An Algorithm for Deciding the Summability of Bivariate Rational Functions
aa r X i v : . [ c s . S C ] J un An Algorithm for Deciding the Summability ofBivariate Rational Functions
Qing-Hu Hou and Rong-Hua Wang Center for Applied MathematicsTianjin University, Tianjin 300072, P.R. [email protected] Center for Combinatorics, LPMC-TJKLCNankai University, Tianjin 300071, P.R. [email protected]
Abstract.
Let ∆ x f ( x, y ) = f ( x + 1 , y ) − f ( x, y ) and ∆ y f ( x, y ) = f ( x, y + 1) − f ( x, y )be the difference operators with respect to x and y . A rational function f ( x, y ) is calledsummable if there exist rational functions g ( x, y ) and h ( x, y ) such that f ( x, y ) = ∆ x g ( x, y ) +∆ y h ( x, y ). Recently, Chen and Singer presented a method for deciding whether a rationalfunction is summable. To implement their method in the sense of algorithms, we need to solvetwo problems. The first is to determine the shift equivalence of two bivariate polynomials.We solve this problem by presenting an algorithm for computing the dispersion sets of anytwo bivariate polynomials. The second is to solve a univariate difference equation in analgebraically closed field. By considering the irreducible factorization of the denominator of f ( x, y ) in a general field, we present a new criterion which requires only finding a rationalsolution of a bivariate difference equation. This goal can be achieved by deriving a universaldenominator of the rational solutions and a degree bound on the numerator. Combiningthese two algorithms, we can decide the summability of a bivariate rational function. Keywords : summability, bivariate rational function, Gosper’s algorithm, dispersion set.
In 1978, Gosper [13] presented the celebrated algorithm which solves the problem of deter-mining whether a given hypergeometric term is equal to the difference of another hyper-geometric term. Based on Gosper’s algorithm, Zeilberger [25, 26] gave a fast algorithm forproving terminating hypergeometric identities. Zeilberger’s method was further extended tothe multivariate case by Wilf and Zeilberger himself in [24]. Paule [20] gave an interpreta-tion of Gosper’s algorithm in terms of the greatest factorial factorizations. Chen, Paule andSaad [12] derived an easy understanding version of Gosper’s algorithm by considering theconvergence of the greatest common divisors of two polynomial sequences.Other approaches to the summability of rational functions were given by Abramov [2–4],Pirastu and Strehl [22], Ash and Catoiu [8]. The key idea of these methods is to rewrite arational function α as α = ∆( β ) + γ , where ∆ is the difference operator, β and γ are rationalfunctions such that the denominator of γ is shift-free. Then α is summable if and only if γ is zero. 1assing from univariate to multivariate, Zeilberger’s algorithm have been discussed byZeilberger himself [7, 19], Koutschan [18], Schneider [23], Chen et. al. [11]. These algorithmsare useful in practice. However, they did not provide a complete answer to the summabilityproblem of bivariate hypergeometric terms. Only very recently, Chen and Singer [10] pre-sented criteria for deciding the summability of bivariate rational functions . When applyingtheir criteria, one will encounter two problems. The first one is how to determine whethertwo bivariate polynomials are shift equivalent. The second one is how to solve univariatedifference equations in algebraically closed fields. The main aim of the present paper is toovercome these problems and to give an algorithm for deciding the summability of bivariaterational functions. We remark that the general question considered in this paper was raisedby Andrews and Paule in [6].For the first problem, we show that the dispersion set of two bivariate polynomials iscomputable. Then two polynomials are shift equivalent if and only if the dispersion set isnot empty. For the second problem, we present a variation of the criteria by considering theirreducible factorization in a general field instead of an algebraically closed field. To applythe new criteria, we need only to find rational solutions of a bivariate difference equation. Bya discussion similar to Gosper’s algorithm, we derive a universal denominator of the rationalsolutions. We further derive a degree bound on the numerator of the rational solutions andthus obtain an algorithm for the new criterion. Combining these two algorithms, we finallyobtain an algorithm for deciding the summability of bivariate rational functions.The paper is organized as follows. In Section 2, we give an algorithm for computing thedispersion set of two bivariate polynomials. In Section 3, we first reduce the summability ofa general rational function to that of a rational function whose denominator is a power of anirreducible polynomial. Then we present a criterion on the summability of this special kind ofrational functions. This criterion reduces the summability problem to the problem of findingrational solutions of a bivariate difference equation. In Section 4, we give an algorithm forsolving the difference equation.Throughout the paper, we take Q , the field of rational numbers, as the ground field. Itshould be mentioned that the discussions work also for other fields, such as the extensionfield Q ( α , . . . , α r ) where α , . . . , α r are either algebraic or transcendental over Q .We follow the notations used in [10]. Let f ( x, y ) ∈ Q ( x, y ) be a bivariate rational function.The shift operators σ x and σ y are given by σ x f ( x, y ) = f ( x + 1 , y ) and σ y f ( x, y ) = f ( x, y + 1) . A function f ∈ Q ( x, y ) is said to be ( σ x , σ y ) -summable if there exist two rational functions g, h ∈ Q ( x, y ) such that f = σ x g − g + σ y h − h. Let Z denote the set of integers. Recall that given two univariate polynomials, say f ( x ) and g ( x ), their dispersion set is defined byDisp x ( f, g ) = { n ∈ Z | f ( x ) = g ( x + n ) } .
