Constructing minimal telescopers for rational functions in three discrete variables
Shaoshi Chen, Qing-Hu Hou, Hui Huang, George Labahn, Rong-Hua Wang
aa r X i v : . [ c s . S C ] M a y Constructing minimal telescopers for rational functionsin three discrete variables ∗ Shaoshi Chen , Qing-Hu Hou , Hui Huang , George Labahn , Rong-Hua Wang KLMM, AMSS, Chinese Academy of Sciences, Beijing, 100190, China Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China Symbolic Computation Group, University of Waterloo, Waterloo, ON, N2L 3G1, Canada School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin, 300387, China [email protected] , qh [email protected] { h2huang, glabahn } @uwaterloo.ca , [email protected] Abstract
We present a new algorithm for constructing minimal telescopers for rational functions inthree discrete variables. This is the first discrete reduction-based algorithm that goes beyond thebivariate case. The termination of the algorithm is guaranteed by a known existence criterionof telescopers. Our approach has the important feature that it avoids the potentially costlycomputation of certificates. Computational experiments are also provided so as to illustrate theefficiency of our approach.
Creative telescoping is a powerful tool used to find closed form solutions for definite sums anddefinite integrals. The method constructs a recurrence (resp. differential) equation satisfied bythe definite sum (resp. integral) with closed form solutions over a specified domain resulting informulas for the sum or integral. Methods for finding such closed form solutions are available formany special functions, with examples given in [2, 35, 4, 8, 25, 16, 39]. Even when no closedform exists the telescoping method often remains useful. For example the resulting recurrence ordifferential equation enables one to determine asymptotic expansions and derive other interestingfacts about the original sum or integral.In the case of summation, specialized to the trivariate case, in order to compute a sum of theform b X y = a b X z = a f ( x, y, z ) , ∗ S. Chen was supported by the NSFC grants 11501552, 11871067, 11688101 and by the Frontier Key Project(QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS.Q. Hou was supported by the NSFC grant 11771330.R. Wang was supported by the NSFC grants 11871067 and 11626170. c , . . . , c ρ , rational functions (or polynomi-als) in x , not all zero, and two functions g ( x, y, z ) , h ( x, y, z ) in the same domain as f such that c f + c σ x ( f ) + · · · + c ρ σ ρx ( f ) = ( σ y ( g ) − g ) + ( σ z ( h ) − h ) , (1.1)where σ x , σ y and σ z denote shift operators in x , y and z , respectively. The number ρ may or maynot be part of the input. If c , . . . , c ρ and g, h are as above, then L = c + · · · + c ρ σ ρx is called a telescoper for f and ( g, h ) is a certificate for L . Additional details, along with classical algorithmsfor computing telescopers and certificates, can be found in [36, 40, 41].Over the past two decades, a number of generalizations and refinements of creative telescopinghave been developed. At the present time reduction-based methods have gained the most supportas they are both efficient and have the important feature of being able to find a telescoper for agiven function without also computing a corresponding certificate. This is desirable in the situationwhere only the telescoper is of interest and its size is much smaller than the size of the certificate.This is often the case, for example, where the right hand side of (1.1) is known to collapse to zero.The reduction-based approach was first developed in the differential case for bivariate rationalfunctions [11], and later generalized to rational functions in several variables [14], to hyperexpo-nential functions [12], to algebraic functions [19] and to D-finite functions [21, 38, 13]. In the shiftcase a reduction-based approach was developed for hypergeometric terms [18, 29] and to multiplebinomial sums [15] (a subclass of the sums of proper hypergeometric terms).In the case of discrete functions having more than two variables no complete reduction-basedcreative telescoping algorithm has been known so far. The goal of the present paper is to take thefirst step towards filling the gap, namely to further extend the approach to the trivariate rationalcase where f, g, h are all rational functions in (1.1). This is a natural follow up to the recentwork [17] which solved the existence problem of telescopers for rational functions in three discretevariables.The basic idea of the general reduction-based approach, formulated for the shift trivariaterational case, is as follows. Let C be a field of characteristic zero. Assume that there is some C ( x )-linear map red( · ) : C ( x, y, z ) C ( x, y, z ) with the property that for all f ∈ C ( x, y, z ), thereexist g, h ∈ C ( x, y, z ) such that f − red( f ) = σ y ( g ) − g + σ z ( h ) − h . In other words, f − red( f )is summable with respect to y, z . Such a map is called a reduction with red( f ) considered as the remainder of f with respect to the reduction red( · ). Then in order to find a telescoper for f , we caniteratively compute red( f ) , red( σ x ( f )) , red( σ x ( f )) , . . . until we find a nontrivial linear dependenceover the field C ( x ). Once we have such a dependence, say c red( f ) + · · · + c ρ red( σ ρx ( f )) = 0for c i ∈ C ( x ) not all zero, then by linearity, red( c f + · · · + c ρ σ ρx ( f )) = 0, that is, c f + · · · + c ρ σ ρx ( f ) = σ y ( g ) − g + σ z ( h ) − h for some g, h ∈ C ( x, y, z ). This yields a telescoper c + · · · + c ρ σ ρx for f .To guarantee the termination of the above process, one possible way is to show that, for everysummable function f , we have red( f ) = 0. If this is the case and L = c + · · · + c ρ σ ρx is atelescoper for f , then L ( f ) is summable by the definition. So red( L ( f )) = 0, and again by thelinearity, red( f ) , . . . , red( σ ρx ( f )) are linear dependent over C ( x ). This means that we will not missany telescoper and that the method will terminate provided that a telescoper is known to exist.This approach was taken in [18]. It requires us to know exactly under what kind of conditionsa telescoper exists, so-called the existence problem of telescopers , and, when these conditions arefulfilled, then it is guaranteed to find one of minimal order ρ . Such existence problems have beenwell studied in the case of bivariate hypergeometric terms [5] and more recently in the triviariaterational case [17].A second, alternate way to ensure termination, used for example in [11, 12], is to show that, for agiven function f , the remainders red( f ) , red( σ x ( f )) , red( σ x ( f )) , . . . form a finite-dimensional C ( x )-vector space. Then, as soon as ρ exceeds this finite dimension, one can be sure that a telescoper oforder at most ρ will be found. This also implies that every bound for the dimension gives rise to anupper bound for the minimal order of telescopers. This approach provides an independent prooffor the existence of a telescoper. However, since such an upper order bound is only of theoreticalinterest and will not affect the practical efficiency of the algorithms, in this paper we will confineourselves with the first approach for termination and leave the second approach for future research.Our starting point is thus to find a suitable reduction for trivariate rational functions. Inparticular we present a reduction red( · ) which satisfies the following properties: (i) red( f ) = 0whenever f ∈ C ( x, y, z ) is summable and (ii) red( f ) is minimal in certain sense. One issue withthis reduction, similar to that encountered in [18], is the difficulty that red( · ) is not a C ( x )-linearmap in general. To overcome this we follow the ideas of [18]. Namely, we introduce the idea ofcongruence modulo summable rational functions and show that red( · ) becomes C ( x )-linear whenit is viewed as a residue class. Using the existence criterion of telescopers established in [17], weare then able to design a creative telescoping algorithm from red( · ) as described in the previousparagraphs.The remainder of the paper proceeds as follows. The next section gives some preliminarymaterial needed for this paper, particularly a review of a reduction method due to Abramov.In Section 3 we extend Abramov’s reduction method to the bivariate case by incorporating aprimary reduction. In Section 4 we show that the reduction remainders introduced in the previoussection are well-behaved with respect to taking linear combinations, followed in Section 5 by anew algorithm for constructing telescopers for trivariate rational functions based on the bivariateextension of Abramov’s reduction method. In Section 6 we provide some experimental tests of ournew algorithm. The paper ends with a conclusion and topics for future research. Throughout the paper we let C denote a field of characteristic zero, with F = C ( x ) and F ( y, z )the field of rational functions in y, z over F . We denote by σ y and σ z the shift operators of F ( y, z ),where for any f ∈ F ( y, z ) we have σ y ( f ( x, y, z )) = f ( x, y + 1 , z ) and σ z ( f ( x, y, z )) = f ( x, y, z + 1) . Let G be the free abelian group generated by the shift operators σ y and σ z . For any τ ∈ G a polynomial p ∈ F [ y, z ] is said to be τ -free if gcd( p, τ ℓ ( p )) = 1 for all nonzero ℓ ∈ Z . A rationalfunction f ∈ F ( y, z ) is called τ -summable if f = τ ( g ) − g for some g ∈ F ( y, z ). The τ -summabilityproblem is then to decide whether a given rational function in F ( y, z ) is τ -summable or not. Ratherthan merely giving a negative answer in case the function is not τ -summable, one could instead seeksolutions for a more general problem, namely the τ -decomposition problem , with the intent to makethe nonsummable part as small as possible. Precisely speaking, the τ -decomposition problem, for agiven rational function f ∈ F ( y, z ), asks for an additive decomposition of the form f = τ ( g ) − g + r ,where g, r ∈ F ( y, z ) and r is minimal in certain sense such that f would be τ -summable if and onlyif r = 0. It is readily seen that any solution to the decomposition problem tackles the correspondingsummability problem as well.In the case where τ = σ y , the decomposition problem was first solved by Abramov in [1] withrefined algorithms in [3, 34, 9, 26]. All these algorithms can be viewed as discrete analogues of theOstrogradsky-Hermite reduction for rational integration. We refer to any of these algorithms asthe reduction of Abramov. Theorem 2.1 (Reduction of Abramov) . Let f be a rational function in F ( y, z ) . Then the reductionof Abramov finds g ∈ F ( y, z ) and a, b ∈ F [ y, z ] with deg y ( a ) < deg y ( b ) and b being σ y -free such that f = σ y ( g ) − g + ab . Moreover, if f admits such a decomposition then • f is σ y -summable if and only if a = 0 ; • b has the lowest possible degree in y when gcd( a, b ) = 1 . That is, if there exist a second g ′ ∈ F ( y, z ) and a ′ , b ′ ∈ F [ y, z ] such that f = σ y ( g ′ ) − g ′ + a ′ /b ′ , then deg y ( b ′ ) ≥ deg y ( b ) . Generalizing to the bivariate case, we consider the ( σ y , σ z )-summability problem of decidingwhether a given rational function f ∈ F ( y, z ) can be written in the form f = σ y ( g ) − g + σ z ( h ) − h for g, h ∈ F ( y, z ). If such a form exists, we say that f is ( σ y , σ z ) -summable , abbreviated as summablein certain instances. The ( σ y , σ z )-decomposition problem is then to decompose a given rationalfunction f ∈ F ( y, z ) into the form f = σ y ( g ) − g + σ z ( h ) − h + r, where g, h, r ∈ F ( y, z ) and r is minimal in certain sense. Moreover, f is ( σ y , σ z )-summable if andonly if r = 0.Recall that an irreducible polynomial f ∈ F [ y, z ] is called ( y, z ) -integer linear over the F if itcan be written in the form f = p ( αy + βz ) for a polynomial p ( Z ) ∈ F [ Z ] and integers α, β ∈ Z . Onemay assume without loss of generality that β ≥ α, β are coprime. A polynomial in F [ y, z ] iscalled ( y, z ) -integer linear over F if all its irreducible factors are ( y, z )-integer linear over F while arational function in F ( y, z ) is called ( y, z ) -integer linear over F if its numerator and denominatorare both ( y, z )-integer linear over F . For simplicity, we just say a rational function is ( y, z )-integerlinear over F of ( α, β )-type if it is equal to p ( αy + βz ) for some p ( Z ) ∈ F ( Z ) and α, β are coprimeintegers with β ≥
0. Algorithms for determining integer linearity can be found in [6, 33, 27].In the context of creative telescoping, we will also need to consider the variable x and its shiftoperator σ x , which for every f ∈ F ( y, z ) maps f ( x, y, z ) to f ( x +1 , y, z ). Recall that two polynomials p, q ∈ C [ x, y, z ] are called ( x, y, z ) -shift equivalent , denoted by p ∼ x,y,z q , if there exist three integers ℓ, m, n such that p = σ ℓx σ my σ nz ( q ). We generalize this notion to the domain F [ y, z ] by saying thattwo polynomials p, q ∈ F [ y, z ] are ( x, y, z )-shift equivalent if p = σ ℓx σ my σ nz ( q ) for integers ℓ, m, n .When ℓ = 0 then p is also called ( y, z ) -shift equivalent to q , denoted by p ∼ y,z q . Clearly, ∼ x,y,z , aswell as ∼ y,z , is an equivalence relation.Let F ( y, z )[S x , S y , S z ] be the ring of linear recurrence operators in x, y, z over F ( y, z ). HereS x , S y , S z commute with each other, and S v ( f ) = σ v ( f ) S v for any f ∈ F ( y, z ) and v ∈ { x, y, z } .The application of an operator P = P i,j,k ≥ p ijk S ix S jy S kz in F ( y, z )[S x , S y , S z ] to a rational function f ∈ F ( y, z ) is then defined as P ( f ) = X i,j,k ≥ p ijk σ ix σ jy σ kz ( f ) . Definition 2.2.
Let f be a rational function in F ( y, z ) . A nonzero linear recurrence operator L ∈ F [S x ] is called a telescoper for f if L ( f ) is ( σ y , σ z ) -summable, or equivalently, if there existrational functions g, h ∈ F ( y, z ) such that L ( f ) = (S y − g ) + (S z − h ) , where denotes the identity map of F ( y, z ) . We call ( g, h ) a corresponding certificate for L . The order of a telescoper is defined to be its degree in S x . A telescoper of minimal order for f is calleda minimal telescoper for f . In this paper, choosing the pure lexicographic order y ≺ z , we say a polynomial in F [ y, z ]is monic if its highest term with respect to y, z has coefficient one. For a nonzero polynomial p ∈ F [ y, z ], its degree and leading coefficient with respect to the variable v ∈ { y, z } are denoted bydeg v ( p ) and lc v ( p ), respectively. We will follow the convention that deg v (0) = −∞ . In this section, we demonstrate how to solve the bivariate decomposition problem (and thusalso the bivariate summability problem) using the reduction of Abramov. To this end, let us firstrecall some key results on the bivariate summability from [28].Based on a theoretical criterion given in [20], Hou and Wang [28] developed a practical algorithmfor solving the ( σ y , σ z )-summability problem. The proof found in [28, Lemma 3.1] contains areduction algorithm with inputs and outputs specified as follows. Primary reduction.
Given a rational function f ∈ F ( y, z ), compute rational functions g, h, r ∈ F ( y, z ) such that f = (S y − g ) + (S z − h ) + r (3.1)and r is of the form r = m X i =1 n i X j =1 a ij b ij d ji (3.2)with m, n i ∈ N , a ij , d i ∈ F [ y, z ] and b ij ∈ F [ y ] satisfying that • deg z ( a ij ) < deg z ( d i ), • d i is a monic irreducible factor of the denominator of r and of positive degree in z , • d i ≁ y,z d i ′ whenever i = i ′ for 1 ≤ i, i ′ ≤ m .Let f be a rational function in F ( y, z ) and assume that applying the primary reduction to f yields (3.1). Deciding if f is ( σ y , σ z )-summable then amounts to checking the summability of r . By[28, Lemma 3.2], this is equivalent to checking the summability of each simple fraction a ij / ( b ij d ji ).Thus the bivariate summability problem for a general rational function is reduced to determiningthe summability of several simple fractions, which in turn can be addressed by the following. Theorem 3.1 ([28, Theorem 3.3]) . Let f = a/ ( b d j ) , where a, d ∈ F [ y, z ] , b ∈ F [ y ] , j ∈ N \ { } and d irreducible with ≤ deg z ( a ) < deg z ( d ) . Then f is ( σ y , σ z ) -summable if and only if(i) d is ( y, z ) -integer linear over F of ( α, β ) -type, R ψ R σ R ψ R σ F ( y, z ) φ α,β F ( y, z ) τ = σ βy σ − αz F ( y, z ) φ α,β F ( y, z ) σ y Figure 1: Commutative diagrams for difference homomorphisms/isomorphisms. (ii) there exists q ∈ F ( y )[ z ] with deg z ( q ) < deg z ( d ) so that ab = σ βy σ − αz ( q ) − q. (3.3)Since d is irreducible, the first condition is easily recognized by comparing coefficients once d is known. In [28, § m -fold Gosper representation. Any polynomial solution of the system gives rise to a desired q for(3.3). For the rest of this section, we show how to detect the second condition via the reduction ofAbramov, without solving any auxiliary recurrence equations. Before that, we need some notions.Let R be a ring and σ : R → R be a homomorphism of R . The pair ( R, σ ) is called a differencering . An element r ∈ R is called a constant of the difference ring ( R, σ ) if σ ( r ) = r . The set ofall constants in R forms a subring of R , called the subring of constants . If R is a field, the ( R, σ )is called a difference field. Let ( R , σ ) and ( R , σ ) be two difference rings and ψ : R → R bea homomorphism. If σ ◦ ψ = ψ ◦ σ , that is, the left diagram in Figure 1 commutes, we call ψ a difference homomorphism , or a difference isomorphism if ψ is an isomorphism. Two differencerings are then said to be isomorphic if there exists a difference isomorphism between them.Let α, β be two integers with β nonzero. We define an F -homomorphism φ α,β : F ( y, z ) → F ( y, z )by φ α,β ( y ) = βy and φ α,β ( z ) = β − z − αy. It is readily seen that φ α,β is an F -isomorphism with inverse φ − α,β given by φ − α,β ( y ) = β − y and φ − α,β ( z ) = βz + αy. We call φ α,β the map for ( α, β ) -shift reduction . Proposition 3.2.
Let α, β ∈ Z with β = 0 and τ = σ βy σ − αz . Then φ α,β is a difference isomorphismbetween the difference fields ( F ( y, z ) , τ ) and ( F ( y, z ) , σ y ) .Proof. Since φ α,β is an F -isomorphism, it remains to show that σ y ◦ φ α,β = φ α,β ◦ τ , namely theright diagram in Figure 1 commutes. This is confirmed by the observations that σ y ( φ α,β ( f ( y, z ))) = σ y ( f ( βy, β − z − αy )) = f ( βy + β, β − z − αy − α )and φ α,β ( τ ( f ( y, z ))) = φ α,β ( f ( y + β, z − α )) = f ( βy + β, β − z − αy − α )for any f ∈ F ( y, z ). ✷ Corollary 3.3.
