accordingly, T + ∂ k − k T is a desingularized one, whichimplies that c + c belongs to I .Since R is a principal ideal domain, I in the above lemmais generated by an element c , which corresponds to a com-pletely desingularized operator.Let ≺ be a term order on (cid:8) x i ∂ j | i, j ∈ N (cid:9) . For any non-zero operator P ∈ R [ x ][ ∂ ], we define the head term of P tobe the highest term appearing in P with respect to ≺ , anddenote it by HT( P ).The next technical lemma serves as a step-stone to con-struct completely desingularized operators. Lemma 5.3.
Let L ∈ R [ x ][ ∂ ] with order r > , and k ≥ r .Then R [ x ][ ∂ ] · M k = R [ x ][ ∂ ] · M k +1 if and only if σ ( I k )= I k +1 . Proof.
Assume that σ ( I k ) = I k +1 . Since M k ⊂ M k +1 , itsuffices to prove that M k +1 ⊂ R [ x ][ ∂ ] · M k .For each T ∈ M k +1 \ M k , we have that lc ∂ ( T ) ∈ σ ( I k ).Thus, there exists F ∈ M k such that σ (lc ∂ ( F )) = lc ∂ ( T ). Inother words, T − ∂F ∈ M k . Consequently, T ∈ R [ x ][ ∂ ] · M k . Conversely, assume that R [ x ][ ∂ ] · M k +1 = R [ x ][ ∂ ] · M k . Itsuffices to prove that I k +1 ⊆ σ ( I k ) because σ ( I k ) ⊆ I k +1 bydefinition. Let B be an R [ x ]-basis of M k . Then B is also abasis of the left ideal R [ x ][ ∂ ] · M k .Let ≺ be the term order such that x ℓ ∂ m ≺ x ℓ ∂ m if ei-ther m Let L ∈ R [ x ][ ∂ ] with order r > . As-sume that the ℓ th submodule M ℓ of Cont( L ) contains a basisof Cont( L ) . Let I ℓ be the ℓ th coefficient ideal of Cont( L ) ,and G a reduced Gr¨obner basis of I ℓ . Let f ∈ G be ofthe lowest degree in x and F be the operator in Cont( L ) with lc ∂ ( F ) = f . Then F is a completely desingularizedoperator for L .Proof. By Lemma 5.2, Cont( L ) contains a completely desin-gularized operator S . Let j = deg ∂ ( S ). Then lc ∂ ( S ) is in I j for some j ≥ ℓ . By Lemma 5.3, σ j − ℓ ( I ℓ ) = I j . It followsthat σ ℓ − j (lc ∂ ( S )) belongs to I ℓ . By (5), we have σ r − j (lc ∂ ( S )) = c S g, where c S ∈ R and g is a primitive polynomial in R [ x ]. Adirect calculation implies that σ ℓ − j (lc ∂ ( S )) = c S σ ℓ − r ( g ).Since σ ℓ − j (lc ∂ ( S )) ∈ I ℓ , so does c S σ ℓ − r ( g ).Note that F is a desingularized operator by Theorem 4.4.By (5), σ r − ℓ ( f ) = c F g, where c F ∈ R . Thus, f = c F σ ℓ − r ( g ).Since G is a reduced Gr¨obner basis of I ℓ , f is the uniquepolynomial in G with minimal degree. Moreover, c S σ l − r ( g )is of minimal degree in I ℓ . So it can be reduced to zeroby f . Thus, c F | c S . On the other hand, c S | c F byDefinition 5.1. Thus, c S and c F are associated to eachother. Consequently, F is a completely desingularized oper-ator for L .he construction in the above theorem leads to the fol-lowing algorithm. Algorithm 5.5. Given L ∈ R [ x ][ ∂ ] , where ∂x = ( x +1) ∂ or ∂x = x∂ + 1 , compute a completely desingularized operatorfor L .(1) Compute a basis A of Cont( L ) by Algorithm 4.6.(2) Set ℓ to be the highest order of the elements in A . Com-pute an R [ x ] -basis B of M ℓ .