Desingularization in the q -Weyl algebra
DDesingularization in the q -Weyl algebra Christoph Koutschan ∗ and Yi Zhang ∗ Johann Radon Institute for Computational and AppliedMathematics (RICAM), Austrian Academy of Sciences, Austria
Abstract
In this paper, we study the desingularization problem in the first q -Weyl algebra. We give an order bound for desingularized operators, andthus derive an algorithm for computing desingularized operators in thefirst q -Weyl algebra. Moreover, an algorithm is presented for computinga generating set of the first q -Weyl closure of a given q -difference oper-ator. As an application, we certify that several instances of the coloredJones polynomial are Laurent polynomial sequences by computing thecorresponding desingularized operator. The desingularization problem has been primarily studied in the context of dif-ferential operators, and more specifically, for linear differential operators withpolynomial coefficients. The solutions of such an operator are called
D-finite [27]or holonomic functions. It is well known [11] that a singularity (e.g., a pole)at a certain point x of one of the solutions must be reflected by the vanish-ing (at x ) of the leading coefficient of the operator. The converse however isnot necessarily true: not every zero of the leading coefficient polynomial in-duces a singularity of at least one function in the solution space. The goal ofdesingularization is to construct another operator, whose solution space con-tains that of the original operator, and whose leading coefficient vanishes onlyat the singularities of the previous solutions. Typically, such a desingularizedoperator will have a higher order, but a lower degree for its leading coefficient.In summary, desingularization provides some information about the solutionsof a given differential equation.For linear ordinary differential and recurrence equations, desingularizationhas been extensively studied in [2, 1, 5, 7, 4]. Moreover, the authors of [6] developalgorithms for the multivariate case. As applications, the techniques of desingu-larization can be used to extend P-recursive sequences [2], certify the integralityof a sequence [1], check special cases of a conjecture of Krattenthaler [28] andexplain order-degree curves [5] for Ore operators.The authors of [7, 28] also give general algorithms for the Ore case. However,from a theoretical point of view, the story is not yet finished, in the sense that ∗ Supported by the Austrian Science Fund (FWF): P29467-N32.E-mail addresses: [email protected], [email protected] a r X i v : . [ c s . S C ] F e b here is no order bound for desingularized operators in the Ore case. In thispaper, we consider the desingularization problem in the first q -Weyl algebra.Our main contribution is to give an order bound (Theorem 4.8) for desingular-ized operators, and thus derive an algorithm (Algorithm 4.13) for computingdesingularized operators in the first q -Weyl algebra. In addition, an algorithm(Algorithm 4.10) is presented for computing a generating set of the first q -Weylclosure of a given q -difference operator.As an example, consider the q -holonomic sequence f ( n ) = [ n ] q := q n − q − q -analog of the natural numbers. The minimal-order homogeneous q -recurrence satisfied by f ( n ) is( q n − f ( n + 1) − ( q n +1 − f ( n ) = 0 , in operator notation: (cid:0) ( x − ∂ − qx + 1 (cid:1) · f ( n ) = 0 , (1)where x = q n and ∂ · f ( n ) = f ( n + 1). When we multiply this operator by asuitable left factor, we obtain a monic (and hence: desingularized) operator oforder 2: 1 qx − (cid:0) ∂ − q (cid:1)(cid:0) ( x − ∂ − qx + 1 (cid:1) = ∂ − ( q + 1) ∂ + q. (2)As it is typically done in the shift case [2], we view a q -difference operator asa tool to define a q -holonomic sequence. Alternatively, one could take the view-point of [1] and study solutions of q -recurrences that are meromorphic functionsin the complex plane (for this, let q ∈ C be transcendental), and whose polesare somehow related to the zeros of the leading coefficient. In that sense, thefactor x − x = q , but in fact, thesolution f ( x ) = x − q − is an entire function and does not have any pole, whichis in agreement with the fact that there exists a desingularized operator (2).However, in contrast to the differential case, in the shift case one also has totake into account poles that are congruent [1] to a zero of the leading coefficient.We expect that the same phenomenon occurs in the q -case, but since our maininterest is in sequences, we do not investigate it in more detail here.As an application, we study several instances of the colored Jones polyno-mial [16, 17, 14], which is a q -holonomic sequence arising in knot theory andwhich is a powerful knot invariant. By inspecting this sequence for a particulargiven knot, one finds that all its entries seem to be Laurent polynomials, andnot, as one would expect, more general rational functions in q . By computingthe corresponding desingularized operator, we can certify that the sequence un-der consideration actually is constituted of Laurent polynomials, and that noother denominators than powers of q can appear. q -difference operators Throughout the paper, we assume that K is a field of characteristic zero, and q istranscendental over K . For instance, K can be the field of complex numbers and2 a transcendental indeterminate. Let K ( q )[ x ] be the ring of usual commutativepolynomials over K ( q ). The quotient field of K ( q )[ x ] is denoted by K ( q, x ). Thenwe have the ring of q -difference operators with rational function coefficients or q -rational algebra K ( q, x )[ ∂ ], in which addition is done coefficient-wise