Let f ∈ F ( y, z ) and assume the conditions of Proposition 3.2. Then f is τ -summable if and only if φ α,β ( f ) is σ y -summable.Proof. By Proposition 3.2, φ α,β is a difference isomorphism between ( F ( y, z ) , τ ) and ( F ( y, z ) , σ y ).It follows that f = τ ( g ) − g ⇐⇒ φ α,β ( f ) = φ α,β ( τ ( g ) − g ) = σ y ( φ α,β ( g )) − φ α,β ( g )for any g ∈ F ( y, z ). The assertion follows. ✷ The problem of deciding whether a rational function f ∈ F ( y )[ z ] satisfies the equation (3.3),that is, the σ βy σ − αz -summability problem for f , is now reduced to the σ y -summability problem for φ α,β ( f ). In fact, there is also a one-to-one correspondence between additive decompositions of f with respect to σ βy σ − αz and additive decompositions of φ α,β ( f ) with respect to σ y . In particular,we have the following proposition. Proposition 3.4.
Let f ∈ F ( y )[ z ] and assume the conditions of Proposition 3.2. Suppose also that φ α,β ( f ) admits the decomposition φ α,β ( f ) = σ y (˜ g ) − ˜ g + ˜ a ˜ b , (3.4) where ˜ g ∈ F ( y )[ z ] , ˜ a ∈ F [ y, z ] and ˜ b ∈ F [ y ] with deg y (˜ a ) < deg y (˜ b ) and ˜ b being σ y -free. Let g = φ − α,β (˜ g ) , a = φ − α,β (˜ a ) and b = φ − α,β (˜ b ) . Then f = τ ( g ) − g + ab (3.5) and g, a, b satisfy the conditions(i) a can be written as P deg z ( a ) i =0 ˆ a i · ( αy + βz ) i for ˆ a i ∈ F [ y ] with deg y (ˆ a i ) < deg y ( b ) ,(ii) b is σ βy -free,with f being τ -summable if and only if a = 0 . Moreover, if gcd( a, b ) = 1 then b has the lowestpossible degree in y .Proof. By Proposition 3.2, σ y ◦ φ α,β = φ α,β ◦ τ and thus φ − α,β ◦ σ y = τ ◦ φ − α,β . Then applying φ − α,β to both sides of (3.4) yields (3.5). Since φ α,β ( f ) is σ y -summable if and only if ˜ a = 0 according toTheorem 2.1, one sees from Corollary 3.3 that f is τ -summable if and only if a = 0. Moreover,if gcd( a, b ) = 1 then gcd(˜ a, ˜ b ) = 1. By Theorem 2.1, ˜ b has the lowest possible degree in y . Theminimality of b then follows from that of ˜ b because deg y ( b ) = deg y (˜ b ).It remains to show that the conditions (i)-(ii) hold. By the definition of φ − α,β , we have that a = φ − α,β (˜ a ) = ˜ a ( β − y, αy + βz ) ∈ F [ y, z ]. It is then evident that there exist the ˆ a i ∈ F [ y ] so that a can be written into the required form described in condition (i). Further observe that deg y (˜ a ) =max i deg y (ˆ a i ) and ˜ b ∈ F [ y ]. Thus the first condition follows by deg y (˜ a ) < deg y (˜ b ) = deg y ( b ). Since˜ b is σ y -free, we have that gcd(˜ b, σ ℓy (˜ b )) = 1 for any nonzero ℓ ∈ Z . Notice that φ − α,β ◦ σ ℓy = τ ℓ ◦ φ − α,β and b ∈ F [ y ], Therefore, gcd( b, τ ℓ ( b )) = gcd( b, σ βℓy ( b )) = 1 by the definition of τ . This implies that b is σ βy -free, proving the second condition. ✷ Remark 3.5.
With the notations and assumptions of Proposition 3.4, one is able to further reducethe size of the polynomial a so that condition (i) is replaced by the condition deg y ( a ) < deg y ( b ) .This is true since every polynomial is τ -summable, which is in turn implied by Corollary 3.3 and thefact that every polynomial is σ y -summable. This reduction, however, does not affect the applicabilityof later algorithms. Proposition 3.4 motivates us to introduce the notions of remainder fractions and remainders, inorder to characterize nonsummable rational functions concretely.
Definition 3.6.
A fraction a/ ( b d j ) with a, d ∈ F [ y, z ] , b ∈ F [ y ] and j ∈ N \{ } is called a remainderfraction if • deg z ( a ) < deg z ( d ) ; • d is monic, irreducible and of positive degree in z ; • conditions (i)-(ii) in Proposition 3.4 hold in case d is ( y, z ) -integer linear over F of ( α, β ) -type. Definition 3.7.
Let f be a rational function in F ( y, z ) . Then r ∈ F ( y, z ) is called a ( σ y , σ z )-remainder of f if f − r is ( σ y , σ z ) -summable and r can be written as r = m X i =1 n i X j =1 a ij b ij d ji , (3.6) where m, n i ∈ N , a ij , d i ∈ F [ y, z ] , b ij ∈ F [ y ] with each a ij / ( b ij d ji ) being a remainder fraction, d i being a factor of the denominator of r , and d i ≁ y,z d i ′ whenever i = i ′ and ≤ i, i ′ ≤ m . Forbrevity, we just say that r is a ( σ y , σ z ) -remainder if f is clear from the context. Sometimes we alsojust say remainder for short unless there is a danger of confusion. We refer to (3.6) , along withthe attached conditions, as the remainder form of r . The uniqueness of partial fraction decompositions (in this case with respect to z ) implies thatthe remainder form for a given ( σ y , σ z )-remainder is unique up to reordering and multiplicationby units of F . Evidently, every single remainder fraction, or part of summands in (3.6), is a( σ y , σ z )-remainder. Remainders not only helps us to recognize summability, but also describes the“minimum” gap between a given rational function and summable rational functions, as shown inthe next two propositions. Proposition 3.8.
Let r ∈ F ( y, z ) be a nonzero ( σ y , σ z ) -remainder with the form (3.6) . Then eachnonzero a ij / ( b ij d ji ) for ≤ i ≤ m and ≤ j ≤ n i , as well as r itself, is not ( σ y , σ z ) -summable.Proof. Since r is a ( σ y , σ z )-remainder, each a ij / ( b ij d ji ) is a remainder fraction. For a particularnonzero a ij / ( b ij d ji ), namely a ij = 0, we claim that it is not ( σ y , σ z )-summable. If d i is not ( y, z )-integer linear over F , then the simple fraction is not ( σ y , σ z )-summable by Theorem 3.1. Otherwise,assume that d i is ( y, z )-integer linear over F of ( α, β )-type. Since a ij / ( b ij d ji ) is a remainder fraction,Definition 3.6 reads that a ij , when viewed as a polynomial in ( αy + βz ), has coefficients of degreesin y less than deg y ( b ij ) and that b ij is σ βy -free. By Proposition 3.4, a ij /b ij is not σ βy σ − αz -summable.The claim then follows by Theorem 3.1.In either case, we have that a ij / ( b ij d ji ) is not ( σ y , σ z )-summable. Since r is nonzero, at leastone of the a ij / ( b ij d ji ) is nonzero. By [28, Lemma 3.2], r is therefore not ( σ y , σ z )-summable. ✷ Proposition 3.9.
Let r ∈ F ( y, z ) be a nonzero ( σ y , σ z ) -remainder with the form (3.6) , in which a ij and b ij d ji are further assumed to be coprime. Assume that there exists another r ′ ∈ F ( y, z ) suchthat r ′ − r is ( σ y , σ z ) -summable. Write r ′ in the form r ′ = p ′ + m ′ X i =1 n ′ i X j =1 a ′ ij b ′ ij d ′ ij , where m ′ , n ′ i ∈ N , p ′ ∈ F ( y )[ z ] , a ′ ij , d ′ i ∈ F [ y, z ] and b ′ ij ∈ F [ y ] with deg z ( a ′ ij ) < deg z ( d ′ i ) and d ′ i beingmonic irreducible factor of the denominator of r ′ and of positive degree in z . For each ≤ i ≤ m ,define Λ i = { i ′ ∈ N | ≤ i ′ ≤ m ′ and d ′ i ′ = σ λ i ′ y σ µ i ′ z ( d i ) for λ i ′ , µ i ′ ∈ Z } . Then Λ i is nonempty for any ≤ i ≤ m . Moreover, m ≤ m ′ , n i ≤ n ′ i ′ for all i ′ ∈ Λ i , deg y ( b ij ) ≤ P i ′ ∈ Λ i deg y ( b ′ i ′ j ) for each ≤ i ≤ m and ≤ j ≤ n i , and the degree in z of the denominator of r is no more than that of r ′ .Proof. Since r ′ − r is ( σ y , σ z )-summable, all the rational function P i ′ ∈ Λ i a ′ i ′ j / ( b ′ i ′ j d ′ i ′ j ) − a ij / ( b ij d ji )are ( σ y , σ z )-summable by [28, Lemma 3.2], and then so are the X i ′ ∈ Λ i σ − λ i ′ y σ − µ i ′ z ( a ′ i ′ j ) σ − λ i ′ y ( b ′ i ′ j ) d ji − a ij b ij d ji . (3.7)Since r is a nonzero ( σ y , σ z )-remainder, we conclude from Proposition 3.8 that each nonzero a ij / ( b ij d ji ) is not ( σ y , σ z )-summable. Notice that for each 1 ≤ i ≤ m , there is at least one in-teger j with 1 ≤ j ≤ n i such that a ij = 0. It then follows from the summability of (3.7) that everyΛ i is nonempty, namely every d i is ( y, z )-shift equivalent to some d ′ i ′ for 1 ≤ i ′ ≤ m ′ , and that n i ≤ n ′ i ′ for any i ′ ∈ Λ i . Notice that the d i are pairwise ( y, z )-shift inequivalent. Thus the Λ i arepairwise disjoint, which implies that m ≤ m ′ . Accordingly, the degree in z of the denominator of r is no more than that of r ′ .It remains to show the inequality for the degree of each b ij . For each 1 ≤ i ≤ m and 1 ≤ j ≤ n i ,by Theorem 3.1, the summability of (3.7) either yields X i ′ ∈ Λ i σ − λ i ′ y σ − µ i ′ z ( a ′ i ′ j ) σ − λ i ′ y ( b ′ i ′ j ) = σ βy σ − αz ( q ) − q + a ij b ij for some q ∈ F ( y )[ z ] , if d i is ( y, z )-integer linear over F of ( α, β )-type or otherwise yields X i ′ ∈ Λ i σ − λ i ′ y σ − µ i ′ z ( a ′ i ′ j ) σ − λ i ′ y ( b ′ i ′ j ) = a ij b ij . The assertion is evident in the latter case. For the former case, the assertion then follows by theminimality of b ij from Proposition 3.4, because a ij / ( b ij d ji ) is a remainder fraction. ✷ σ y , σ z )-decomposition problem. Bivariate reduction of Abramov.
