On the Structure of Compatible Rational Functions
aa r X i v : . [ c s . S C ] J a n On the Structure of Compatible Rational Functions ∗ Shaoshi Chen , , , Ruyong Feng , Guofeng Fu , Ziming Li Key Lab of Math.-Mech. AMSS, Chinese Academy of Sciences, Beijing 100190, (China) Algorithms Project-Team, INRIA, Paris-Rocquencourt, 78513 Le Chesnay, (France) RISC, Johannes Kepler University, 4040 Linz, (Austria) { schen, ryfeng } @amss.ac.cn, { fuguofeng, zmli } @mmrc.iss.ac.cn ABSTRACT
A finite number of rational functions are compatible if theysatisfy the compatibility conditions of a first-order linearfunctional system involving differential, shift and q -shift op-erators. We present a theorem that describes the structureof compatible rational functions. The theorem enables us todecompose a solution of such a system as a product of a ra-tional function, several symbolic powers, a hyperexponentialfunction, a hypergeometric term, and a q -hypergeometricterm. We outline an algorithm for computing this product,and present an application. Categories and Subject Descriptors
I.1.2 [
Computing Methodologies ]: Symbolic and Alge-braic Manipulation—
Algebraic Algorithms
General Terms
Algorithms, Theory
Keywords
Compatibility conditions, compatible rational functions, hy-perexponential function, ( q -)hypergeometric term
1. INTRODUCTION
A linear functional system consists of linear partial dif-ferential, shift and q -shift operators. The commutativity ofthese operators implies that the coefficients of a linear func-tional system satisfy compatibility conditions. ∗ This work was supported in part by two grants of NSFCNo. 60821002/F02 and No. 10901156. The first author wasa PhD student in the Chinese Academy of Sciences and IN-RIA, Paris-Rocquencourt when the first draft of this paperwas written. He is now a post doctoral fellow at RISC-Linz,and acknowledges the financial support by Austrian FWFgrant Y464-N18
Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.
ISSAC XXXX
Copyright 2011 ACM ...$10.00.
A nonzero solution of a first-order linear partial differen-tial system in one unknown function is called a hyperexpo-nential function. Christopher and Zoladek [9, 21] use thecompatibility (integrability) conditions to show that a hy-perexponential function can be written as a product of arational function, finitely many power functions, and an ex-ponential function. Their results generalize a well-knownfact, namely, for a rational function r ( t ),exp (cid:18)Z r ( t ) dt (cid:19) = f ( t ) r ( t ) e · · · r m ( t ) e m exp( g ( t )) , where e , . . . , e m are constants, and f, r , . . . , r m , g are ra-tional functions. The generalization is useful to computeLiouvilian first integrals.A nonzero solution of a first-order linear partial differ-ence system in one unknown term is called a hypergeometricterm. The Ore-Sato Theorem [16, 18] states that a hyper-geometric term is a product of a rational function, severalpower functions and factorial terms. A q -analogue of theOre-Sato theorem is given in [11, 8]. All these results arebased on compatibility conditions. The Ore-Sato theoremwas rediscovered in one way or another, and is importantfor the proofs of a conjecture of Wilf and Zeilberger aboutholonomic hypergeometric terms [2, 4, 17]. This theoremand its q -analogue also play a crucial role in deriving crite-ria on the existence of telescopers for hypergeometric and q -hypergeometric terms, respectively [1, 8].Consider a first-order