andmultiplication is defined by associativity via the commutation rule ∂f ( x ) = f ( qx ) ∂ for each f ( x ) ∈ K ( q, x ) . The variable x acts on a function g ( x ) by the usual multiplication, and the q -difference operator ∂ acts on it by the q -dilation with respect to x : ∂ ( g ( x )) = g ( qx ) . This ring is an Ore algebra [26, 10].Another ring is K ( q )[ x ][ ∂ ], which is a subring of K ( q, x )[ ∂ ]. We call it the ringof q -difference operators with polynomial coefficients or the q-Weyl algebra [13,Section 2.1].Given P ∈ K ( q )[ x ][ ∂ ] \ { } , we can uniquely write it as P = (cid:96) r ∂ r + (cid:96) r − ∂ r − + · · · + (cid:96) with (cid:96) , . . . , (cid:96) r ∈ K ( q )[ x ] and (cid:96) r (cid:54) = 0. We call r the order , and (cid:96) r the leadingcoefficient of P . They are denoted by deg ∂ ( P ) and lc ∂ ( P ), respectively. Wecall (cid:96) the trailing coefficient of P . Without loss of generality, we assume that (cid:96) (cid:54) = 0 throughout the paper. Otherwise, let t be the minimal index such that (cid:96) t (cid:54) = 0. Set ˜ P = ∂ − t P . Then the trailing coefficient of ˜ P is ∂ − t ( (cid:96) t ), which is anonzero polynomial in K ( q )[ x ]. As a matter of convention, we say that the zerooperator in K ( q )[ x ][ ∂ ] has order − σ : K ( q )[ x ] → K ( q )[ x ] be a ring automorphism that leaves the elementsof K ( q ) fixed and σ ( x ) = qx . Assume that Q ∈ K ( q )[ x ][ ∂ ] is of order k . Arepeated use of the commutation rule yieldslc ∂ ( QP ) = lc ∂ ( Q ) σ k (lc ∂ ( P )) . (3)Assume that S is a subset of K ( q, x )[ ∂ ], then the left ideal generated by S is de-noted by K ( q, x )[ ∂ ] S . For an operator P ∈ K ( q )[ x ][ ∂ ], we define the contractionideal or q -Weyl closure of P :Cont( P ) := K ( q, x )[ ∂ ] P ∩ K ( q )[ x ][ ∂ ] . q -case In this section, we define the dispersion of two polynomials in K ( q )[ x ] andpresent an algorithm for computing it, based on irreducible factorizations overthe ring K [ q ][ x ]. The dispersion in the q -case will be used in the next sectionfor giving an order bound of a desingularized operator (Definition 4.7). Lemma 3.1.
The following claims hold:(i) If p ( x ) is an irreducible polynomial in K ( q )[ x ] of positive degree with p (0) (cid:54) =0 , so is p ( q α x ) for each α ∈ Z . ii) Let p ( x ) be an irreducible polynomial in K ( q )[ x ] of positive degree with p (0) (cid:54) = 0 . Then gcd( p ( q α x ) , p ( x )) = 1 for each α ∈ Z \{ } . (iii) Let f ( x ) , g ( x ) be two polynomials in K ( q )[ x ] with f (0) (cid:54) = 0 . Then the set { α ∈ N | deg x (gcd( f ( q α x ) , g ( x ))) > } (4) is a finite set.Proof. (i) It follows from [24, Proposition 3].(ii) Suppose that there exists α ∈ Z \{ } such thatgcd( p ( q α x ) , p ( x )) (cid:54) = 1 . Since p ( x ) is an irreducible polynomial in K ( q )[ x ], we have that p ( x ) | p ( q α x ).We may write p ( x ) = c d x d + c d − x d − + · · · + c , (5)where c i ∈ K ( q ), 0 ≤ i ≤ d with d >
0, and c , c d (cid:54) = 0. Then p ( q α x ) = ( c d q dα ) x d + ( c d − q ( d − α ) x d − + · · · + c . (6)Since p ( x ) | p ( q α x ), we conclude from (5) and (6) that p ( q α x ) = q dα p ( x ) . Comparing the constant coefficients of both sides in the above equation, itfollows that c q dα = c . Since c (cid:54) = 0, we have that q dα = 1, a contradiction to the fact that q is not aroot of unity of K .(iii) Suppose that (4) is an infinite set. Then there exists an irreduciblefactor p ( x ) of f ( x ) such thatgcd( p ( q α x ) , g ( x )) (cid:54) = 1 for infinitely many α ∈ N . Since g ( x ) only has finitely many distinct irreducible factors, it follows from (i)that gcd( p ( q α x ) , p ( q α x )) (cid:54) = 1 for some α (cid:54) = α ∈ N . Therefore, we have gcd( p ( q α − α x ) , p ( x )) (cid:54) = 1 , a contradiction to (ii).Based on the above lemma and [24, Definition 1], we give the followingdefinition. Definition 3.2.
Let f ( x ) , g ( x ) be two polynomials in K ( q )[ x ] with f (0) (cid:54) = 0 .The dispersion of f ( x ) and g ( x ) is given by dis( f ( x ) , g ( x )) := max { α | α ∈ N , deg x (gcd( f ( q α x ) , g ( x ))) > } ∪ { } .
4e include 0 in the above definition in order to guarantee that the dispersionis always defined, even for constant polynomials. The dispersion in the q -case isthe largest integer q -shift such that the greatest common divisor of the shiftedpolynomial and the unshifted one is nontrivial. Specifically, assume that f ( x )has the following factorization: f ( x ) = p e · · · p e m m , where p , . . . , p m ∈ K ( q )[ x ] \ K ( q ) are irreducible and pairwise coprime. It isstraightforward to see from Definition 3.2 thatdis( f ( x ) , g ( x )) = max { dis( p i , g ) | ≤ i ≤ m } . For example, the dispersiondis(( x + 1)(4 x + q ) , ( q x + 1)( q x + q + 1)) = 2 , because dis( x + 1 , q x + 1) = 2.Similar to the shift case, the dispersion in the q -case can be computed by aresultant-based algorithm [3, Example 1]. We have implemented it in Mathe-matica, but experiments suggest that it is inefficient in practice. For instance,consider f ( x ) = 5( qx + 1)( x − q )( x + 2)( x − qx + 1)(2 qx + 5) ,g ( x ) = f ( q x ) . The polynomial f has coefficients in Z [ q ], and has degree 9 in x . The dispersionof f and g is 4. Below is a table for the timings (in seconds) for the com-putation of dispersion of f and g by the resultant-based ( res ) algorithm andthe factorization-based ( fac ) algorithm, respectively. For this purpose, the twopolynomials were given in fully expanded form.System Mathematicares 43.6006fac 0.011015Like [24], we also give an algorithm based on irreducible factorization over K [ q ][ x ]. Proposition 3.3.