Given a rational function f ∈ F ( y, z ), compute three rationalfunctions g, h, r ∈ F ( y, z ) such that r is a ( σ y , σ z )-remainder of f and f = (S y − g ) + (S z − h ) + r. (3.8)1. apply the primary reduction to f to find g, h ∈ F ( y, z ), m, n i ∈ N , a ij , d i ∈ F [ y, z ] and b ij ∈ F [ y ] such that (3.1) holds.2. for i = 1 , . . . , m doif d i is ( y, z )-integer linear over F of ( α i , β i )-type then2.1 compute ˜ a ij / ˜ b ij = φ α i ,β j ( a ij /b ij ) with φ α i ,β i being the map for ( α i , β i )-shift reduction;2.2 for j = 1 , . . . , n i do2.2.1 apply the reduction of Abramov to ˜ a ij / ˜ b ij with respect to y to get ˜ q ij , ˜ r ij ∈ F ( y )[ z ]such that ˜ a ij ˜ b ij = σ y (˜ q ij ) − ˜ q ij + ˜ r ij . φ − α i ,β i to both sides of the previous equation to get a ij b ij = σ β i y σ − α i z ( q ij ) − q ij + r ij , (3.9)where q ij = φ − α i ,β i (˜ q ij ) and r ij = φ − α i ,β i (˜ r ij ).2.2.3 update a ij and b ij to be the numerator and denominator of r ij , respectively.2.3 update g = g + n i X j =1 β i − X k =0 σ ky σ − α i z q ij d ji ! and h = h + P n i j =1 P α i k =1 σ − kz (cid:18) − q ij d ji (cid:19) α i ≥ P n i j =1 P − α i − k =0 σ kz (cid:18) q ij d ji (cid:19) α i < .
3. set r = P mi =1 P n i j =1 a ij / ( b ij d ji ), and return g, h, r . Theorem 3.10.
Let f be a rational function in F ( y, z ) . Then the bivariate reduction of Abramovcomputes two rational functions g, h ∈ F ( y, z ) and a ( σ y , σ z ) -remainder r ∈ F ( y, z ) such that (3.8) holds. Moreover, f is ( σ y , σ z ) -summable if and only if r = 0 .Proof. Applying the primary reduction to f yields (3.1). For any nonzero a ij / ( b ij d ji ) obtained instep 1, if d i is not ( y, z )-integer linear over F then we know from Theorem 3.1 that a ij / ( b ij d ji ) isnot ( σ y , σ z )-summable and is thus already a remainder fraction. Otherwise, there exist coprimeintegers α i , β i with β i > d i = p i ( α i y + β i z ) for some p i ( Z ) ∈ F [ Z ]. By Proposition 3.4,for each 1 ≤ j ≤ n i , steps 2.2.1-2.2.2 correctly find q ij and r ij such that (3.9) holds and r ij /d ji is a1remainder fraction. After step 2.2, plugging all (3.9) into (3.1) then gives (with a slight abuse ofnotation): f = (S y − g ) + (S z − h ) + X i : d i = p i ( α i y + β i z ) n i X j =1 σ β i y σ − α i z ( q ij ) − q ij d ji + r, where the index i runs through all ( y, z )-integer linear d i ’s and r = P mi =1 P n i j =1 a ij / ( b ij d ji ) is a( σ y , σ z )-remainder by Definition 3.7. The assertions then follow from Proposition 3.8 and theobservation that σ β i y σ − α i z ( q ij ) − q ij d ji = (S y − β i − X k =0 σ ky σ − α i z q ij d ji !! + (S z − (cid:18)P α i k =1 σ − kz (cid:18) − q ij d ji (cid:19)(cid:19) if α i ≥ z − (cid:18)P − α i − k =0 σ kz (cid:18) q ij d ji (cid:19)(cid:19) if α i < d i = p i ( α i y + β i z ). ✷ Example 3.11.
Consider the rational function f admitting the partial fraction decomposition f = f + f + f with f = x z + 1( x + y )( x + z ) + 1 | {z } d , f = ( x + xy + 3 x − z − x − y + 3( x + y )( x + y + 3)(( x + 2 y + 3 z ) + 1 | {z } d ) and f = 1 x − y + z | {z } d . Note that d , d , d are ( y, z ) -shift inequivalent to each other. Hence f remains unchanged afterapplying the primary reduction. Since d is not ( y, z ) -integer linear, we leave f untouched andproceed to deal with f . Notice that d is ( y, z ) -integer linear of (2 , -type. Then applying thereduction of Abramov to φ , ( f d ) yields φ , ( f d ) = (S y − (cid:18) z − xy − x y + 2 x x + 3 y ) (cid:19) + xz + x + 1 x + 3 y , which, when applied by φ − , , leads to f d = (S y S − z − q ) + x (2 y + 3 z ) + x + 1 x + y with q = 3(2 y + 3 z ) − xy − x y + 6 x x + y ) . Using (3.10) , we decompose f as f = (S y − X k =0 σ ky σ − z (cid:16) q d (cid:17)! + (S z − X k =1 σ − kz (cid:16) − q d (cid:17)! + x (2 y + 3 z ) + x + 1( x + y )(( x + 2 y + 3 z ) + 1) | {z } r . One sees that r is a ( σ y , σ z ) -remainder of f , and thus f is not ( σ y , σ z ) -summable by Theorem 3.10.Along the same lines as above, we have f = (S y − (cid:18) yx − y + z + 1 (cid:19) + (S z − (cid:18) yx − y + z (cid:19) , implying that f is ( σ y , σ z ) -summable. Combining everything together, f is finally decomposed as f = (S y − g ) + (S z − h ) + f + r with g = P k =0 σ ky σ − z (cid:16) q d (cid:17) + y/ ( x − y + z + 1) and h = P k =1 σ − kz (cid:16) − q d (cid:17) + y/ ( x − y + z ) . Therefore, f is not ( σ y , σ z ) -summable by Theorem 3.10. We will use f as a running example in this paper. As mentioned in the introduction, we expect our reduction algorithm to induce a linear map,that is, the sum of two remainders was expected to also be a remainder. Unfortunately, this isnot always the case in our setting, because some requirements in Definition 3.7 may be broken bythe addition among remainders, as seen in the following examples. This prevents us from applyingthe bivariate reduction of Abramov developed in the previous section to construct a telescoperin a direct way as was done in the differential case. However, observe that a rational functionin F ( y, z ) may have more than one ( σ y , σ z )-remainder and any two of them differ by a ( σ y , σ z )-summable rational function. This suggests a possible way to circumvent the above difficulty. Thatis, choosing proper members from the residue class modulo summable rational functions of thegiven remainders so as to make the linearity become true. The goal of this section is to show thatthis direction always works and it can be accomplished algorithmically. We note that a similar ideawas used in [18, § Example 4.1.
Let r = f and s = σ x ( f ) with f being given in Example 3.11. Then r and s are both ( σ y , σ z ) -remainders since both denominators d and σ x ( d ) are not ( y, z ) -integer linear.However their sum is not a ( σ y , σ z ) -remainder since d is ( y, z ) -shift equivalent to σ x ( d ) , namely d = σ − y σ − z σ x ( d ) . Nevertheless, we can find another ( σ y , σ z ) -remainder t of s such that r + t hasthis property. For example, let t = (S y − (cid:0) − σ − y ( s ) (cid:1) + (S z − (cid:0) − σ − y σ − z ( s ) (cid:1) + s = ( x + 1) ( z −
1) + 1( x + y )( x + z ) + 1 , and then r + t = (2 x + 2 x + 1) z − x − x + 1( x + y )( x + z ) + 1 is a ( σ y , σ z ) -remainder by definition. Alternatively, one can compute a ( σ y , σ z ) -remainder ˜ t of r ,say ˜ t = (S y −
1) ( r ) + (S z −
1) ( σ y ( r )) + r = x ( z + 1) + 1( x + y + 1)( x + z + 1) + 1 so that ˜ t + s = (2 x + 2 x + 1) z + x + 2( x + y + 1)( x + z + 1) + 1 is a ( σ y , σ z ) -remainder. Example 4.2.