(3) Set B ′ = { B ∈ B | deg ∂ ( B ) = ℓ } . Compute a reducedGr¨obner basis G of h{ lc ∂ ( B ) | B ∈ B ′ }i . (4) Set f to be the polynomial in G whose degree is the low-est one in x . Tracing back to the computation of step 3,one can find u B ∈ R [ x ] such that f = P B ∈B ′ u B lc ∂ ( B ) . (5) Output P B ∈B ′ u B B . The termination of this algorithm is evident. Its correct-ness follows from Theorem 5.4. Example 5.6. Consider two sequences ( a n ) n ≥ and ( b n ) n ≥ satisfying the following two recurrence equations [10] na n = a n − + a n − and nb n = b n − + b n − , respectively. The sequence c n = n ! a n b n has an annihilator L ∈ Z [ n ][ ∂ ] with deg ∂ ( L ) = 10 and lc ∂ ( L ) = ( n +10)( n +47 n + · · · +211696) . In step 1 of the above algorithm, Cont( L ) = R [ x ][ ∂ ] · M .In steps 2 and 3, we observe that I is generated by n +14 .In other words, we obtain a completely desingularized oper-ator T of order with lc ∂ ( T ) = n + 14 . Translating intothe recurrence equations of c n , we arrive at nc n = α c n − + · · · + α c n − , where α i ∈ Z [ n ] , i = 1 , . . . , . This verifies Krattenthaler’sconjecture for the sequences a n and b n .Note that it is impossible to have a completely desingular-ized operator of order less than . In fact, for some lowerorders, one can obtain σ − ( I ) = h n, n ( n − , n ( n − n + 1336) i ,σ − ( I ) = h n, n ( n − i ,σ − ( I ) = h n, n ( n − i . They cannot produce a leading coefficient whose degree andcontent are both minimal. Example 5.7. Consider the following recurrence equations: na n = (31 n − a n − + (49 n − a n − + (9 n − a n − ,nb n = (4 n + 13) b n − + (69 n − b n − + (36 n − b n − . Let c n = n ! a n b n , which has an annihilator L ∈ Z [ n ][ ∂ ] of or-der with lc ∂ ( L )=( n +9) α , where α ∈ Z [ n ] and deg n ( α )=20 .By the known algorithms for desingularization in [2, 1, 6,7], we find that c n satisfies the recurrence equation βnc n = β c n − + . . . + β c n − , where β is an 853-digit integer, β i ∈ Z [ n ] , i = 1 , . . . , .On the other hand, Algorithm 5.5 finds a completely desin-gularized operator T for L of order whose leading coef-ficient is n + 14 . Translating into the recurrence equationof c n yields nc n = γ c n − + · · · + γ c n − , where γ i ∈ Z [ n ] . Let L ∈ R [ x ][ ∂ ] with positive order and T a desingularizedoperator for L . Then the degree of lc ∂ ( L ) in x is equal to d := deg x (lc ∂ ( L )) − ( k + · · · + k m ) , where k , . . . , k m are given in Definition 3.1. Hence, Cont( L )cannot contain any operator whose leading coefficient hasdegree lower than d .We provide a lower bound for the content of the lead-ing coefficients of operators in Cont( L ) with respect to thedivisibility relation on R . To this end, we write L = a k f k ( x ) ∂ k + a k − f k − ( x ) ∂ k − + · · · + a f ( x )where a i ∈ R and f i ( x ) ∈ R [ x ] is primitive, i = 0 , , . . . , k .We say that L is R -primitive if gcd( a , a , . . . , a k ) = 1.Gauss’s lemma in the commutative case also holds for R -primitive polynomials. Lemma 5.8. Let P and Q be two operators in R [ x ][ ∂ ] . If P and Q are R -primitive, so is P Q .Proof. First, we recall a result in [18, Theorem 3.7, Corollary3.8]. Assume that A is a ring with endomorphism σ : A → A and σ -derivation δ : A → A . Let I ⊆ A be a σ - δ -ideal, thatis, an ideal such that σ ( I ) ⊆ I and δ ( I ) ⊆ I . Then thereexists a unique ring homomorphism χ : A [ ∂ ; σ, δ ] → ( A/I )[ ˜ ∂ ; ˜ σ, ˜ δ ]such that χ | A : A → A/I is the canonical homomorphism,and χ ( ∂ ) = ˜ ∂ , where ˜ σ and ˜ δ are the homomorphism and˜ σ -derivation induced by σ and δ , respectively.Let p be a prime element of R and I = h p i be the