2t is known that unless f and g are the same constant polynomial, the dispersion setDisp x ( f, g ) is finite and is computable. For the algorithm, see [21, page 79]. We can ex-tend this concept to the bivariate case. Definition 2.1.
Let f, g be two bivariate polynomials in Q [ x, y ] and σ x , σ y be the shift oper-ators. The dispersion set of f and g is defined by Disp( f, g ) = { ( m, n ) ∈ Z | f = σ mx σ ny g } . If Disp( f, g ) is not empty, we say f and g are shift equivalent .In particular, when f = σ mx g (resp. f = σ ny g ), we say f, g in the same σ x - orbit (resp. σ y - orbit ), denoted by f ∼ x g and f ∼ y g respectively. We remark that testing shift equivalence over fields have been considered by Grigorievand Karpinski [14–16]. More precisely, they gave algorithms to find shifts ( α , . . . , α r ) ∈ F r such that f ( x + α , . . . , x r + α r ) = g ( x , . . . , x r ) , where F is a field and f, g ∈ F [ x , . . . , x r ]. Instead of considering shifts over a field, we focuson integer shifts, i.e., m, n ∈ Z .In the univariate case, the dispersion set of any two polynomials is computable. Thefollowing theorem shows that the dispersion set is also computable in the bivariate case. Theorem 2.2.
Let f, g ∈ Q [ x, y ] be two polynomials. Then we can determine the dispersionset Disp( f, g ) .Proof. Since the shift operators σ x and σ y preserve the degree, we get that Disp( f, g ) = ∅ unless deg x f = deg x g .When f = 0 or deg x ( f ) = 0, the computation of Disp( f, g ) reduces to the univariate case.More precisely, we have Disp( f, g ) = Z × Disp y ( f, g ) . Now assume that deg x f = d > f, g as f = d X k =0 a k ( y ) x k , g = d X k =0 b k ( y ) x k . Suppose that ( m, n ) ∈ Disp( f, g ). By comparing the leading coefficient with respect to x , wesee that n falls in the dispersion set N = Disp y ( a d ( y ) , b d ( y )) . If N is a finite set, we then haveDisp( f, g ) = [ n ∈N Disp x ( f ( x, y ) , g ( x, y + n )) × { n } . a d ( y ) = b d ( y ) = c , where c is a non-zero constant. By comparingthe second leading coefficient with respect to x , we see that a d − ( y ) = d · c · m + b d − ( y + n ) . (2.1)According to the degree of a d − ( y ) in variable y , there are three cases. Case . deg a d − ( y ) >
1. Then Disp( f, g ) = ∅ unless the leading term of a d − ( y ) and that of b d − ( y ) coincide. Assume a d − ( y ) = h X j =0 p j y j and b d − ( y ) = h X j =0 q j y j . By comparing the coefficients of y h − in the expansions of a d − ( y ) and b d − ( y + n ), we seethat n is uniquely determined by hq h n + q h − = p h − . (2.2)Suppose that n is an integer solution of (2.2). We then haveDisp( f, g ) = Disp x ( f ( x, y ) , g ( x, y + n )) × { n } . Case . deg a d − ( y ) = 1. We also have Disp( f, g ) = ∅ unless the leading term of a d − ( y ) andthat of b d − ( y ) coincide. Assume a d − ( y ) = p y + p and b d − ( y ) = p y + q . Then (2.1) leads to ( d · c ) · m + p · n = p − q , (2.3)which is a linear Diophantine equation in unknowns m, n . Either there is no solution, or thesolutions are of the form m = ut + v, and n = u ′ t + v ′ , where u, v, u ′ , v ′ are explicit integers and t runs over Z . Now by setting all coefficients of x, y in the expansion of f ( x, y ) − g ( x + ut + v, y + u ′ t + v ′ ) to be zeros, we arrive at a system ofpolynomial equations in t . The set of integer solutions of each equation is computable (see,for example [21, page 79]). The final dispersion set of f and g is the intersection of thesesolution sets. Case . deg a d − ( y ) = 0 or a d − ( y ) = 0. If deg y b d − ( y ) >
0, we then have Disp( f, g ) = ∅ .Otherwise, m is uniquely determined by (2.1). Suppose m is an integer solution of (2.1), wehave Disp( f, g ) = { m } × Disp y ( f ( x, y ) , g ( x + m , y )) . This completes the proof.Based on the proof as above, we can describe an algorithm for computing the dispersionset of two polynomials in Q [ x, y ]. Algorithm DispSetInput:
Two polynomials f = P d k =0 a k ( y ) x k and g = P d k =0 b k ( y ) x k . Output:
The dispersion set Disp( f, g ). 4. If d = d , return ∅ . Else set d = d = d .2. If d ≤