mixed system (cid:26) ∂z ( t, x ) ∂t = u ( t, x ) z ( t, x ) , z ( t, x + 1) = v ( t, x ) z ( t, x ) (cid:27) , where u and v are rational functions with v = 0. Its compat-ibility condition is ∂v ( t, x ) /∂t = v ( t, x )( u ( t, x + 1) − u ( t, x )) . By Proposition 5 in [10], a nonzero solution of the abovesystem can be written as a product f ( t, x ) r ( t ) x E ( t ) G ( x ),where f is a bivariate rational function in t and x , r isa univariate rational function in t , E is a hyperexponen-tial function in t , and G is a hypergeometric term in x .This proposition is used to compute Liouvillian solutionsof difference-differential systems.In fact, the above proposition is also fundamental for thecriteria on the existence of telescopers when both differen-tial and shift operators are involved [7]. This motivates us togeneralize the proposition to include differential, differenceand q -difference cases. Such a generalization will enable usto establish the existence of telescopers when both differen-tial (shift) and q -shift operators appear. Next, the proof ofthe Wilf-Zeilberger conjecture for hypergeometric terms isased on the Ore-Sato theorem. So it is reasonable to expectthat a structural theorem on compatible rational functionswith respect to differential, shift and q -shift operators helpsus study the conjecture in more general cases.The main result of this paper is Theorem 5.4 which re-veals a special structure of compatible rational functions.By the theorem, a hyperexponential-hypergeometric solu-tion, defined in Section 2, is a product of a rational func-tion, several symbolic powers, a hyperexponential function,a hypergeometric term, and a q -hypergeometric term (seeProposition 6.1). This paves the way to decompose suchsolutions by Christopher-Zoladek’s generalization, the Ore-Sato Theorem, and its q -analogue.This paper is organized as follows. The notion of com-patible rational functions is introduced in Section 2. Thebivariate case is studied in Section 3. After presenting a fewpreparation lemmas in Section 4, we prove in Section 5 atheorem that describes the structure of compatible rationalfunctions. Section 6 is about algorithms and applications.
2. COMPATIBLE RATIONAL FUNCTIONS
In the rest of this paper, F is a field of characteristic zero.Let t = ( t , . . . , t l ), x = ( x , . . . , x m ) and y = ( y , . . . , y n ).Assume that q , . . . , q n ∈ F are neither zero nor roots ofunity. For an element f of F ( t , x , y ), define δ i ( f ) = ∂f/∂t i for all i with 1 ≤ i ≤ l , σ j ( f ( t , x , y )) = f ( t , x , . . . , x j − , x j + 1 , x j +1 , . . . , x m , y )for all j with 1 ≤ j ≤ m , and τ k ( f ( t , x , y )) = f ( t , x , y , . . . , y k − , q k y k , y k +1 , . . . , y n )for all k with 1 ≤ k ≤ n . They are called derivations, shiftoperators, and q -shift operators, respectively.Let ∆ = { δ , . . . , δ l , σ , . . . , σ m , τ , . . . , τ n } . These oper-ators commute pairwise. The field of constants w.r.t. anoperator in ∆ consists of all rational functions free of theindeterminate on which the operator acts nontrivially.By a first-order linear functional system over F ( t , x , y ),we mean a system consisting of δ i ( z ) = u i z, σ j ( z ) = v j z, τ k ( z ) = w k z (1)for some rational functions u i , v j , w k ∈ F ( t , x , y ) and forall i, j, k with 1 ≤ i ≤ l , 1 ≤ j ≤ m and 1 ≤ k ≤ n .System (1) is said to be compatible if v · · · v m w · · · w n = 0 (2)and the conditions listed in (3)-(8) hold: δ i ( u j ) = δ j ( u i ) , ≤ i < j ≤ l , (3) σ i ( v j ) /v j = σ j ( v i ) /v i , ≤ i < j ≤ m , (4) τ i ( w j ) /w j = τ j ( w i ) /w i , ≤ i < j ≤ n , (5) δ i ( v j ) /v j = σ j ( u i ) − u i , ≤ i ≤ l and 1 ≤ j ≤ m , (6) δ i ( w k ) /w k = τ k ( u i ) − u i , ≤ i ≤ l and 1 ≤ k ≤ n , (7) σ j ( w k ) /w k = τ k ( v j ) /v j , ≤ j ≤ m and 1 ≤ k ≤ n . (8)Compatibility conditions (3)-(8) are caused by the commu-tativity of the maps in ∆. A sequence of rational func-tions: u , . . . , u l , v , . . . , v m , w , . . . , w n is said tobe compatible w.r.t. ∆ if (2)-(8) hold. By a ∆-extension of F ( t , x , y ), we mean a ring exten-sion R of F ( t , x , y ) s.t. every derivation and automorphismin ∆ can be extended to a derivation and a monomorphismfrom R to R , and, moreover, the extended maps are commu-tative with each other. Given a finite number of first-ordercompatible systems, one can construct a Picard-Vessiot ∆-extension of F ( t , x , y ) that contains “all” solutions of thesesystems. Moreover, every nonzero solution is invertible.Details on Picard-Vessiot extensions of compatible systemsmay be found in [5]. More general and powerful extensionsare described in [12]. By a hyperexponential-hypergeometricsolution h over F ( t , x , y ), we mean a nonzero solution of thesystem (1). The coefficients u i , v j and w k in (1) are called δ i -, σ j -, and τ k -certificates of h , respectively. For brevity,we abbreviate “hyperexponential-hypergeometric solution”as “ H -solution”. An H -solution is a hyperexponential func-tion when m = n = 0 in (1), it is a hypergeometric termif l = n = 0, and a q -hypergeometric term if l = m = 0. Remark
We opt for the word “solution” rather than“function”, since all the t i , x j and y k are regarded as indeter-minates. It is more sophisticated to regard hypergeometricterms as functions of integer variables [17, 4, 3]. As a matter of notation, for an element f ∈ F ( t , x , y ),the denominator and numerator of f are denoted den( f )and num( f ), respectively. Note that den( f ) and num( f )are coprime. For a ring A , A × stands for A \{ } , and fora field E , E stands for the algebraic closure of E . For ev-ery φ ∈ ∆ and f ∈ F ( t , x , y ) × , we denote by ℓφ ( f ) thefraction φ ( f ) /f . When φ is a derivation δ i , ℓδ i ( f ) standsfor the logarithmic derivative of f with respect to t i . Thisnotation allows us to avoid stacking fractions and subscripts.Let E be a field and t an indeterminate. A nonzero el-ement f of E ( t ) can be written uniquely as f = p + r ,where p ∈ E [ t ] and r is a proper fraction. We say that p is the polynomial part of f w.r.t. t . Remark
Let z ∈{ t , . . . , t l , x , . . . , x m , y , . . . , y n } and f ∈ F ( t , x , y ) × . For all i with ≤ i ≤ l , the polynomialpart of ℓδ i ( f ) w.r.t. z has degree at most zero in z .
3. BIVARIATE CASE
In this section, we assume that l = m = n = 1. Forbrevity, set t = t , x = x , y = y , δ = δ , σ = σ , τ = τ ,and q = q . By (2), (6), (7) and (8), three rational func-tions u, v, w in F ( t, x, y ) are ∆-compatible if vw = 0, ℓδ ( v ) = σ ( u ) − u, (9) ℓδ ( w ) = τ ( u ) − u, (10) ℓσ ( w ) = ℓτ ( v ) . (11)Other compatibility conditions become trivial in this case. Example
Let α ∈ F ( t, y ) × . The system consistingof δ ( z ) = ℓδ ( α ) x z and σ ( z ) = α z is compatible w.r.t. δ and σ . Denote a solution of this system by α x , which isirrational if α = 1 . The next lemma is immediate from [10, Proposition 5].