Let f ( x ) be a primitive polynomial in K [ q ][ x ] of positivedegree with respect to x , and f (0) (cid:54) = 0 . Then for each α ∈ Z , we have(i) f ( q α x ) = q e g ( x ) , where g ( x ) is a primitive polynomial in K [ q ][ x ] with thesame degree as f ( x ) , g (0) (cid:54) = 0 and e ∈ N .(ii) Let f ( x ) = (cid:80) di =0 a i x i and g ( x ) = (cid:80) di =0 b i x i be two polynomials such that f ( q α x ) = q e g ( x ) for some e ∈ N . Then q dα = a b d b a d . Proof. (i) Assume that f ( x ) = (cid:80) di =0 a i x i with a d , a (cid:54) = 0, gcd( a d , . . . , a ) = 1in K [ q ]. Then f ( q α x ) = ( a d q dα ) x d + ( a d − q ( d − α ) x d − + · · · + a . (7)5ince gcd( a d , . . . , a ) = 1 in K [ q ], we have thatgcd( a d q dα , a d − q ( d − α , . . . , a ) = q e for some e ∈ N . Thus, we can write f ( q α x ) = q e g ( x ), where g ( x ) is a primitive polynomialin K [ q ][ x ] with the same degree as f ( x ) and g (0) (cid:54) = 0.(ii) Since f ( q α x ) = q e g ( x ), it follows from (7) that a a d q dα = q e b q e b d = b b d . Thus, we conclude that q dα = a b d b a d . Given f ( x ) , g ( x ) ∈ K ( q )[ x ], we may further assume that f ( x ) , g ( x ) are twopolynomials in K [ q ][ x ] by clearing their denominators. The above propositiongives a method to compute the dispersion of two primitive irreducible polyno-mials in K [ q ][ x ]. Below is the corresponding algorithm. Algorithm 3.4.
Given two primitive irreducible polynomials f, g ∈ K [ q ][ x ] ofpositive degrees with respect to x and f (0) (cid:54) = 0 . Compute dis( f, g ) .(1) Compute d = deg x ( f ) , d = deg x ( g ) . If d (cid:54) = d , then return . Other-wise, set d = d .(2) Let f = (cid:80) di =0 a i x i and g = (cid:80) di =0 b i x i . If a b d b a d is not a nonnegative powerof q d , then return . Otherwise, set α to be the natural number such that q dα = a b d b a d .(3) Compute h = f ( q α x ) a d q dα − g ( x ) b d . If h is not the zero polynomial, return .Otherwise, return α . The termination of the above algorithm is obvious. The correctness followsfrom Proposition 3.3.
Example 3.5.
Consider the following two primitive polynomials in K [ q ][ x ] : f ( x ) = qx − ,g ( x ) = q x − . Using the above algorithm, we find that dis( f ( x ) , g ( x )) = 2 . Using the irreducible factorization over K [ q ][ x ], we derive the following al-gorithm to compute the dispersion of two arbitrary polynomials in K [ q ][ x ]: Algorithm 3.6.
Given f ( x ) , g ( x ) ∈ K [ q ][ x ] with f (0) (cid:54) = 0 , compute dis( f, g ) .(1) [Initialize] If deg x ( f ) < or deg x ( g ) < then return . Otherwise, set dispersion = 0 .
2) [Factorization] Compute the set { f i ( x ) } and { g j ( x ) } of distinct primitiveirreducible factors over K [ q ] of positive degree in x for f ( x ) and g ( x ) ,respectively.(3) For each pair ( f i ( x ) , g j ( x )) of these factors, use Proposition 3.3 to compute α = dis( f i ( x ) , g j ( x )) . If α > dispersion , then set dispersion = α .(4) Return dispersion . The termination of the above algorithm is obvious. The correctness followsfrom Definition 3.2 and Proposition 3.3. It is implemented in Mathematica.
Example 3.7.
Consider the following two polynomials in K [ q ][ x ] : f ( x ) = ( qx − qx + 1)( qx − ,g ( x ) = q x ( q x − q x + 1)( q x − . They are already in factored form. Using the above algorithm, we find that dis( f ( x ) , g ( x )) = 2 . q -Weyl algebra We are now going to present algorithms for the q -Weyl closure (Algorithm 4.10)and for the desingularization of a q -difference operator (Algorithm 4.13). Thesealgorithms are analogs of those in [28] and use Gr¨obner basis computations.Hence, in practice, they are slower than algorithms based on linear algebra [5, 7](see also Section 5), but their advantage is that also the degree with respectto q can be taken into account—a feature that will be essential for the examplespresented in the next section.In this section, we consider the desingularization for the leading coefficient ofa given q -difference operator. The trailing coefficient can be handled in a similarway. We summarize some terminologies given in [5, 7, 28] by specializing thegeneral Ore ring setting to the q -Weyl algebra. Definition 4.1.