Let r = x (2 y + 3 z ) + x + 1( x + y )(( x + 2 y + 3 z ) + 1) and s = ( x + 1)(2 y + 3 z ) + ( x + 1) + 2 x + ( x + y + 5)(( x + 2 y + 3 z + 1) + 1) . Then both r and s are ( σ y , σ z ) -remainders, but again their sum is not since ( x + 2 y + 3 z ) + 1 is ( y, z ) -shift equivalent to ( x + 2 y + 3 z + 1) + 1 . As in Example 4.1, we find a ( σ y , σ z ) -remainder ˜ s = a/b ( x + 2 y + 3 z ) + 1 with ab = ( x + 1)(2 y + 3 z ) + x + 3 x + 4 x + y + 6 such that s − ˜ s is ( σ y , σ z ) -summable. However, the sum r + ˜ s is still not a ( σ y , σ z ) -remainder since ( x + y )( x + y + 6) is not σ βy -free (here β = 3 in this case). Notice that ab = (S y S − z − X k =1 σ − ky σ kz (cid:16) ab (cid:17)! + ( x + 1)(2 y + 3 z ) + x + 3 x + 4 x + y , so (3.10) enables us to find a new ( σ y , σ z ) -remainder t = ( x + 1)(2 y + 3 z ) + x + 3 x + 4( x + y )(( x + 2 y + 3 z ) + 1) such that s − t is ( σ y , σ z ) -summable and r + t = ( x + 1)(2 y + 3 z ) + x + 3 x + 5( x + y )(( x + 2 y + 3 z ) + 1) is a ( σ y , σ z ) -remainder. Another possible choice is to find a ( σ y , σ z ) -remainder ˜ r of r such that ˜ r + s is a ( σ y , σ z ) -remainder. In order to achieve the linearity of remainders, we need to develop two lemmas. The first oneis an immediate result of Theorem 5.6 in [18] based on Proposition 3.4.
Lemma 4.3.
Let α, β ∈ Z with β = 0 . Let a ∈ F [ y, z ] and b ∈ F [ y ] \ { } with b being σ βy -free. Thenfor any given σ βy -free polynomial b ∗ ∈ F [ y ] , one finds q ∈ F ( y )[ z ] , a ′ ∈ F [ y, z ] and b ′ ∈ F [ y ] with a ′ of the form P deg z ( a ′ ) i =0 ˆ a ′ i ( αy + βz ) i for ˆ a ′ i ∈ F [ y ] satisfying deg y (ˆ a ′ i ) < deg y ( b ′ ) , b ′ being σ βy -free and gcd( b ∗ , σ βℓy ( b ′ )) = 1 for any nonzero ℓ ∈ Z , such that ab = (S βy S − αz − q ) + a ′ b ′ . Proof.
Let τ = σ βy σ − αz . Then b is τ -free, since b ∈ F [ y ] and it is σ βy -free. By Proposition 3.2, wehave that σ y ◦ φ α,β = φ α,β ◦ τ . Thus σ ℓ ′ y ◦ φ α,β = φ α,β ◦ τ ℓ ′ for any integer ℓ ′ . It follows thatgcd( φ α,β ( b ) , σ ℓ ′ y ( φ α,β ( b ))) = gcd( φ α,β ( b ) , φ α,β ( τ ℓ ′ ( b ))) = 1 for any nonzero ℓ ′ ∈ Z , implying that φ α,β ( b ) is σ y -free. Similarly, one shows that φ α,β ( b ∗ ) is also σ y -free.Now by [18, Theorem 5.6], there exist ˜ q ∈ F ( y )[ z ], ˜ a ∈ F [ y, z ] and ˜ b ∈ F [ y ] with deg y (˜ a ) < deg y (˜ b ), ˜ b being σ y -free and gcd( φ α,β ( b ∗ ) , σ ℓy (˜ b )) = 1 for any nonzero ℓ ∈ Z such that φ α,β (cid:16) ab (cid:17) = σ y (˜ q ) − ˜ q + ˜ a ˜ b . q = φ − α,β (˜ q ), a ′ = φ − α,β (˜ a ) and b ′ = φ − α,β (˜ b ). Then gcd( b ∗ , σ βℓ ( b ′ )) = 1 for any nonzero ℓ ∈ Z ,since φ − α,β ◦ σ ℓy = τ ℓ ◦ φ − α,β and τ ( b ′ ) = σ βy ( b ′ ). The assertion thus follows from Proposition 3.4. ✷ Lemma 4.4.
Let a/ ( b d j ) with a, d ∈ F [ y, z ] , b ∈ F [ y ] and j ∈ N \ { } be a remainder fraction.Then for any integer pair ( λ, µ ) , one finds g, h ∈ F ( y, z ) such that ab d j = (S y − g ) + (S z − h ) + σ λy σ µz ( a ) σ λy ( b ) σ λy σ µz ( d ) j . (4.1) Furthermore,(i) if d is not ( y, z ) -integer linear over F , then σ λy σ µz ( a ) / ( σ λy ( b ) σ λy σ µz ( d ) j ) is a remainder fraction;(ii) if d is ( y, z ) -integer linear over F of ( α, β ) -type, then for any given σ βy -free polynomial b ∗ ∈ F [ y ] , one further finds g ′ , h ′ ∈ F ( y, z ) , a ′ ∈ F [ y, z ] and b ′ ∈ F [ y ] with a ′ / ( b ′ σ λy σ µz ( d ) j ) being aremainder fraction and gcd( b ∗ , σ βℓy ( b ′ )) = 1 for any nonzero ℓ ∈ Z , such that ab d j = (S y − g ′ ) + (S z − h ′ ) + a ′ b ′ σ λy σ µz ( d ) j . (4.2) Proof.
The equation (4.1) follows by iteratively applying the formulas st = (S v − − i − X j =0 σ jv (cid:16) st (cid:17) + σ iv ( s ) σ iv ( t ) = (S v − i X j =1 σ − jv (cid:16) st (cid:17) + σ − iv ( s ) σ − iv ( t )for any s, t ∈ F [ y, z ], i ∈ N and v ∈ { y, z } .To see (i) note that deg z ( a ) < deg z ( d ) and d is monic, irreducible and of positive degree in z , as a/ ( b d j ) is a remainder fraction. Shifting polynomials in F [ y, z ] with respect to y or z preservesthese properties. Since σ λy σ µz ( d ) is again not ( y, z )-integer linear over F , the assertion holds byDefinition 3.6.For (ii) applying Lemma 4.3 to σ λy σ µz ( a ) /σ λy ( b ) in (4.1) with respect to b ∗ yields σ λy σ µz ( a ) σ λy ( b ) = (S βy S − αz − q ) + a ′ b ′ , where q ∈ F ( y )[ z ], a ′ ∈ F [ y, z ] and b ′ ∈ F [ y ] with a ′ / ( b ′ σ λy σ µz ( d ) j ) being a remainder fraction andgcd( b ∗ , σ βℓy ( b ′ )) = 1 for any nonzero ℓ ∈ Z . Moreover, (3.10) enables one to compute g ′ , h ′ ∈ F ( y, z )so that σ λy σ µz ( a ) σ λy ( b ) σ λy σ µz ( d ) j = (S y − g ′ ) + (S z − h ′ ) + a ′ b ′ σ λy σ µz ( d ) j . Plugging this into (4.1) and updating g ′ = g + g ′ , h ′ = h + h ′ gives (4.2). ✷ Remainder linearization.
Given two ( σ y , σ z )-remainders r, s ∈ F ( y, z ), compute g, h ∈ F ( y, z )and a ( σ y , σ z )-remainder t ∈ F ( y, z ) such that s = (S y − g ) + (S z − h ) + t with r + t being a ( σ y , σ z )-remainder.1. write r and s in the remainder forms r = ¯ m X i =1 ¯ n i X j =1 ¯ a ij ¯ b ij ¯ d ji and s = m X i =1 n i X j =1 a ij b ij d ji . (4.3)2. set g = h = 0.for i = 1 , . . . , m doif there exists k ∈ { , , . . . , ¯ m } such that ¯ d k = σ λy σ µz ( d i ) for some λ, µ ∈ Z , then2.1 for j = 1 , . . . , n i , apply Lemma 4.4 to a ij / ( b ij d ji ) to find g ij , h ij ∈ F ( y, z ), a ′ ij ∈ F [ y, z ]and b ′ ij ∈ F [ y ] with a ′ ij / ( b ′ ij ¯ d jk ) being a remainder fraction and – gcd(¯ b kj , σ βℓy ( b ′ ij )) = 1 for any nonzero ℓ ∈ Z if d i is ( y, z )-integer linear over F of( α, β )-type, – a ′ ij = σ λy σ µz ( a ij ) and b ′ ij = σ λy ( b ij ) otherwise,such that a ij b ij d ji = (S y − g ij ) + (S z − h ij ) + a ′ ij b ′ ij ¯ d jk ; (4.4)and update a ij , b ij and d i to be a ′ ij , b ′ ij and ¯ d k , respectively.2.2 update g = g + P n i j =1 g ij and h = h + P n i j =1 h ij .3. set t = P mi =1 P n i j =1 a ij / ( b ij d ji ), and return g, h, t . Theorem 4.5.