cor-responding ideal in R [ x ]. Then I is a σ - δ -ideal. From theabove paragraph, there exists a unique homomorphism χ : R [ x ][ ∂ ; σ, δ ] → ( R [ x ] /I )[ ˜ ∂ ; ˜ σ, ˜ δ ]such that χ | R [ x ] : R [ x ] → R [ x ] /I is the canonical homo-morphism, and χ ( ∂ ) = ˜ ∂ . Note that σ − ( I ) ⊂ I , because,for pf ∈ I with f ∈ R [ x ], σ − ( pf ) = pσ − ( f ) ∈ I . It fol-lows that ˜ σ is an injective endomorphism of A/I . On theother hand, R [ x ] /I is a domain because I is a prime ideal.Thus, ( R [ x ] /I )[ ˜ ∂ ; ˜ σ, ˜ δ ] is a domain because R [ x ] /I is a do-main and ˜ σ is injective. Since P and L are R -primitive, wehave that χ ( P ) χ ( L ) = 0. So χ ( P L ) = 0, because χ is ahomomorphism. Consequently, P L is R -primitive.There are more sophisticated variants of Gauss’s lemmafor Ore operators in [17, Proposition 2] and [8, Lemma 9.5]. Theorem 5.9. Let L ∈ R [ x ][ ∂ ] with positive order and p bea non-unit element of R . If L is R -primitive and p | lc ∂ ( L ) ,then p is non-removable.Proof. Assume that p is removable, then there exists a p -removing operator P such that P L ∈ R [ x ][ ∂ ]. By Lemma 2.4,we can write P = p p d + p p d ∂ + · · · + p k p d k ∂ k where p i ∈ R [ x ], gcd( p i , p ) = 1 in R [ x ], i = 0 , . . . , k and d k ≥ 1. Let d = max ≤ i ≤ k d i and P = p d P . Thenthe content c of P with respect to ∂ is gcd( p , . . . , p k ) be-cause gcd( p i , p ) = 1, i = 0 , . . . , k. Let P = cP . Then P is the primitive part of P . In particular, P is R -primitive.Then cP L = p d P L . Since gcd( c, p ) = 1 and P L ∈ R [ x ][ ∂ ], p ivides the content of P L with respect to ∂ . Since p is anon-unit element of R , P L is not R -primitive, a contradi-tion to Lemma 5.8. Example 5.10. In the shift Ore algebra Z [ n ][ ∂ ] , consider a Z -primitive operator L = 3( n + 2)(3 n + 4)(3 n + 5)(7 n + 3) (cid:0) n + 21 n + 2 (cid:1) ∂ + ( − n − n − n − n − n − n − ∂ + 24(2 n + 1)(4 n + 1)(4 n + 3)(7 n + 10) (cid:0) n + 71 n + 48 (cid:1) , which annihilates (cid:0) nn (cid:1) + 3 n . We observe that is a constantfactor of lc ∂ ( L ) . By Theorem 5.9, is non-removable. 6. CONCLUDING REMARKS In this paper, we determine a basis of a contraction idealdefined by an Ore operator in R [ x ][ ∂ ], and compute a com-pletely desingularized operator whose leading coefficient isminimal in terms of both degree and content. A more chal-lenging topic is to consider the corresponding problems inthe multivariate case.Our algorithms rely heavily on the computation of Gr¨ob-ner bases over a principal ideal domain R . At present, thecomputation of Gr¨obner bases over R is not fully availablein a computer algebra system. So the algorithms in this pa-per are not yet implemented. To improve their efficiency, weneed to use linear algebra over R as much as possible. 7. ACKNOWLEDGEMENT I am grateful to my supervisors Manuel Kauers and Zim-ing Li for initiating this research project, stimulating inspi-rational discussions and helping me revise the paper. I thankShaoshi Chen for suggestions on the proof of Lemma 5.8 andvaluable information on literatures. 8. REFERENCES [1] S. A. Abramov, M. Barkatou, and M. van Hoeij.Apparent singularities of linear difference equationswith polynomial coefficients. AAECC , 117–133, 2006.[2] S. A. Abramov and M. van Hoeij. 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