Let P ∈ K ( q )[ x ][ ∂ ] with positive order, and p be a divisorof lc ∂ ( P ) in K ( q )[ x ] .(i) We say that p is removable from P at order k if there exist Q ∈ K ( q, x )[ ∂ ] with order k , and w, v ∈ K ( q )[ x ] with gcd( p, w ) = 1 in K ( q )[ x ] such that QP ∈ K ( q )[ x ][ ∂ ] and σ − k (lc ∂ ( QP )) = wvp lc ∂ ( P ) . We call Q a p -removing operator for P over K ( q )[ x ] , and QP the corre-sponding p -removed operator .(ii) A polynomial p ∈ K ( q )[ x ] is simply called removable from P if it is re-movable at order k for some k ∈ N . Otherwise, p is called non-removable from P . Note that every p -removed operator lies in Cont( P ).7 xample 4.2. Consider the following q -difference operator [9, Example 4.9] oforder in K ( q )[ x ][ ∂ ] : P = q x ( q − x ) ∂ − (1 − x )(1 − qx ) . Set Q = q x − ∂ + q + q − q − q x − ∂ + q − q − q + 1 x − . Let L = QP . Then L = q x∂ + q ( q x + q x + q x − qx − x − ∂ +( q − q ( q + 1)( q + q + 1)( q x + qx − x − ∂ +( q − ( q + 1)( q + q + 1)( qx − , is a ( q − x ) -removed operator for P of order . The following proposition provides a convenient form of p -removing opera-tors over K ( q )[ x ]. It is a special case of [28, Lemma 2.4] and also included in [5].In Corollary 4.6, we will use it to prove that x -removing operators do not exist. Proposition 4.3.
Let P ∈ K ( q )[ x ][ ∂ ] be a q -difference operator with positiveorder. Assume that p ∈ K ( q )[ x ] is removable from P at order k . Then thereexists a p -removing operator for P over K ( q )[ x ] of 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 K ( q )[ x ] , gcd( p i , σ k ( p )) = 1 in K ( q )[ x ] or p i = 0 for each i = 0 , , . . . , k , and d k ≥ . In [5, Lemma 4], the authors give an order bound for a p -removing operatorin the shift case. We find that the proof also applies to the q -difference caseprovided that p is an irreducible polynomial in K ( q )[ x ] and p (0) (cid:54) = 0. Wesummarize it in the following lemma. Lemma 4.4.
Let P be a nonzero operator in K ( q )[ x ][ ∂ ] of positive order withtrailing coefficient (cid:96) . Assume that p is an irreducible factor of lc ∂ ( P ) such that p (0) (cid:54) = 0 and p k is removable from P for some k ≥ . Then p k is removablefrom P at order dis( p, (cid:96) ) .Proof. It is literally the same as [5, Lemma 4].Let P = (cid:80) ri =0 (cid:96) i ∂ i be a nonzero operator in K ( q )[ x ][ ∂ ] of positive order. Wesay that P is x -primitive if x (cid:45) gcd( (cid:96) , . . . , (cid:96) r ) in K ( q )[ x ]. Gauß’ lemma in thecommutative case also holds for x -primitive operators. The proof is similar tothat of [29, Lemma 3.4.8]. Here, we give an independent proof. Lemma 4.5.
Let P and Q be two operators in K ( q )[ x ][ ∂ ] . If P and Q are x -primitive, so is QP .Proof. Suppose that QP is not x -primitive. We may write P = r (cid:88) i =0 a i ∂ i , Q = s (cid:88) i =0 b i ∂ i and QP = r + s (cid:88) i =0 c i ∂ i , a i , b i , c i are polynomials in K ( q )[ x ]. By assumption, wehave x | gcd( c , . . . , c r + s ). Since P and Q are x -primitive, there exists 0 ≤ i ≤ r and 0 ≤ j ≤ s such that x (cid:45) a i and x (cid:45) b j . We may further assume that i and j are maximal with this property. Consider c i + j = (cid:88) i + j = i + j a i σ i ( b j ) , (8)By the maximality of i and j , we have that x | a i and x | b j for i > i and j > j . Note that x also divides σ i ( b j ) for j > j and i = i + j − j because σ i ( x ) = q i x . Therefore, in the right side of equation (8), each summand isdivisible by x except a i σ i ( b j ). By assumption, x divides c i + j . Thus, x divides a i σ i ( b j ). It implies that x | a i or x | σ i ( b j ). Since x (cid:45) a i , we havethat x | σ i ( b j ). If follows that x | σ − i ( σ i ( b j )) = b j , a contradiction. Corollary 4.6.
Let P be a nonzero operator in K ( q )[ x ][ ∂ ] of positive order. If x divides lc ∂ ( P ) , then x is non-removable from P .Proof. Suppose that x is removable from P . By Definition 4.1, there exists an x -removing operator Q such that QP ∈ K ( q )[ x ][ ∂ ]. By Proposition 4.3, we canwrite Q = p x d + p x d ∂ + · · · + p k x d k ∂ k , where p i ∈ K ( q )[ x ], gcd( p i , x ) = 1 in K ( q )[ x ], i = 0 , . . . , k and d k ≥
1. Let d = max ≤ i ≤ k d i and Q = x d Q. Then the content w of Q with respect to ∂ is gcd( p , . . . , p k ) becausegcd( p i , x ) = 1 for each i = 0 , . . . , k. Let Q = wQ . Then Q is the primitive part of Q . In particular, Q is x -primitive. Then wQ P = x d QP.
Since gcd( w, x ) = 1 and QP ∈ K ( q )[ x ][ ∂ ], we have that x divides the contentof Q P with respect to ∂ . It follows that Q P is not x -primitive, a contradictionto Lemma 4.5.Next, we give the definition of desingularized operators in the q -case, whichis a special case of [28, Definition 3.1]. Definition 4.7.