Let r and s be two ( σ y , σ z ) -remainders. Then the remainder linearization correctlyfinds two rational functions g, h ∈ F ( y, z ) and a ( σ y , σ z ) -remainder t ∈ F ( y, z ) such that s = (S y − g ) + (S z − h ) + t. (4.5) and c r + c t is a ( σ y , σ z ) -remainder for all c , c ∈ F .Proof. Since both r and s are ( σ y , σ z )-remainders, they can be written into the remainder forms(4.3). DefineΛ s = { i ∈ N | ≤ i ≤ m and d i = σ − λ i y σ − µ i z ( ¯ d k ) for some λ i , µ i ∈ Z and 1 ≤ k ≤ ¯ m } , Λ cs = { , , . . . , m } \ Λ s , and Λ cr = { k ∈ N | ≤ k ≤ ¯ m and ¯ d k ≁ y,z d i for all 1 ≤ i ≤ m } . i ∈ Λ s , denote by k i the integer so that 1 ≤ k i ≤ ¯ m and ¯ d k i ∼ y,z d i . Let { d ′ , . . . , d ′ m } be aset of polynomials in F [ y ] with d ′ i = ¯ d k i if i ∈ Λ s and d ′ i = d i if i ∈ Λ cs . Then d ′ , . . . , d ′ m are pairwise( y, z )-shift inequivalent since s is a ( σ y , σ z )-remainder. For all i ∈ Λ s and all 1 ≤ j ≤ n i , onesees from Lemma 4.4 that step 2.1 correctly finds the g ij , h ij , a ′ ij , b ′ ij satisfying described conditionssuch that (4.4) holds. It then follows from Definition 3.7 that t = P mi =1 P n i j =1 a ′ ij / ( b ′ ij d ′ ij ) is a( σ y , σ z )-remainder. Substituting all (4.4) into (4.3), together with step 2.2, immediately yields theequation (4.5).Let c , c ∈ F . Then it remains to prove that c r + c t is a ( σ y , σ z )-remainder. A straightforwardcalculation yields that c r + c t = X i ∈ Λ s max { ¯ n ki ,n i } X j =1 a ∗ ij b ∗ ij d ′ ij + X k ∈ Λ cr n k X j =1 c ¯ a kj ¯ b kj ¯ d jk + X i ∈ Λ cs n i X j =1 c a ′ ij b ′ ij d ′ ij , in which b ∗ ij is the least common multiple of { ¯ b k i j , b ′ ij } and a ∗ ij = c ¯ a k i j ( b ∗ ij / ¯ b k i j ) + c a ′ ij ( b ∗ ij /b ′ ij )with (¯ a k i j , ¯ b k i j ) (resp. ( a ′ ij , b ′ ij )) being specified to be (0 ,
1) in case j > ¯ n k i (resp. j > n i ). Ob-serve that polynomials in { ¯ d , ¯ d , . . . , ¯ d ¯ m } = { ¯ d k i | i ∈ Λ s } ∪ { ¯ d k | k ∈ Λ cr } , as well as those in { d ′ , d ′ , . . . , d ′ m } = { d ′ i | i ∈ Λ s } ∪ { d ′ i | i ∈ Λ cs } , are pairwise ( y, z )-shift inequivalent, as both r and t are ( σ y , σ z )-remainders. Since d ′ i = ¯ d k i for i ∈ Λ s and d ′ i = d i for i ∈ Λ cs , polynomials in { d ′ i | i ∈ Λ s } ∪ { ¯ d k | k ∈ Λ cr } ∪ { d ′ i | i ∈ Λ cs } are pairwise ( y, z )-shift inequivalent as well by the definition of Λ cr . Since r and t are both ( σ y , σ z )-remainders, each c ¯ a kj / (¯ b kj ¯ d jk ) for k ∈ Λ cr and 1 ≤ j ≤ n k , as well as each c a ′ ij / ( b ′ ij d ′ ij ) for i ∈ Λ cs and 1 ≤ j ≤ n i , is a remainder fraction. Thus it amounts to showing that for i ∈ Λ s and 1 ≤ j ≤ max { ¯ n k i , n i } , each a ∗ ij / ( b ∗ ij d ′ ij ) is also a remainder fraction with the conclusion thenfollowing by Definition 3.7. Let i ∈ Λ s and 1 ≤ j ≤ max { ¯ n k i , n i } . Obviously, a ∗ ij ∈ F [ y, z ]and deg z ( a ∗ ij ) ≤ max { deg z (¯ a k i j ) , deg z ( a ′ ij ) } < deg z ( d ′ i ). If d i (and then d ′ i = ¯ d k i ) is ( y, z )-integerlinear over F of ( α, β )-type, then ¯ a k i j (resp. a ′ ij ) can be viewed as a polynomial in ( αy + βz ) withcoefficients all having degrees in y less than deg y (¯ b k i j ) (resp. deg y ( b ′ ij )). Notice that b ∗ ij is theleast common multiple of { ¯ b k i j , b ′ ij } . Thus a ∗ ij can be viewed as a polynomial in ( αy + βz ) withcoefficients all having degrees in y less than deg y ( b ∗ ij ). Moreover, b ∗ ij is σ βy -free as ¯ b k i j , b ′ ij both areand gcd(¯ b k i j , σ βℓy ( b ′ ij )) = 1 for any nonzero ℓ ∈ Z by step 2.2. Therefore, each a ∗ ij / ( b ∗ ij d ′ ij ) is aremainder fraction by definition. This concludes the proof. ✷ Recall that a telescoper L , for a given rational function f ∈ F ( y, z ), is a nonzero operator in F [S x ] such that L ( f ) is ( σ y , σ z )-summable. For discrete trivariate rational functions, telescopersdo not always exist. Recently, a criterion for determining the existence of telescopers in this casewas presented in the work [17]. We summarize this in the following theorem. In order to adapt itinto our setting, we will consider primitive parts of polynomials in F [ y ]. Recall that the primitivepart of p ∈ F [ y ] with respect to y , denoted by prim y ( p ), is the primitive part with respect to y ofthe numerator (with respect to x ) of p . Then prim y ( p ) is a primitive polynomial in C [ x, y ] whosecoefficients with respect to y have no nonconstant common divisors in C [ x ].7 Theorem 5.1 (Existence criterion) . Let f be a rational function in F ( y, z ) . Assume that applyingthe bivariate reduction of Abramov to f yields (3.8) , where g, h, r ∈ F ( y, z ) and r is a ( σ y , σ z ) -remainder with the remainder form (3.6) . Then f has a telescoper if and only if for each ≤ i ≤ m and ≤ j ≤ n i ,(i) there exists a positive integer ξ i such that σ ξ i x ( d i ) = σ ζ i y σ η i z ( d i ) for some integers ζ i , η i ;(ii) and b ij is ( x, y ) -integer linear over C , in particular, σ ξ i x (prim y ( b ij )) = σ ζ i y (prim y ( b ij )) if d i isnot ( y, z ) -integer linear over F . With termination guaranteed by the above criterion, we now use the bivariate reduction ofAbramov to develop a telescoping algorithm in the spirit of the general reduction-based approach.
Algorithm ReductionCT.
Given a rational function f ∈ F ( y, z ), compute a minimal telescoper L ∈ F [S x ] for f and a corresponding certificate ( g, h ) ∈ F ( y, z ) when telescopers exist.1. apply the bivariate reduction of Abramov to f to find g , h ∈ F ( y, z ) and a ( σ y , σ z )-remainder r ∈ F ( y, z ) such that f = (S y − g ) + (S z − h ) + r . (5.1)2. if r = 0 then set L = 1, ( g, h ) = ( g , h ) and return.3. if conditions (i)-(ii) in Theorem 5.1 are not satisfied, then return “No telescopers exist”.4. set R = c r , where c is an indeterminate.for ℓ = 1 , , . . . do4.1 apply the remainder linearization to σ x ( r ℓ − ) with respect to R to find g ℓ , h ℓ ∈ F ( y, z )and a ( σ y , σ z )-remainder r ℓ ∈ F ( y, z ) such that σ x ( r ℓ − ) = (S y − g ℓ ) + (S z − h ℓ ) + r ℓ , (5.2)and that R + c ℓ r ℓ is a ( σ y , σ z )-remainder, where c ℓ is an indeterminate.4.2 update R = R + c ℓ r ℓ and update g ℓ = g ℓ + σ x ( g ℓ − ), h ℓ = h ℓ + σ x ( h ℓ ) so that σ ℓx ( f ) = (S y − g ℓ ) + (S z − h ℓ ) + r ℓ . (5.3)4.3 if there exist nontrivial c , . . . , c ℓ ∈ F such that R = 0, then set L = P ℓi =0 c i S ix and( g, h ) = ( P ℓi =0 c i g i , P ℓi =0 c i h i ), and return. Theorem 5.2.