Lemma
Let u, v ∈ F ( t, x, y ) with v = 0 . If (9) holds,then u = ℓδ ( f ) + ℓδ ( α ) x + β and v = ℓσ ( f ) αλ for some f in F ( t, x, y ) , α, β in F ( t, y ) , and λ in F ( x, y ) . ssume that an H -solution h has δ -certificate u and σ -certificate v . By Lemma 3.1, h = cfα x EG in some ∆-ring,where c is a constant w.r.t. δ and σ , E is hyperexponentialwith certificate β , and G is hypergeometric with certificate λ .We shall prove two similar results: one is about differentialand q -shift variables; the other about shift and q -shift ones.To this end, we recall some terminologies from [2, 4, 12].Let A = F ( t, y ) and p ∈ A [ x ] × . The σ -dispersion of p isdefined to be the largest nonnegative integer i s.t. for some r in A , r and r + i are roots of p . Let f ∈ A ( x ) × . We saythat f is σ -reduced if den( f ) and σ i (num( f )) are coprimefor every integer i ; and that f is σ -standard if zero is the σ -dispersion of num( f )den( f ). A σ -standard rational functionis a σ -reduced one, but the converse is false. By Lemma 6.2in [12], f = ℓσ ( a ) b for some a, b in A ( x ) with b being σ -standard or σ -reduced.Let B = F ( t, x ) and p ∈ B [ y ] × . The τ -dispersion of p isdefined to be the largest nonnegative integer i s.t. for some nonzero r ∈ B , r and q i r are roots of p . In addition, the τ -dispersion of p is set to be zero if p = cy k for some c ∈ B .Let f ∈ B ( y ) × . The polar τ -dispersion is the τ -dispersionof den( f ). The notion of τ -reduced and τ -standard rationalfunctions are defined likewise. One can write f = ℓτ ( a ) b ,where a, b ∈ B ( y ) × and b is τ -standard or τ -reduced.Now, we prove a q -analogue of Lemma 3.1. Lemma
Let u, w ∈ F ( t, x, y ) with w =0 . If (10) holds,then u = ℓδ ( f )+ a and w = ℓτ ( f ) b for some f in F ( t, x, y ) , a in F ( t, x ) , and b in F ( x, y ) . Proof.
Set w = ℓτ ( f ) b for some f, b in F ( t, x, y ) with b being τ -standard. Set b = y k P/Q , where
P, Q ∈ F ( x )[ t, y ]are coprime, and neither is divisible by y . Since b is τ -standard, so is P/Q . Assume u = ℓδ ( f ) + a . By (10), ℓδ ( P/Q ) = τ ( a ) − a. (12)Since P/Q is τ -standard, the τ -dispersion of P Q is zero,and so is the polar τ -dispersion of the left-hand side in (12),which, together with [12, Lemma 6.3], implies that a belongsto F ( t, x )[ y, y − ]. Moreover, a is free of positive powers of y by Remark 2.2 (setting z = y ); and a is free of negativepowers of y , because neither P nor Q is divisible by y . Weconclude that a is in F ( t, x ). Consequently, τ ( a ) = a . Itfollows from (12) that δ ( P/Q ) = 0, i.e., b is in F ( x, y ).By the above lemma, an H -solution h can be written as aproduct of a constant w.r.t. δ and τ , a rational function, ahyperexponential function, and a q -hypergeometric term.The last lemma is a q -analogue of [4, Theorem 9]. Ourproof is based on an easy consequence of [20, Lemma 2.1]. Fact
Let a, b ∈ F ( t, x, y ) × . If σ ( a ) = ba , and P isan irreducible factor of den( b ) with deg x P > , then σ i ( P ) is a factor of num( b ) for some nonzero integer i .The same is true if we swap den( b ) and num( b ) in theabove assertion. Lemma
Let v, w ∈ F ( t, x, y ) × . If (11) holds, then v = ℓσ ( f ) a and w = ℓτ ( f ) b for some f in F ( t, x, y ) , a in F ( t, x ) , and b in F ( t, y ) . Proof.