Let P ∈ K ( q )[ x ][ ∂ ] with order r > , and lc ∂ ( P ) = p e · · · p e m m , (9) where p , . . . , p m ∈ K ( q )[ x ] \ K ( q ) are irreducible and pairwise coprime. Anoperator L ∈ K ( q )[ x ][ ∂ ] \ { } of order k is called a desingularized operator for P if L ∈ Cont( P ) and σ r − k (lc ∂ ( L )) = abp k · · · p k m m lc ∂ ( P ) , (10) where a, b ∈ K ( q ) with b (cid:54) = 0 , and p d i i is non-removable from P for each d i > k i , i = 1 , . . . , m . heorem 4.8. Let P be a nonzero operator in K ( q )[ x ][ ∂ ] of order r > .Assume that (cid:96) r and (cid:96) are the leading and trailing coefficient of P , respectively.Set (cid:96) r = x e ˜ (cid:96) r for some e ∈ N and ˜ (cid:96) r (0) (cid:54) = 0 . Then there exists a desingularizedoperator of P of order r + dis( ˜ (cid:96) r , (cid:96) ) .Proof. Assume that ˜ (cid:96) r = p e · · · p e m m , where p , . . . , p m ∈ K ( q )[ x ] \ K ( q ) areirreducible, pairwise coprime. For each i ∈ { , . . . , m } , let k i be the naturalnumber such that p k i i is removable from P , but p d i i is non-removable from P for each d i > k i . It follows from Lemma 4.4 that p k i i is removable from P atorder dis( p i , (cid:96) ). On the other hand, if e ≥
1, then it follows from Corollary 4.6that x d is non-removable from P for each 1 ≤ d ≤ e . Above all, we concludefrom [7, Lemma 4] that there exists a desingularized operator of P of order r + max { dis( p i , (cid:96) ) | ≤ i ≤ m } , which is equal to r + dis( ˜ (cid:96) r , (cid:96) ). Example 4.9.
Consider the q -difference operator from Example 4.2: P = q x ( q − x ) ∂ − (1 − x )(1 − qx ) . By the above theorem, we find that P has a desingularized operator of order q ( q − x ) , (1 − x )(1 − qx )) = 4 . Actually, a desingularized operator of P with minimal order is L as specified inExample 4.2, which is of order 3. In the above example, the order bound given by Theorem 4.8 is overshooting.However, we will see in the next section that it is tight in all examples fromknot theory that we looked at.The first application of Theorem 4.8 is to derive an algorithm for computingthe first q -Weyl closure of a q -difference operator.Let P be a nonzero operator in K ( q )[ x ][ ∂ ] of order r >
0. For each k ≥ r ,we set M k ( P ) = { T ∈ Cont( P ) | deg ∂ ( T ) ≤ k } . It is straightforward to see that M k ( P ) is a finitely generated left K ( q )[ x ]-submodule of Cont( P ). We call it the k -th submodule of Cont( P ). If the op-erator P is clear from the context, then we denote M k ( P ) simply by M k . Agenerating set of M k can be derived by a syzygy computation over K ( q )[ x ] [29,Section 3.3.2]. Algorithm 4.10.
Given a q -difference operator P ∈ K ( q )[ x ][ ∂ ] of positive or-der. Compute a generating set of the q -Weyl closure of P .(1) Derive an order bound k for a desingularized operator of P by using The-orem 4.8.(2) Compute a generating set S of M k by using Gr¨obner bases [29, Section3.3.2].(3) Return S . Example 4.11.
Consider the q -difference operator in Example 4.2: P = q x ( q − x ) ∂ − (1 − x )(1 − qx ) . From Example 4.9, we know that an order bound for a desingularized operatorof P is . Using Gr¨obner bases, we can find a generating set of M . Since thesize for the generating set of M is large, we do not display it here. Instead,it follows from Example 4.9 that P has a desingularized operator with order 3.By [29, Theorem 3.2.3, Corollary 3.2.4], the q -Weyl closure of P is also gen-erated by M . Through computation, we find that M is generated by { P, L } ,where L is specified in Example 4.2. The second application of Theorem 4.8 is to give an algorithm for computinga desingularized operator of a given q -difference operator.Let P be a nonzero operator in K ( q )[ x ][ ∂ ] of order r >
0. For each k ≥ r ,let I k = (cid:8) [ ∂ k ] T | T ∈ M k ( P ) (cid:9) , where [ ∂ k ] T denotes the coefficient of ∂ k in T . It is straightforward to see that I k is an ideal of K ( q )[ x ]. We call I k the k -th coefficient ideal of Cont( P ). By [29,Lemma 3.3.3], we can compute a generating set of I k if a generating set of M k is given.Assume that k is an order bound for desingularized operators of P . From [29,Theorem 3.3.6], an element in I k \ { } with minimal degree in x will give riseto a desingularized operator of P . In [29, Remark 3.3.7], the author describeshow to use the Euclidean algorithm over K ( q )[ x ] to find an element s in I k \ { } with minimal degree in x . However, this will in general introduce a polynomialin K [ q ] when we clear the denominators in s . In the next section, we will need tofind desingularized operators of some q -difference operators from knot theory,whose leading coefficient is of the form q a x b , where a, b ∈ N . Thus, we shall alsominimize the degree of q among leading coefficients of desingularized operatorsof a given q -difference operator. Assume that B ⊂ K [ q ][ x ] is a generating setof I k . Next, we give a method that finds an element in K [ q ][ x ] of I k \ { } with minimal degree in x , which also has minimal degree in q among nonzeroelements of (cid:104) B (cid:105) in K [ q ][ x ] with minimal degree in x . Proposition 4.12.