0, return the set Z × Disp y ( f, g ). Else continue the following steps.3. If deg a d ( y ) >
0, compute N = { n | a d ( y ) = b d ( y + n ) } and for each n ∈ N , computethe set S n of integers m such that f = σ mx σ n y g . Return the set [ n ∈N S n × { n } . Else set a d ( y ) = c and continue the following steps.4. If deg y a d − ( y ) >
1, compute the unique n according to (2.2). If n is an integer, thenreturn Disp x ( f ( x, y ) , g ( x, y + n )) × { n } . Else return ∅ .5. If deg y a d − ( y ) = 1. If the leading terms of a d − ( y ) and b d − ( y ) are different, thenreturn ∅ . Else solve the linear Diophantine equation (2.3). Suppose that the solutionsare of the form m = ut + v and n = u ′ t + v ′ . Substituting m by ut + v and n by u ′ t + v ′ in f = σ mx σ ny g and comparing the coefficientsof each power of x and y to get a system of polynomial equations in t . Return all integersolutions if there are. Else return ∅ .6. If deg y a d − ( y ) = 0 or a d − ( y ) = 0. If deg y b d − ( y ) > ∅ . Else computethe unique m satisfying (2.1). If m is not an integer, then return ∅ . Else return theset { m } × Disp y ( f ( x, y ) , g ( x + m , y )) . The following is an example which shows how to determine the shift equivalence of anytwo given bivariate polynomials.
Example 2.3.
Let f = 2 x + 2 xy + y + y + 1 and g = 2 x + 2 xy + y + 2 x + y + 1 . We try to determine whether f and g are shift equivalent according to the proof of Theo-rem 2.2. Rewrite f, g as f = 2 x + (2 y ) x + ( y + y + 1) , and g = 2 x + (2 y + 2) x + ( y + y + 1) . It’s easy to check that this meets Case in the proof. Thus m, n satisfy the linear equation m + n = − whose solutions are m = t and n = − t − , t ∈ Z . Now by setting all coefficients of x , y in the expansion of f ( x, y ) − g ( x + t, y − t − to bezeros, we obtain an integer solution t = − . It means that f ( x, y ) = g ( x − , y + 1) and thus f, g are shift equivalent. Summability criterion
As stated in the introduction, one can decompose a univariate rational function α into theform α = ∆ β + γ . The goal of this section is to introduce a bivariate variant of such additivedecomposition and thus reduce the bivariate summability problem of a general rational func-tion to that of a rational function whose denominator is a power of an irreducible polynomial.We then present a criterion for the summability of this kind of special rational functions.Let f ∈ Q ( x, y ) be a bivariate rational function. Assume that the irreducible factorizationof the denominator D ( x, y ) of f ( x, y ) is D ( x, y ) = m Y i =1 d n i i ( x, y ) , where d i ( x, y ) are irreducible polynomials and n i are positive integers. Viewing f as a rationalfunction of y over the field Q ( x ), we have the partial fraction decomposition f = P + m X i =1 n i X j =1 a i,j d ji , (3.1)where P ∈ Q ( x )[ y ], a i,j ∈ Q ( x )[ y ] and deg y ( a i,j ) < deg y ( d i ). It is well known that thepolynomial P is the difference of a polynomial.Now suppose that d i ( x, y ) = d k ( x + m, y + n ) for some index i = k . Then we have a i,j d ji = σ x ( g ) − g + σ y ( h ) − h + σ − mx σ − ny ( a i,j ) d jk , where g = m − P ℓ =0 σ ℓ − mx ( a i,j ) σ ℓx σ ny ( d jk ) , if m ≥ , − − m − P ℓ =0 σ ℓx ( a i,j ) σ m + ℓx σ ny ( d jk ) , if m < , and h = n − P ℓ =0 σ ℓ − ny σ − mx ( a i,j ) σ ℓy ( d jk ) , if n ≥ , − − n − P ℓ =0 σ ℓy σ − mx ( a i,j ) σ n + ℓy ( d jk ) , if n < . Repeating the above transformation, we arrive at the following decomposition.
Lemma 3.1.
For a rational function f ∈ Q ( x, y ) , we can decompose it into the form f = ∆ x ( g ) + ∆ y ( h ) + r, where g, h ∈ Q ( x, y ) and r is of the form r = m X i =1 n i X j =1 a i,j ( x, y ) d ji ( x, y ) , (3.2)6 ith a i,j ∈ Q ( x )[ y ] , deg y ( a i,j ) < deg y ( d i ) , d i ∈ Q [ x, y ] are irreducible polynomials, and d i and d i ′ are not shift equivalent for any ≤ i = i ′ ≤ m . From Lemma 3.1, we see that f is ( σ x , σ y )-summable if and only if r is ( σ x , σ y )-summable.The following lemma shows that the summability of r is equivalent to the summability ofeach summand of r . Lemma 3.2.