Let f be a rational function in F ( y, z ) . Then the algorithm ReductionCT termi-nates and returns a minimal telescoper for f when such a telescoper exists.Proof. By Theorems 3.10 and 5.1, steps 2-3 are correct. Observe from Definition 3.7 that σ x ( r )is a ( σ y , σ z )-remainder as r is. By Theorem 4.5, step 4.1 correctly finds g , h ∈ F ( y, z ) and a( σ y , σ z )-remainder r ∈ F ( y, z ) such that (5.2) holds for ℓ = 1 and R + c r = c r + c r is a( σ y , σ z )-remainder for all c , c ∈ F . Applying σ x to both sides of (5.1), together with step 4.1, onesees that step 4.2 gives (5.3) for ℓ = 1. The correctness of step 4.2 for each iteration of the loop instep 4 then follows by induction on ℓ .8 If f does not have a telescoper then the algorithm halts after step 3. Otherwise, telescopers for f exist by Theorem 5.1. Let L = P ρℓ =0 c ℓ S ℓx ∈ F [S x ] be a telescoper for f of minimal order. Then c ρ = 0 and by (5.3), applying L to f gives L ( f ) = ρ X ℓ =0 c ℓ σ ℓx ( f ) = (S y − ρ X ℓ =0 c ℓ g ℓ ! + (S z − ρ X ℓ =0 c ℓ h ℓ ! + ρ X ℓ =0 c ℓ r ℓ . Notice that P ρℓ =0 c ℓ r ℓ is a ( σ y , σ z )-remainder by step 4.1. It follows from Theorem 3.10 that L ( f )is ( σ y , σ z )-summable, namely L is a telescoper for f , if and only if P ρℓ =0 c ℓ r ℓ = 0. This implies thatthe linear system over F obtained by equating P ρℓ =0 c ℓ r ℓ to zero has a nontrivial solution, whichyields a telescoper of minimal order. The algorithm thus terminates. ✷ Recall that an operator L ∈ F [S x ] is a common left multiple of operators L , . . . , L m ∈ F [S x ] if thereexist operators L ′ , . . . , L ′ m ∈ F [S x ] such that L = L ′ L = · · · = L ′ m L m . Amongst all such commonleft multiples, the monic one of lowest possible degree with respect to S x is called the least commonleft multiple . In view of the computation, many efficient algorithms are available, see [7] and thereferences therein.In analogy to [32, Theorem 2], we have the following lemma. Lemma 5.3.
Let r = r + · · · + r m with r i ∈ F ( y, z ) and let L , . . . , L m ∈ F [S x ] be the respectiveminimal telescopers for r , . . . , r m . Then the least common left multiple L of the L i is a telescoperfor r . Moreover, if any telescoper for r is also a telescoper for each r i with ≤ i ≤ m , then L is aminimal telescoper for r . Then one sees that the least common multiple is a minimal telescoper for the given sum providedthat the denominators of distinct summands comprise distinct ( x, y, z )-shift equivalence classes.
Proposition 5.4.
Let r ∈ F ( y, z ) be a rational function of the form r = r + r + · · · + r m , where r i = a i /d i with a i , d i ∈ F [ y, z ] satisfying the conditions(i) deg z ( a i ) < deg z ( d i ) ;(ii) any monic irreducible factor of d i of positive degree in z is ( x, y, z ) -shift inequivalent to allfactors of d i ′ whenever ≤ i, i ′ ≤ m and i = i ′ .Let L , . . . , L m ∈ F [S x ] be respective minimal telescopers for r , . . . , r m . Then the least common leftmultiple L of the L i is a minimal telescoper for r . Moreover, for each ≤ i ≤ m , let ( g i , h i ) be acorresponding certificate for L i and let L ′ i ∈ F [S x ] be the cofactor of L i so that L = L ′ i L i . Then m X i =1 L ′ i ( g i ) , m X i =1 L ′ i ( h i ) ! is a corresponding certificate for L . Proof.
Let ˜ L ∈ F [S x ] be a telescoper for r . In order to show the first assertion, by Lemma 5.3,it suffices to verify that ˜ L is also a telescoper for each r i with 1 ≤ i ≤ m . Notice that theapplication of a nonzero operator from F [S x ] does not change the ( x, y, z )-shift equivalence classes,with representatives being monic irreducible polynomials of positive degrees in z , that appear in agiven polynomial in F [ y, z ]. Hence condition (ii) remains valid when d i and d i ′ are replaced by ˜ L ( d i )and ˜ L ( d i ′ ), respectively. It then follows that any two monic irreducible factors of positive degreesin z from distinct d i are ( y, z )-shift inequivalent to each other. By the definition of telescopers,˜ L ( r ) is ( σ y , σ z )-summable, and then so is each ˜ L ( r i ) according to [28, Lemma 3.2]. This impliesthat ˜ L is indeed a telescoper for each r i . The second assertion follows by observing that (S y − z −
1) both commute with operators from F [S x ]. ✷ The above proposition provides an alternative way to construct a minimal telescoper for a givenrational function.
Example 5.5.
Consider the rational function f given in Example 3.11. Note that f is a remainderfraction and satisfies conditions (i)-(ii) in Theorem 5.1. So telescopers for f exist. Applying thealgorithm ReductionCT to f , we obtain in step 4 that σ ℓx ( f ) = (S y − g ℓ ) + (S z − h ℓ ) + r ℓ for ℓ = 0 , , , where r = f , r = ( x + 1) ( z −
1) + 1( x + y )( x + z ) + 1 , r = ( x + 2) ( z −
2) + 1( x + y )( x + z ) + 1 and g ℓ , h ℓ ∈ F ( y, z ) are not displayed here to keep things neat. By finding a F -linear dependencyamong r , r , r , we see that L = ( x + 2 x + x + 2 x + 1) S x − x + 4 x + 4 x + 2 x + 2) S x +( x + 6 x + 13 x + 14 x + 7) is a minimal telescoper for f . Example 5.6.
Consider the rational function f given in Example 3.11. Then it can be decomposedas f = (S y − g ) + (S z − h ) + x (2 y + 3 z ) + x + 1( x + y )(( x + 2 y + 3 z ) + 1) | {z } r for some g , h ∈ F ( y, z ) . Note that r is a remainder fraction and satisfies conditions (i)-(ii) in Theorem 5.1. Thus tele-scopers for f exist. Applying the algorithm ReductionCT to f , we obtain in step 4 that σ ℓx ( f ) = (S y − g ℓ ) + (S z − h ℓ ) + r ℓ for ℓ = 0 , , . . . , , where g ℓ , h ℓ ∈ F ( y, z ) are again not displayed due to the large sizes, and r = ( x +1)(2 y +3 z )+ x + x + ( x + y +2)(( x +2 y +3 z ) +1) , r = ( x + )(2 y +3 z )+ x +2 x + ( x + y +4)(( x +2 y +3 z ) +1) , r = ( x +1)(2 y +3 z )+ x +4( x + y )(( x +2 y +3 z ) +1) ,r = ( x + )(2 y +3 z )+ x +4 x + ( x + y +2)(( x +2 y +3 z ) +1) , r = ( x + )(2 y +3 z )+ x +5 x + ( x + y +4)(( x +2 y +3 z ) +1) , r = ( x +2)(2 y +3 z )+ x +6 x +13( x + y )(( x +2 y +3 z ) +1) . Then one finds a F -linear dependency among r , r , r which yields a minimal telescoper L = ( x + 3 x −
3) S x − x + 6 x −
3) S x + x + 9 x + 15 . The following illustrates the result of Proposition 5.4.
Example 5.7.
Consider the same rational function f as in Example 3.11. Then we know that f is ( σ y , σ z ) -summable. Thus L = 1 is a minimal telescoper for f . Let L , L ∈ F [S x ] be theoperators computed in Examples 5.5-5.6. It then follows that the least common left multiple L of { L , L , L } , given by L = S x − x +5 x +1)(3 x +24 x +31)( x +7 x +7)(3 x +21 x +19) S x + ( x +3 x − x +27 x +43)( x +7 x +7)(3 x +21 x +19) S x − x +10 x +13) x +7 x +7 S x + x +24 x +31)( x +8 x +4)( x +7 x +7)(3 x +21 x +19) S x − x +6 x − x +27 x +43)( x +7 x +7)(3 x +21 x +19) S x + x +13 x +37 x +7 x +7 S x − x +11 x +25)(3 x +24 x +31)( x +7 x +7)(3 x +21 x +19) S x + ( x +9 x +15)(3 x +27 x +43)( x +7 x +7)(3 x +21 x +19) , is a telescoper for f . On the other hand, by directly applying the algorithm ReductionCT to f ,one sees that L is in fact a minimal telescoper for f . The efficiency of Algorithm
ReductionCT can be enhanced by incorporating two modificationsin the algorithm.