Let B ⊂ K [ q ][ x ] be a generating set of I k . Assume that G is a reduced Gr¨obner basis of the ideal generated by B over K [ q ][ x ] with respectto the lexicographic order q ≺ x . Set g to be the element in G with minimaldegree in x . Then g is also an element in I k with minimal degree in x .Proof. Assume that f ∈ K ( q )[ x ] is an element in I k with minimal degree in x .Since B = { b , . . . , b (cid:96) } is a generating set of I k , we have f = c b + . . . + c (cid:96) b (cid:96) , where c , . . . , c (cid:96) ∈ K ( q )[ x ]. By clearing denominators in the above equation, itfollows that ˜ f = ˜ c b + . . . + ˜ c (cid:96) b (cid:96) , f = cf and c, ˜ c i ∈ K [ q ], i = 1 , . . . , (cid:96) . Since G is Gr¨obner basis of theideal generated by B over K [ q ][ x ], the head term of ˜ f is divisible by g i forsome g i ∈ G . By the choice of the term order, it is straightforward to see thatdeg x ( g i ) ≤ deg x ( ˜ f ). On the other hand, the degree of f in x is equal to thatof ˜ f . Thus, deg x ( g i ) ≤ deg x ( f ). Since g is the element in G with minimaldegree in x , we have deg x ( g ) ≤ deg x ( g i ) ≤ deg x ( f ) . Algorithm 4.13.
Given a q -difference operator P ∈ K ( q )[ x ][ ∂ ] of positive or-der. Compute a desingularized operator of P .(1) Derive an order bound k for a desingularized operator of P by using The-orem 4.8.(2) Compute a generating set S of M k by using Gr¨obner bases [29, Section3.3.2].(3) Compute a generating set in K [ q ][ x ] of I k by using [29, Lemma 3.3.3].(4) Compute an element g ∈ K [ q ][ x ] of I k with minimal degree in x by usingProposition 4.12.(5) Tracing back to the computation of steps (3) and (4), one can find a q -difference operator L ∈ K [ q ][ x, ∂ ] of Cont( P ) such that lc ∂ ( L ) = g . Out-put L . The termination of the above algorithm is evident. The correctness followsfrom [29, Theorem 3.3.6].
Example 4.14.
Consider the q -difference operator in Example 4.2: P = q x ( q − x ) ∂ − (1 − x )(1 − qx ) . (1) By Example 4.9, we know that the minimal order for a desingularizedoperator of P is .(2) Using Gr¨obner bases, we can find a generating set of M . Since the sizefor the generating set of M is large, we do not display it here.(3) By [29, Lemma 3.3.3], we find that I is generated by q x .(4) It is straightforward to see that q x is the element in I with minimaldegree in x .(5) Tracing back to the computation of steps (3) and (4), we find a q -differenceoperator L ∈ K [ q ][ x, ∂ ] of Cont( P ) , which is exactly the operator in Ex-ample 4.2. By computation, we also find that I = (cid:104) q x (cid:105) . This is not a contradiction because I = σ ( I ) in K ( q )[ x ]. ♣ ✰✶✲✶ ❂❂ Figure 1: Knot diagram of the twist knot K twist p (left), where the box representsrepeated half-twists, according to the legend on the right. In the past years, q -difference equations arose naturally in quantum topologyand knot theory. During the quest for better and better knot invariants—theideal invariant would allow to distinguish all knots—the so-called colored Jonespolynomial was discovered. The name polynomial is somewhat misleading, asthis invariant consists actually of an infinite sequence of rational functions in Q ( q ) or Laurent polynomials in Q [ q, q − ]. For the precise definition of thecolored Jones polynomial we refer to [16], where it is proven that for each knotthis infinite sequence satisfies a linear q -difference equation with polynomialcoefficients, i.e., that the colored Jones polynomial is always a q -holonomicsequence. The same author formulated the following conjecture. Conjecture 5.1 ([12]) . Let J K ( n ) ∈ Q ( q ) denote the Jones polynomial of aknot K , colored by the n -dimensional irreducible representation of sl and nor-malized by J Unknot ( n ) = 1 . Then for the colored Jones polynomial, i.e., for thesequence (cid:0) J K ( n ) (cid:1) n ∈ N the following holds:(1) (1 − q n ) J K ( n ) satisfies a bimonic recurrence relation,(2) J K ( n ) does not satisfy a monic recurrence relation. Here, the notion bimonic refers to the property that both the leading andthe trailing coefficient are monic (in the sense of Corollary 4.6, i.e., of the form q an + b ). Using desingularization, we can construct such bimonic recurrences,thereby confirming part (1) of the conjecture in some particular instances. Thisshows that the colored Jones polynomial is actually a sequence of Laurent poly-nomials, even when the sequence is extended to the negative integers, by apply-ing the recurrence into the other direction. The knot-theoretic interpretation ofthis phenomenon is that the substitution q → q − corresponds to reversing theorientation of the knot.We investigate the colored Jones polynomials of two families of knots thatappeared previously in the literature: twist knots [17] and pretzel knots [14], seeFigures 1 and 2. While it is very difficult to compute the colored Jones poly-nomial for an arbitrary given knot, one can give simpler formulas for these two13 ✷ ✸ ✷ ♣ ✰ ✸ Figure 2: Knot diagram of the ( − , , p + 3)-pretzel knot K pretz p ; again theboxes represent repeated half-twists as described in Fig. 1.families. For example, the n -th entry J twist p ( n ) of the colored Jones polynomialfor the p -th twist knot K twist p is given by the double sum n (cid:88) k =0 k (cid:88) j =0 ( − j +1 q k + pj ( j +1)+ j ( j − / (cid:0) q j +1 − (cid:1) (cid:0) q − n ; q (cid:1) k (cid:0) q n ; q (cid:1) k (cid:0) q k − j +1 ; q (cid:1) j (cid:0) q ; q (cid:1) k + j +1 . From this representation it is a routine task (but possibly computationally ex-pensive) to compute a q -holonomic recurrence equation for J twist p ( n ) when p is a fixed integer. This can be done either by q -holonomic summation meth-ods (as implemented in the qMultiSum package [25] or HolonomicFunctions package [20]) or by guessing (as implemented in the
Guess package [19]). Forexample, for p = − q -recurrence q n +2 (cid:0) q n +2 − (cid:1) (cid:0) q n +1 − (cid:1) J twist − ( n + 2) + (cid:0) q n +1 − (cid:1) (cid:0) q n +1 + 1 (cid:1) (cid:0) q n +1 ++ q n +1 + q n +3 + q n +3 − q n +4 − (cid:1) J twist − ( n + 1) + q n +2 ( q n − × (cid:0) q n +3 − (cid:1) J twist − ( n ) = q n +1 (cid:0) q n +1 + 1 (cid:1) (cid:0) q n +1 − (cid:1) (cid:0) q n +3 − (cid:1) . Garoufalidis and Sun have computed such an inhomogeneous q -recurrenceequation for each twist knot K twist p with − ≤ p ≤
15; the recurrences areavailable in electronic form from [17]. Similarly, the q -recurrences satisfied by J pretz p ( n ) for − ≤ p ≤ f ( n + d ) has (among others) a factor ( q n + d − f ( n ) → f ( n ) / ( q n − HolonomicFunctions [20] and
Singular [18]; the source code anda demo notebook are freely available as part of the supplementary electronic ma-terial [21]. Note that we also modify Algorithm 4.13 for desingularization of thetrailing coefficient of a given q -difference operator in the corresponding packageand notebook. We give an example about finding desingularized operators inthe context of knot theory. 14 xample 5.2. We consider the q -difference operators that correspond to thehomogeneous parts of the recurrences for the colored Jones polynomials of theknots K twist − , K twist2 , K pretz − , and K pretz2 . For example, the operator P twist − cor-responds, after normalization, to the left-hand side of the above q -recurrence for J twist − ( n ) : P twist − = q x (cid:0) qx − (cid:1) ∂ − ( qx − qx + 1) (cid:0) q x − q x − q x − qx − qx + 1 (cid:1) ∂ + q x (cid:0) q x − (cid:1) For space reasons, the other three operators are displayed in abbreviated formonly: P twist2 = ( qx − qx + 1)( qx − ∂ + (cid:96) , ∂ + (cid:96) , ∂ + (cid:96) , ,P pretz − = ( qx − qx + 1)( qx − ∂ + (cid:96) , ∂ + (cid:96) , ∂ + (cid:96) , ,P pretz2 = q ( qx − q x + 1) ∂ + (cid:96) , ∂ + (cid:96) , ∂ + · · · + (cid:96) , , where (cid:96) i,j ∈ K [ q ][ x ] . We now apply our desingularization algorithm to each ofthe four operators.(1) By using Theorem 4.8, we obtain an order bound b for a desingularizedoperator (see Table 1).(2) Using Gr¨obner bases, we can find a generating set of M b . Since the sizeof this generating set is large, we do not display it here.(3) By [29, Lemma 3.3.3], we find the generator of I b (see Table 1).(4) It is straightforward to see that in each of the four cases, this single gen-erator is the element in I b with minimal degree in x .(5) Tracing back to the computation of steps (3) and (4), we find a q -differenceoperator L ∈ K [ q ][ x, ∂ ] of Cont( P ) , which is of the following form: L twist − = q x ∂ − (cid:0) q x − q x − q x − q x − q x − q x + 1 (cid:1) ∂ − q x (cid:0) q x − q x − q x − q x − q x − x + q (cid:1) ∂ + q x ,L twist2 = ∂ + p , ∂ + p , ∂ + · · · + p , ,L pretz − = ∂ + p , ∂ + p , ∂ + · · · + p , ,L pretz2 = ∂ + p , ∂ + p , ∂ + · · · + p , , where p i,j ∈ K [ q ][ x ] .We observe that in all four examples the minimal order for desingularized op-erators matches with the predicted order bound, i.e., the bound is tight in thesecases. This can be seen by inspecting the ( b − -st coefficient ideal I b − (seeTable 1). We conclude that the sequences that are annihilated by the four op-erators, respectively, consist indeed of (Laurent) polynomials, provided that theinitial values have this property as well. twist − P twist2 P pretz − P pretz2 order bound b I b I = (cid:104) x (cid:105) I = (cid:104) (cid:105) I = (cid:104) (cid:105) I = (cid:104) (cid:105) generator of I b − (cid:104) q x − (cid:105) (cid:104) q x − (cid:105) (cid:104) q x − (cid:105) Table 1: Computations for Example 5.2
Example 5.3.