Let r ∈ Q ( x, y ) be of the form (3.2) . Then r is ( σ x , σ y ) -summable if and onlyif a i,j ( x,y ) d ji ( x,y ) is ( σ x , σ y ) -summable for all ≤ i ≤ m and ≤ j ≤ n i .Proof. The sufficiency follows from the linearity of the difference operators ∆ x and ∆ y . Itsuffices to prove the necessity. Assume that r is ( σ x , σ y )-summable, then there exist g, h ∈ Q ( x, y ) such that r = σ x ( g ) − g + σ y ( h ) − h . We can always decompose g, h as g = A D + A D and h = B C + B C , where A i , B i , C i , D i ( i = 1 ,
2) are polynomials in y over Q ( x ), deg y ( A ) < deg y ( D ), deg y ( B ) < deg y ( C ), D (resp. C ) contains only irreducible factors that are shift equivalent to d i , while D (resp. C ) contains no such factors. Let r i = P n i j =1 a i,j ( x,y ) d ji ( x,y ) . We then have r i − (cid:18) σ x A D − A D + σ y B C − B C (cid:19) = σ x A D − A D + σ y B C − B C − X j = i r j . Note that σ x , σ y preserve the ( σ x , σ y )-equivalence. Therefore, we have r i = σ x A D − A D + σ y B C − B C , which means r i is ( σ x , σ y )-summable.By the same observation as in [10, Page 330], we see that σ x and σ y preserve the multi-plicities of the fractions a i,j /d ji . This implies that r i is ( σ x , σ y )-summable if and only if eachsummand a i,j /d ji is ( σ x , σ y )-summable. This concludes the proof.Now we only need to study the summability problem of rational functions of the form a/d j , where d ∈ Q [ x, y ] is irreducible, a ∈ Q ( x )[ y ], and deg y ( a ) < deg y ( d ). For this kind ofrational functions, we have the following criterion for their summability. Theorem 3.3.
Let f = a ( x,y ) d j ( x,y ) , where d ( x, y ) ∈ Q [ x, y ] is an irreducible polynomial, a ∈ Q ( x )[ y ] is non-zero and deg y ( a ) < deg y ( d ) . Then f is ( σ x , σ y ) -summable if and only if (1) there exist integers t, ℓ with t = 0 such that σ tx d ( x, y ) = σ ℓy d ( x, y ) , (3.3)(2) for the smallest positive integer t such that (3.3) holds, we have a = σ tx σ − ℓy p − p, (3.4) for some p ∈ Q ( x )[ y ] with deg y ( p ) < deg y ( d ) .
7e can adapt the argument used in [10, Theorem 3.7] to complete the proof of Theo-rem 3.3. The details are elaborated in the appendix.The criterion (3.3) can be tested by computing the dispersion set Disp( d, d ). In the nextsection, we will give an algorithm for solving the equation (3.4). Then combining Lemma 3.1,Lemma 3.2 and Theorem 3.3, we will obtain an algorithm for determining whether a bivariaterational function is summable.
Let d be a positive integer and u be a polynomial in y over Q ( x ) with deg y ( u ) < d . Inthis section, we present a method of finding solutions p ∈ Q ( x )[ y ] with deg y ( p ) < d to thefollowing difference equation u = σ mx σ − ny p − p, (4.1)where m, n are given integers and m > y ( p ) < d , we may assume p = p ( x ) + p ( x ) y + · · · + p d − ( x ) y d − . Then comparing the coefficients of each power of y on both sides of (4.1), we obtain a systemof linear difference equations in p i ( x ). Abramov-Barkatou [5] and Abramov-Khmelnov [1]presented algorithms for solving such systems.We will rewrite p as the ratio c ( x, y ) /d ( x ) and estimate the denominator d ( x ) directly.Then we give an upper bound on the x -degree of the numerator c ( x, y ) and thus solve for p by the method of undetermined coefficients.Assume that u = a ( x, y ) /b ( x ), where a, b are polynomials in x and y . We notice that onecan give an estimation of d ( x ) by using the convergence argument introduced by Chen, Pauleand Saad [12]. More precisely, we have d ( x ) (cid:12)(cid:12) gcd (cid:0) b ( x ) b ( x + m ) . . . , b ( x − m ) b ( x − m ) · · · (cid:1) . Note also that one can give an estimation by using an argument similar to [17]. Here we giveanother estimation based on Gosper representation [21, page 80].Rewrite Equation (4.1) as a ( x, y ) = b ( x ) b ( x + m ) σ mx σ − ny ( b ( x ) p ( x, y )) − b ( x ) p ( x, y ) . (4.2)Let b ( x ) b ( x + m ) = A ( x ) B ( x ) C ( x + m ) C ( x ) , (4.3)be the Gosper representation. That is,gcd( A ( x ) , B ( x + hm )) = 1 , ∀ h ∈ N . (4.4)Then the denominator of p can be given by the following theorem.8 heorem 4.1. Let ( A ( x ) , B ( x ) , C ( x )) be the Gosper representation of b ( x ) b ( x + m ) , and b ( x ) p ( x, y ) be a rational solution of (4.2) . Then b ( x ) p ( x, y ) must be of the form b ( x ) p ( x, y ) = B ( x − m ) p ( x, y ) C ( x ) , where p ( x, y ) is a polynomial in both x and y .Proof. Assume that b ( x ) p ( x, y ) = g ( x, y ) q ( x ) C ( x ) , where g ( x, y ) ∈ Q [ x, y ], q ( x ) ∈ Q [ x ] is a monic polynomial and ( q ( x ) , g ( x, y )) = 1. Accordingto Equation (4.2), we deduce that a ( x, y ) B ( x ) C ( x ) q ( x ) q ( x + m ) = A ( x ) g ( x + m, y − n ) q ( x ) − B ( x ) g ( x, y ) q ( x + m ) . (4.5)It’s easy to check that q ( x ) | g ( x, y ) B ( x ) q ( x + m ) . Since ( q ( x ) , g ( x, y )) = 1, we obtain q ( x ) | B ( x ) q ( x + m ) . Using this divisibility repeatedly, we can get q ( x ) | B ( x ) B ( x + m ) · · · B ( x + ( r − m ) q ( x + rm ) . When r > max Disp x ( q ( x ) , q ( x )), we have ( q ( x ) , q ( x + rm )) = 1, and thus q ( x ) | B ( x ) B ( x + m ) · · · B ( x + ( r − m ) . From Equation (4.5), we also derive that q ( x + m ) | g ( x + m, y − n ) A ( x ) q ( x ) . By a similar discussion, we arrive at q ( x ) | A ( x − m ) A ( x − m ) · · · A ( x − rm ) . By the definition of Gosper representation, we know that gcd( A ( x ) , B ( x + hm )) = 1 for any h ∈ N . Thus the only opportunity for q ( x ) is q ( x ) = 1.When q ( x ) = 1, Equation (4.5) will be reduced to a ( x, y ) B ( x ) C ( x ) = A ( x ) g ( x + m, y − n ) − B ( x ) g ( x, y ) . It’s easy to see that B ( x ) | A ( x ) g ( x + m, y − n ) , and hence B ( x ) | g ( x + m, y − n ). Setting g ( x, y ) = B ( x − m ) p ( x, y ) concludes the proof.9ubstituting (4.3) and b ( x ) p ( x, y ) = B ( x − m ) p ( x,y ) C ( x ) into (4.2), we obtain a ( x, y ) C ( x ) = A ( x ) p ( x + m, y − n ) − B ( x − m ) p ( x, y ) . (4.6)Notice that deg y ( p ) < d . Therefore, in order to solve for p ( x, y ), it suffices to find anupper bound on deg x ( p ).From the Gosper representation (4.3), we see that deg x ( A ) = deg x ( B ) and their leadingcoefficients coincide. Now we write A ( x ) = d X k =0 a k x k , B ( x − m ) = d X k =0 b k x k , C ( x ) = d X k =0 c k x k ,a ( x, y ) = d X k =0 α k ( y ) x k , p ( x, y ) = d X k =0 p k ( y ) x k . Theorem 4.2.
Suppose A ( x ) , B ( x ) and C ( x ) are given in (4.3) , deg y ( a ) < d and they havethe above expansions. If p ( x, y ) is a polynomial that satisfies (4.6) , then d ≤ max (cid:26) d + d − d + d , b d − − a d − ma d + d − (cid:27) . Proof.