Simplification of remainder step 4.1
For each iteration of the loop in step 4, rather than using the overall ( σ y , σ z )-remainder R = P ℓ − k =0 c k r k in step 4.1, we can apply the remainder linearization to the shift value σ x ( r ℓ − ) withrespect to the initial ( σ y , σ z )-remainder r only. This is sufficient as, for any ( σ y , σ z )-remainder r ℓ of σ x ( r ℓ − ) with ℓ ≥
1, if r + r ℓ is a ( σ y , σ z )-remainder then so is R + c ℓ r ℓ , provided that thealgorithm proceeds in the described iterative fashion.The intuition for this simplification is as follows. Notice that if the algorithm continues afterpassing through step 3 then r = 0. Since distinct ( y, z )-shift equivalence classes can be tackledseparately, we restrict ourselves to the case where the denominator of r is of the form d σ i x ( d ) · · · σ i m x ( d )with d ∈ F [ y, z ] being monic, irreducible and of positive degree in z , i , . . . , i m being distinct positiveintegers such that d, σ i x ( d ) , . . . , σ i m x ( d ) are ( y, z )-shift inequivalent to each other. For simplicity,we call (0 , i , . . . , i m ) the x -shift exponent sequence of d in r . By Theorem 5.1, there exists apositive integer ξ such that σ ξx ( d ) ∼ y,z d and so we let ξ be the smallest one with such a property.Then there are only ξ many ( y, z )-shift equivalence classes produced by shifting d with respect to x , with d, σ x ( d ) , . . . , σ ξ − x ( d ) as respective representatives. Without loss of generality, we furtherassume that 0 < i < · · · < i m < ξ . For ℓ ≥
1, let r ℓ be the output of the remainder linearizationwhen applied to σ x ( r ℓ − ) with respect to r . By induction on ℓ , one sees that the x -shift exponentsequence of d in r ℓ is given by ( ℓ, i + ℓ, . . . , i m + ℓ ) mod ξ, m + 1)-subset of { , , . . . , ξ − } . It thus follows from Definition 3.7 that R + c ℓ r ℓ is also a ( σ y , σ z )-remainder. Simplification of remainder step 4.3
Our second modification is in step 4.3, where we first individual derive from R = 0 the equationfor each remainder fraction a/ ( b d j ) appearing in the remainder form of R , and then build a linearsystem over F from the coefficients of the numerator of the equation with respect to y and Z = αy + βz , instead of y and z , in the case where d is ( y, z )-integer linear of ( α, β )-type. Noticethat R = c r + c r + · · · + c ℓ r ℓ at the stage of step 4.3. Let d , . . . , d m be all monic irreduciblepolynomials of positive degrees in z that appear in the denominator of R , with multiplicities n , . . . , n m , respectively. For 1 ≤ i ≤ m , 1 ≤ j ≤ n i and 0 ≤ k ≤ ℓ , let a ( k ) ij ∈ F [ y, z ] and b ( k ) ij ∈ F [ y ] be such that a ( k ) ij / ( b ( k ) ij d ji ) is a remainder fraction appearing in the remainder form of r k .By coprimeness among the d i , one gets that R = 0 ⇐⇒ ℓ X k =0 c k · a ( k ) ij b ( k ) ij = 0 for all i = 1 , . . . , m and j = 1 , . . . , n i . If d i is ( y, z )-integer linear of ( α i , β i )-type, then by Definition 3.6, every a ( k ) ij can be viewed as apolynomial in Z i = α i y + β i z with coefficients all having degrees in y less than deg y ( b ( k ) ij ). In thiscase, rather than naively considering the coefficients with respect to y and z , we instead force allthe coefficients with respect to y and Z i of the numerator of P ℓk =0 c k · ( a ( k ) ij /b ( k ) ij ) to zero. This wayensures that the resulting linear system over F typically has smaller size than the naive one. We have implemented our new algorithm
ReductionCT in the computer algebra system
Maple 2018 . Our implementation includes the two enhancements to step 4 discussed in theprevious subsection. In order to get an idea about the efficiency of our algorithm, we appliedour implementation to certain examples and tabulated their runtime in this section. All timingswere measured in seconds on a Linux computer with 128GB RAM and fifteen 1.2GHz Dual coreprocessors. The computations for the experiments did not use any parallelism.We considered trivariate rational functions of the form f ( x, y, z ) = a ( x, y, z ) d ( x, y, z ) · d ( x, y, z ) , (6.1)where • a ∈ Z [ x, y, z ] of total degree m ≥ || a || ∞ ≤
5, in other words, the maximalabsolute value of the coefficients of a with respect to x, y, z are no more than 5; • d i = p i · σ ξx ( p i ) with p = P ( ξy − ζx, ξz + ζx ) and p = P ( ζx + ξy + 2 ξz ) for two nonzerointegers ξ, ζ and two integer polynomials P ( y, z ) ∈ Z [ y, z ], P ( z ) ∈ Z [ z ], both of which havetotal degree n > m, n, ξ, ζ ),Table 1 collects the timings of four variants of the algorithm ReductionCT from Section 5: forthe columns RCT and RCT , we both compute the telescoper as well as the certificate, but thedifference lies in that the former brings the certificate to a common denominator while the latterleaves the certificate as an unnormalized linear combination of rational functions; for RCT weonly compute the telescoper and neglect almost everything related to the certificate; for RCTLM ,RCTLM and RCTLM , we perform the same functionality as RCT , RCT and RCT but basedon the idea from Proposition 5.4. As indicated by the table, the timings for RCT (resp. RCTLM )are virtually the same as for RCT (resp. RCTLM ).( m, n, ξ, ζ ) RCT RCT RCT RCTLM RCTLM RCTLM order(1, 1, 1, 1) 0.196 0.098 0.979 0.220 0.109 0.110 1(1, 1, 1, 5) 7.319 0.112 0.123 9.483 0.131 0.123 1(1, 1, 1, 9) 105.548 0.123 0.121 104.514 0.128 0.125 1(1, 1, 1, 13) 2586.295 0.114 0.136 3078.043 0.133 0.126 1(1, 1, 1, 3) 0.574 0.098 0.097 0.712 0.107 0.104 1(1, 2, 1, 3) 17.812 0.258 0.256 17.299 0.268 0.263 1(1, 3, 1, 3) 266.206 2.008 1.999 220.209 2.027 1.997 1(1, 4, 1, 3) 2838.827 37.052 37.358 3039.199 33.599 30.547 1(1, 5, 1, 3) 19403.916 1085.659 1074.295 18309.000 1111.333 1119.393 1(2, 3, 1, 3) 31678.706 2.257 2.540 15825.876 2.295 2.224 3(3, 3, 1, 3) 44243.254 5.106 5.378 16869.097 4.512 4.295 3(3, 2, 1, 3) 710.810 0.480 0.492 670.501 0.522 0.487 3(3, 2, 2, 3) 1314.809 0.751 0.701 941.009 0.792 0.756 6(3, 2, 4, 3) 1558.440 1.528 1.525 1121.624 1.598 1.550 12(3, 2, 8, 3) 1878.424 4.567 4.215 986.017 4.133 4.245 24(3, 2, 16, 3) 2800.050 25.027 21.136 1317.603 38.399 38.504 48Table 1: Timings for sixth variants of the algorithm ReductionCT . In this paper, we have studied the class of trivariate rational functions and presented a creativetelescoping algorithm for this class. The procedure is based on a bivariate extension of Abramov’sreduction method initiated in [1]. Our algorithm finds a minimal telescoper for a given trivariaterational function without also needing to compute an associated certificate. This in turn providesa more efficient way to deal with rational double summations in practice.We are interested in the more important problem of computing hypergeometric multiple sum-mations or proving identities which involve such summations. A function f ( x, y , . . . , y n ) is calleda multivariate hypergeometric term if the quotients f ( x + 1 , y , . . . , y n ) f ( x, y , . . . , y n ) , f ( x, y + 1 , . . . , y n ) f ( x, y , . . . , y n ) , . . . , f ( x, y , . . . , y n + 1) f ( x, y , . . . , y n )are all rational functions in x, y , . . . , y n . The problem of hypergeometric multiple summationstends to appear more often than the rational case, particularly in combinatorics [10, 15], and it is3also more challenging.Since a large percent of hypergeometric terms falls into the class of holonomic functions, theproblem of hypergeometric multiple summations can also be considered in a more general frameworkof multivariate holonomic functions. In this context, several creative telescoping approaches havealready been developed in [40, 37, 24, 23, 31]. The algorithms in the first three papers are based onelimination and suffer from the disadvantage of inefficiency in practice. The algorithm in [23], alsoknown as Chyzak’s algorithm, deals with single sums (and single integrals) and can only be used tosolve multiple ones in a recursive manner. A fast but heuristic approach was given in [31] in orderto eliminate the bottleneck in Chyzak’s algorithm of solving a coupled first-order system. Thisapproach generalizes to multiple sums (and multiple integrals). We refer to [30] for a detailed andexcellent exposition of these approaches. We remark that all these approaches find the telescoperand the certificate simultaneously, with the exception of Takayama’s algorithm in [37] where naturalboundaries have to be assured a priori. Note also that holonomicity is a sufficient but not necessarycondition for the applicability of creative telescoping applied to hypergeometric terms (cf. [5, 17]).Restricted to the hypergeometric setting, partial solutions for the problem of multiple sum-mations were proposed in [22] and [15]. In the former paper, the authors presented a heuristicmethod to find telescopers for trivariate hypergeometric terms, through which they also managedto prove certain famous hypergeomeric double summation identities. In the second paper, theauthors mainly focused on a subclass of proper hypergeometric summations – multiple binomialsums. They first showed that the generating function of a given multiple binomial sum is alwaysthe diagonal of a rational function and vice versa. They then constructed a differential equationfor the diagonal by a reduction-based telescoping approach. Finally the differential equation istranslated back into a recurrence relation satisfied by the given binomial sum. In the future, wehope to explore this topic further and aim at developing a complete reduction-based telescopingalgorithm for hypergeometric terms in three or more variables. References [1] Sergei A. Abramov. The rational component of the solution of a first order linear recurrencerelation with rational right hand side.
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