By applying our desingularization algorithm to the unnormalized q -recurrences of J twist p ( n ) for the same values of p as in the previous example,we can prove that in these instances the operators are not completely desingu-larizable, therefore confirming part (2) of Conjecture 5.1. Since Algorithms 4.10 and 4.13 involve Gr¨obner bases computations, it israther inefficient to find desingularized operators when the size of the given q -difference operator is large. Alternatively, we may apply guessing [19] tocompute a desingularized operator of a given q -difference operator, once wederive an order bound by Theorem 4.8.In order to illustrate the guessing approach, we focus on a slightly modi-fied problem, namely that of finding bimonic recurrence equations : we want tocompletely desingularize both the leading and the trailing coefficient, i.e., afterdesingularization these two coefficients should have the form q an + b for someintegers a, b ∈ N . The existence of such a recurrence equation certifies that thebi-infinite sequence (cid:0) f ( n ) (cid:1) n ∈ Z has only Laurent polynomial entries. Note thatthis approach is also suited for inhomogeneous recurrences.It works as follows: assume we are given a (possibly inhomogeneous) recur-rence p − ( q, q n ) + r (cid:88) i =0 p i ( q, q n ) f ( n + i ) (cid:124) (cid:123)(cid:122) (cid:125) =: R ( n ) = 0 , with r ≥ p i ∈ K [ q, q n ] for − ≤ i ≤ r . Define the polynomial c ( q, q n ) ∈ K [ q, q n ] by c ( q, q n ) = lcm (cid:0) p ( q, q n ) , p r ( q, q n ) (cid:1) q an + b with integers a, b ∈ N chosen such that c ( q, q n ) is neither divisible by q norby q n . The goal is to determine polynomials u i ( q, q n ) ∈ K [ q, q n ] such that thecoefficients (cid:96) i ( q, q n ), − ≤ i ≤ r + s , in the linear combination s (cid:88) i =0 u i ( q, q n ) R ( n + i ) = (cid:96) − ( q, q n ) + r + s (cid:88) i =0 (cid:96) i ( q, q n ) f ( n + i )are all divisible by c ( q, q n ). Hence, we make an ansatz for the coefficients ofthe linear combination, instead of trying to guess the desingularized operatordirectly. The latter would be much more costly to compute (compare the numberof green dots with the number of blue dots in Figure 3). The procedure issketched in Algorithm 5.4. We have implemented it in Mathematica; the sourcecode and a demo notebook are freely available as part of the supplementaryelectronic material [22]. 16
10 1520406080100
Figure 3: q n -support of the coefficients p − , . . . , p ∈ Q [ q, q n ] of the inhomo-geneous q -recurrence for K pretz3 (red), q n -support of the coefficients u , . . . , u (green), and q n -support of the resulting bimonic recurrence (blue), representedby the coefficients (cid:96) − , . . . , (cid:96) ; the horizontal axis gives the index of the coeffi-cient, the vertical axis the exponent of q n . Algorithm 5.4.
Given a recurrence R ( n ) = p − ( q, q n ) + (cid:80) ri =0 p i ( q, q n ) f ( n + i ) and a factor c ( q, q n ) that is to be removed. Compute u i ∈ K [ q, q n ] such that (cid:80) si =0 u i ( q, q n ) R ( n + i ) = c ( q, q n ) (cid:0) (cid:96) − ( q, q n ) + (cid:80) r + si =0 (cid:96) i ( q, q n ) f ( n + i ) (cid:1) for somepolynomials (cid:96) i ∈ K [ q, q n ] .(1) Make an ansatz of the form A = (cid:80) si =0 (cid:80) d i j = e i c i,j ( q ) q jn R ( n + i ) (one maynote that the coefficients c ,j and c s,j are already prescribed (up to a con-stant multiple in K ( q ) ) by the choice of c ( q, q n ) .(2) Write A in the form A = a − ( q, q n ) + (cid:80) r + si =0 a i ( q, q n ) f ( n + i ) .(3) For − ≤ i ≤ r + s compute the remainder of the polynomial division of a i ( q, q n ) by c ( q, q n ) , regarded as polynomials in q n .(4) Perform coefficient comparison in these remainders with respect to q n .(5) Solve the resulting linear system over K ( q ) for the unknowns c i,j ∈ K [ q ] (we may clear denominators since the system is homogeneous).(6) Return u i ( q, q n ) = (cid:80) d i j = e i c i,j ( q ) q jn . It is interesting to note that our computed bimonic recurrences reveal certainsymmetries in their coefficients, more precisely, they are kind of palindromic.For example, the bimonic q -recurrence that we found for J pretz − ( n ), written inthe form (cid:88) j =0 (cid:96) − ,j ( q ) q jn + (cid:88) i =0 9 (cid:88) j =0 (cid:96) i,j ( q ) q jn f ( n + i )17 f ( n ) f ( n + 1) f ( n + 2) f ( n + 3) f ( n + 4) f ( n + 5) q n − · · q n − · · − · · q n · · · · q n · · · · · q n · − · · · · q n · · · − · q n − · · · · · q n − · · · · q n − · · − · · − · q n · − · Table 2: Coefficients of the bimonic q -recurrence for J pretz − ( n ); for space reasonsonly the evaluations for q = 2 are given. In order to reveal the underlyingsymmetry, common powers of q are kept as powers of 2; for example, the entry24 · in the last-but-one column comes from the coefficient of q n f ( n +4) whichis q ( q + q − q + 2). The first column corresponds to the inhomogeneous part.has the following palindromicity properties (0 ≤ i ≤
5, 0 ≤ j ≤ (cid:96) i,j = q j − i − (cid:96) − i, − j and (cid:96) − ,j = − q j − (cid:96) − , − j . This phenomenon is illustrated in Table 2. It is also visible in Figure 3 buton a different example. The occurrence of palindromic operators in the contextof knot theory has been studied in more detail in [15]. Indeed, if we use thebimonic recurrence to define the sequence f ( n ) = ( q n − J pretz − ( n ) for n ≤ f ( n ) = − q n f ( − n ) for all n ∈ N . We have applied Algorithm 5.4 to all recurrences associated to the twistknots K twist p for − ≤ p ≤
15 and to some of the pretzel knots K pretz p . Allthese results can be found in the supplementary electronic material [22]. In this paper, we determine a generating set of the q -Weyl closure of a givenunivariate q -difference operator, and compute a desingularized operator whoseleading coefficient has minimal degree in q . Moreover, we use our algorithms tocertify that several instances of the colored Jones polynomial are Laurent poly-nomial sequences. A challenging topic for future research would be to considerthe corresponding problems in the multivariate case.Another direction of research we want to consider in the future is the desin-gularization problem for linear Mahler equations [23], which attracted quitesome interest in the computer algebra community recently, see for example [8].Mahler equations arise in the study of automatic sequences, in the complex-ity analysis of divide-and-conquer algorithms, and in some number-theoreticquestions. 18 cknowledgment The authors would like to thank Stavros Garoufalidis forproviding the examples of Section 5 and for enlightening discussions on theknot theory part. We are also grateful to the anonymous referee for the detailedreport and for numerous valuable comments.
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