There are two cases concerning n : Case . n = 0.Since the degrees and the leading coefficients of A ( x ) and B ( x − m ) coincide, the leadingterm of the right hand of (4.6) is canceled. By considering the second leading term, weencounter two cases. Case a. The second leading term is not canceled. We then have d = d + d − d + 1 . Case b. The second leading term is also canceled. We must have a d ( p d ( y ) d m + p d − ( y )) + a d − p d ( y ) = a d p d − ( y ) + b d − p d ( y ) , which leads to d = b d − − a d − ma d . Case . n = 0.Starting from i = 0, we consider whether the ( i + 1)th leading term of the right hand of(4.6) is canceled consequently.For i = 0, we have the cases 2 a and 2 b . Case a. The leading term is not canceled. We have d = d + d − d . ase b. The leading term is canceled. Then we have p d ( y − n ) = p d ( y ) , which implies that p d ( y ) is a constant.In general, we have the cases 2 a i and 2 b i . Case a i . The ( i + 1)th leading term is not canceled. Then we have d = d + d − d + i. Case b i . The ( j + 1)th leading term are all canceled for j = 0 , , , . . . , i . We claim thatdeg y ( p d − j ) ≤ j for j = 0 , , , . . . , i . When i = 0, the claim holds by the discussion in Case b . Suppose we have known that deg y ( p d − j ) ≤ j for j = 0 , , . . . , i −
1. Now we consider theinduction step from i − i . By the condition that the ( i + 1)th leading term is canceled,we have a d ( p d − i ( y − n ) − p d − i ( y ))= b d − p d − i +1 ( y ) + . . . + b d − i p d − a d (cid:18) p d (cid:18) d i (cid:19) m i + p d − ( y − n ) (cid:18) d − i − (cid:19) m i − + . . . + p d − i +1 ( y − n )( d − i + 1) (cid:19) − a d − (cid:18) p d (cid:18) d i − (cid:19) m i − + p d − ( y − n ) (cid:18) d − i − (cid:19) m i − + . . . + p d − i +1 ( y − n ) (cid:19) − a d − (cid:18) p d (cid:18) d i − (cid:19) m i − + p d − ( y − n ) (cid:18) d − i − (cid:19) m i − + . . . + p d − i +2 ( y − n ) (cid:19) − . . . − a d − i p d . (4.7)Since deg y ( p d − j ) ≤ j for j = 0 , . . . , i −
1, we know that deg y ( p d − i ) ≤ i . This proves theclaim.Now consider the above process. Since deg y ( p ( x, y )) < d , there exists an integer i ≤ d such that deg y ( p d − i ) < i . Without loss of generality, we assume that i is the smallest suchinteger. If we are in case 2 a i , we will get an upper bound d = d + d − d + i ≤ d + d − d + d . Otherwise, we have deg y ( p d − i ( y − n ) − p d − i ( y )) ≤ i − , which means that the coefficient of y i − on the right hand side of (4.7) is canceled. Thus, b d − − a d ( d − i + 1) m − a d − = 0 , leading to the upper bound: d = b d − − a d − ma d + i − ≤ b d − − a d − ma d + d − . This completes the proof. 11ow we are ready to describe an algorithm for testing the summability of a rationalfunction f ( x, y ) of the form f = a ( x,y ) b ( x ) d j ( x,y ) . Algorithm SumTest:Input:
A rational function f = a ( x,y ) b ( x ) d j ( x,y ) . Output:
True, if f is ( σ x , σ y )-summable and return p such that Equation (4.1) holds; False,otherwise.1. Compute Disp( d ( x, y ) , d ( x, y )) by algorithm DispSet . If it is the set { (0 , } , thenreturn False. Otherwise, let ( m, n ) be the element in the dispersion set such that m is theminimum positive integer.2. Compute the Gosper representation ( A ( x ) , B ( x ) , C ( x )) of b ( x ) b ( x + m ) .3. Set d = deg y ( d ( x, y )) , d = deg x ( A ( x )) , d = deg x ( C ( x )) , and d = deg x ( a ( x, y )) . Suppose a d = coeff( A ( x ) , x, d ) ,a d − = coeff( A ( x ) , x, d − ,b d − = coeff( B ( x − m ) , x, d − , where coeff( f ( x ) , x, m ) denotes the coefficient of x m in the expansion of f ( x ).4. Let d = max { d + d − d + d , j b d − − a d − ma d k + d − } . Set p ( x, y ) = d X i =0 d − X j =0 c i,j x i y j , and plug it into a ( x, y ) C ( x ) = A ( x ) p ( x + m, y − n ) − B ( x − m ) p ( x, y ) . By comparing all the coefficients of x and y on both sides, we determine whether the aboveequation has a solution p ( x, y ). If it has no solution, then return False. Otherwise, return asolution p ( x, y ) and p ( x, y ) = B ( x − m ) p ( x,y ) b ( x ) C ( x ) .Finally, we give some examples to illustrate how to use our criterion for deciding thesummability of some rational functions. Example 4.3.
Let f ( x, y ) = − ( x + y + 4)( x + 2 x + 2 xy − y + y )( x + 2 xy + y − . Denote d ( x, y ) = x + 2 xy + y − x + y ) − . y computing the dispersion set, we find that x + 2 x + 2 xy − y + y = d ( x + 1 , y ) . By partial fraction decomposition, we derive that f ( x, y ) = X l =0 a l ( x, y ) d ( x + l, y ) , where a ( x, y ) = − x − y, a ( x, y ) = x + y + 2 . Using ( σ x , σ y ) -reduction, we can write f ( x, y ) as f ( x, y ) = ∆ x ( g ) + r ( x, y ) , (4.8) where g ( x, y ) = x + y + 1( x + y ) − and r ( x, y ) = 1( x + y ) − . It is easy to see that σ x d ( x, y ) = σ y d ( x, y ) . What left now is to check whether there existsa polynomial p ( x, y ) such that σ x σ − y p ( x, y ) − p ( x, y ) . (4.9) The x -degree bound and y -degree bound of p are and respectively. By the method ofundetermined coefficients, we find a solution p ( x, y ) = − y − . From the proof of Lemma 4.5,we find out r ( x, y ) = σ x g ( x, y ) − g ( x, y ) + σ y h ( x, y ) − h ( x, y ) , where g ( x, y ) = − y − x + y ) − , h ( x, y ) = y ( x + y ) − . Substituting into (4.8) , we finally derive that f ( x, y ) = σ x g ( x, y ) − g ( x, y ) + σ y h ( x, y ) − h ( x, y ) , where g ( x, y ) = x ( x + y ) − , and h ( x, y ) = y ( x + y ) − . Example 4.4.
Let f ( x, y ) = x + x y + y + 1( x + y )( x + 2 xy + xy + y ) . We can decompose it into f ( x, y ) = x + x − x + x − x + x y − xy − y ( x + 2 xy + xy + y ) x ( x −
2) + − x + y ( x + y ) x ( x − . Note that σ mx ( x + y ) = σ ny ( x + y ) , for any ( m, n ) = (0 , . Then Theorem 3.3 implies that − x + y ( x + y ) x ( x − is not ( σ x , σ y ) -summable, which leads to theresult that f ( x, y ) is not ( σ x , σ y ) -summable in Q ( x, y ) . Acknowledgments.
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Appendix
In Theorem 3.3, we provide a criterion for the summability of rational functions of theform a ( x,y ) d j ( x,y ) , where a ∈ Q ( x )[ y ] and d ∈ Q [ x, y ] is an irreducible polynomial. In this appendix,we present the proof of this criterion.Firstly, we give the following lemma which proves the sufficiency of Theorem 3.3. Lemma 4.5.
Let f ∈ Q ( x, y ) be of the form f = a ( x,y ) d j ( x,y ) , where d ∈ Q [ x, y ] is an irreduciblepolynomial, a ∈ Q ( x )[ y ] , and deg y ( a ) < deg y ( d ) . Suppose that there exist integers t, l with t > and a polynomial p ( x, y ) ∈ Q ( x )[ y ] such that σ tx d ( x, y ) = σ ℓy d ( x, y ) , nd a = σ tx σ − ℓy p ( x, y ) − p ( x, y ) . Then f is ( σ x , σ y ) -summable in Q ( x, y ) .Proof. Let g = P t − k =0 σ kx ( p ) σ kx ( d j ) , then ad j − ( σ x g − g ) = ad j − σ tx pσ tx d j + pd j = a + pd j − σ tx pσ ℓy d j = − σ ℓy σ tx σ − ℓy pd j ! + σ tx σ − ℓy pd j . The rest part of the appendix is devoted to proving the necessity of Theorem 3.3.Analogue to the discrete residue given by Chen and Singer [10], we introduce the conceptof polynomial residue. Let K be a field and f ∈ K ( x ). By partial fraction decomposition, f can be written as f = p ( x ) + m X i =1 n i X j =1 k i,j X ℓ =0 a i,j,ℓ ( x ) σ ℓx d ji ( x ) , (4.10)where p ( x ) ∈ K [ x ], m, n i , k i,j ∈ N , deg x ( a i,j,ℓ ) < deg x ( d i ), and d i ( x ) ( i = 1 , . . . , m ) areirreducible polynomials that in distinct σ x -orbits. The summation k i,j X ℓ =0 σ − ℓx ( a i,j,ℓ )is called the polynomial residue of f at the σ x -orbit of d i ( x ) of multiplicity j , denoted byres x ( f ( x ) , d i ( x ) , j ).It is easy to check that the summability of rational functions in K ( x ) can be given viapolynomial residue. The proof is similar to the case of discrete residue [9, 10] and is omited. Proposition 4.6.
Let f ( x ) ∈ K ( x ) be of the form (4.10) . Then f ( x ) is σ x -summable in K ( x ) if and only if the polynomial residue res x ( f ( x ) , d i ( x ) , j ) is zero for any polynomial d i ( x ) andany multiplicity j . Now we are ready to prove the necessity of Theorem 3.3.Suppose that f = a/d j is ( σ x , σ y )-summable and assume that f = σ x g − g + σ y h − h, (4.11)where g, h ∈ Q ( x, y ). As a univarite analogue to Lemma 3.1, we can decompose g into theform g = σ y g − g + g + λ σ µ x d j + · · · + λ s σ µ s x d j , where g , g ∈ Q ( x, y ) with g containing no term of the form λσ ux d j in its partial fractiondecomposition with respect to y , µ ℓ ∈ Z , λ ℓ ∈ Q ( x )[ y ], and σ µ ℓ x d ( ℓ = 1 , . . . , s ) are irreduciblepolynomials in distinct σ y -orbits. Claim 1.
Let Λ := { σ µ x d, . . . , σ µ s x d, σ µ +1 x d, . . . , σ µ s +1 x d } . Then 16a) At least one element of Λ is in the same σ y -orbit as d .(b) For each element η ∈ Λ, there is one element of Λ \{ η } S { d } that is in the same σ y -orbitas η . Proof of Claim 1. (a) Suppose there is no element of Λ that is in the same σ y -orbit as d . Since f = a/d j , we have res y ( f, d, j ) = a = 0. While by (4.11) and Proposition 4.6, we deduce thatres y ( f, d, j ) = res y ( σ x g − g, d, j ) = 0 , which is a contradiction.(b) The assertion follows from the same argument when considering the polynomialresidues of η on both sides of (4.11).Claim 1 implies that either d ∼ y σ µ ′ x d or d ∼ y σ µ ′ +1 x d for some µ ′ ∈ { µ , . . . , µ s } . Wewill only consider the first case. The second case can be treated similarly. Claim 2.
Assume d ∼ y σ µ ′ x d . We have the following assertions.(a) Suppose k ≥ σ lx d ≁ y d for 1 ≤ l ≤ k −
1. Then there exist µ ′ , . . . , µ ′ k ∈ { µ , . . . , µ s } such that σ µ ′ +1 x d ∼ y σ µ ′ x d, σ µ ′ +1 x d ∼ y σ µ ′ x d, . . . , σ µ ′ k − +1 x d ∼ y σ µ ′ k x d, and σ k − x d ∼ y σ µ ′ k x d. (b) There exists a positive integer t ≤ s such that σ tx d ∼ y d . Proof of Claim 2. (a) From Claim 1(b), we derive that σ µ ′ +1 x d is σ y -equivalent to an elementof Λ \{ σ µ ′ +1 x d } S { d } . If σ µ ′ +1 x d ∼ y d , then σ µ ′ +1 x d ∼ y σ µ ′ x d and thus σ x d ∼ y d , whichcontradicts to the hypothese on k . If σ µ ′ +1 x d ∼ y σ µ ′ l +1 x d , then σ µ ′ x d ∼ y σ µ ′ l x d for some l , whichcontradicts to the assumption that σ µ ℓ x are in distinct σ y -orbits. Therefore we are left withthe only possibility that σ µ ′ +1 x d ∼ y σ µ ′ x d for some µ ′ ∈ { µ , . . . , µ s } \ { µ ′ } . Continue thisprocess, we will find µ ′ , . . . , µ ′ k such that σ µ ′ +1 x d ∼ y σ µ ′ x d, . . . , σ µ ′ k − +1 x d ∼ y σ µ ′ k x d. Finally, we have σ µ ′ k x d ∼ y σ µ ′ + k − x d ∼ y σ k − x d. (b) If such t does not exist, then one could find { µ ′ , . . . , µ ′ s +1 } satisfying the constraintsin (a). Thus, it holds that µ ′ r = µ ′ t for some r > t . Hence σ µ ′ + rx d ∼ y σ µ ′ + tx d , which leads to σ r − tx d ∼ y d , a contradiction.Suppose t is the smallest integer such that σ tx d ∼ y d . Then taking k = t in Claim 2(a),we derive that there exist µ ′ , . . . , µ ′ t ∈ { µ , . . . , µ s } such that σ µ ′ +1 x d ∼ y σ µ ′ x d, σ µ ′ +1 x d ∼ y σ µ ′ x d, . . . , σ µ ′ t − +1 x d ∼ y σ µ ′ t x d, σ µ ′ t +1 x d ∼ y σ tx d ∼ y d. Recall that σ µ ′ x d ∼ y d . By the definition of ∼ y , there exist integers s , s , . . . , s t such that σ µ ′ k +1 x d = σ µ ′ k +1 x σ s k +1 y d, ≤ k ≤ t − , σ µ ′ t +1 x d = σ s y d, and σ µ ′ x d = σ s y d. Hence, σ s y d = σ µ ′ x d = σ µ ′ − x σ s y d = σ µ ′ − x σ s + s y d = · · · = σ µ ′ t − t +1 x σ s + s + ··· + s t y d = σ s + ··· + s t y σ − tx d. Setting ℓ = s + · · · + s t − s , we then have σ tx d = σ ℓy d .Now we compare the polynomial residues on both sides of (4.11). We list the residues σ y -orbit Comparison of two sides of (4.11) d, σ µ ′ t +1 x d a = σ x σ − s y λ ′ t − σ − s y λ ′ σ µ ′ t − +1 x d, σ µ ′ t x d σ x σ − s t y λ ′ t − − λ ′ t σ µ ′ t − +1 x d, σ µ ′ t − x d σ x σ − s t − y λ ′ t − − λ ′ t − ... ... σ µ ′ +1 x d, σ µ ′ x d σ x σ − s y λ ′ − λ ′ σ µ ′ +1 x d, σ µ ′ x d σ x σ − s y λ ′ − λ ′ Table 1: Orbits and their corresponding polynomial residues.in Table 1, where the first column consists of the σ y -orbits of elements in Λ and the secondcolumn consists of the equations obtained by equating the corresponding polynomial residueson both sides of (4.11). By investigating the equations in Table 1 from bottom to top, wefind that a = σ tx σ − ℓy p − p, where p = σ − s y λ ′ ( x, y ). Since deg y λ ′ < deg y d , we have deg y p < deg y dd