In this proof, P | Q means that P, Q ∈ F ( t )[ x, y ] × and Q = P R for some R ∈ F ( t )[ x, y ].Set v = ℓσ ( f ) a , where f, a ∈ F ( t, x, y ) and a is σ -reduced.Assume w = ℓτ ( f ) b . By (11), ℓσ ( b ) = ℓτ ( a ), that is, σ ( b ) = gb, where g = τ (num( a )) den( a ) τ (den( a )) num( a ) . (13) First, we show that a is the product of an element in F ( t, x )and an element in F ( t, y ). Suppose the contrary. Then thereis an irreducible polynomial P ∈ F ( t )[ x, y ] with deg x P > y P > P divides den( a )num( a ) in F ( t )[ x, y ]. As-sume that P | num( a ). If P ∤ den( g ), then P | τ (num( a ))since num( a ) and den( a ) are coprime. So τ − ( P ) | num( a ).If P | den( g ), then σ i ( P ) | num( g ) for some integer i by (13)and Fact 3.1. Thus, σ i ( P ) | τ (num( a )), because num( g ) isa factor of τ (num( a ))den( a ) and a is σ -reduced. This im-plies σ i τ − ( P ) | num( a ). In either case, we have that σ j τ − ( P ) | num( a ) for some integer j .Assume P | den( a ). Then the same argument implies σ k τ − ( P ) | den( a ) for some integer k .Hence, there exists an integer m s.t. P := σ m τ − ( P ) isan irreducible factor of den( a )num( a ), where P = P . Arepeated use of the above reasoning leads to an infinite se-quence of irreducible polynomials P , P , P , . . . in F ( t )[ x, y ]s.t. P i = σ m i τ − ( P i − ) and P i | den( a )num( a ) . Therefore,there are two F ( t )-linearly dependent members in the se-quence. Using these two members, we get P = cσ m τ n ( P )for some c in F ( t ) and m, n in Z with n = 0. Write P = p d ( x ) y d + p d − ( x ) y d − + · · · + p ( x ) , where d > p i ∈ F ( t )[ x ] and p d = 0. Then p d ( x ) = cp d ( x + m ) q − dn and p ( x ) = cp ( x + m ) . Since P is irreducible and of positive degree in x , p is alsononzero. We see that 1 = cq − dn and 1 = c when comparingthe leading coefficients in the above two equalities. Conse-quently, q is a root of unity, a contradiction. This proves thatall irreducible factors of den( a )num( a ) are either in F ( t )[ x ]or F ( t )[ y ]. Therefore, a is a product of an element in F ( t, x )and an element in F ( t, y ).So we can write a = a a for some a in F ( t, x ) and a in F ( t, y ). By ℓσ ( b ) = ℓτ ( a ), the equation σ ( z ) = ℓτ ( a ) z has a rational solution b . Since ℓτ ( a ) is a constant w.r.t. σ ,we conclude ℓτ ( a ) = 1, for otherwise, σ ( z ) = ℓτ ( a ) z wouldhave no rational solution. So b ∈ F ( t, y ) and a ∈ F ( t, x ).Similar to Lemmas 3.1 and 3.2, the above lemma impliesthat an H -solution h can be written as a product of a con-stant w.r.t. σ and τ , a rational function, a hypergeometricterm, and a q -hypergeometric term.We shall extend these lemmas to multivariate cases in Sec-tion 5. Before closing this section, we present three examplesto illustrate calculations involving compatibility conditions.These calculations are useful in Section 5. Example
Assume u = ℓδ ( f ) + ℓδ ( a ) x + b and v = ℓσ ( f ) a c, where f, c ∈ F ( t, x, y ) × , a ∈ F ( t, y ) × , and b ∈ F ( t, x, y ) . Bythe logarithmic derivative identity: for all r, s in F ( t, x, y ) × , ℓδ ( r s ) = ℓδ ( r ) + ℓδ ( s ) , we get ℓδ ( v ) = ℓδ ◦ ℓσ ( f ) + ℓδ ( a ) + ℓδ ( c ) . Since ℓδ ( a ) is constant w.r.t. σ , and σ ◦ ℓδ = ℓδ ◦ σ , we have σ ( u ) − u = σ ◦ ℓδ ( f ) − ℓδ ( f ) + ℓδ ( a ) + σ ( b ) − b = ℓδ ◦ ℓσ ( f ) + ℓδ ( a ) + σ ( b ) − b. f (9) holds, then ℓδ ( c )= σ ( b ) − b . Hence, δ ( c )=0 iff σ ( b ) = b ,i.e., c ∈ F ( x, y ) iff b ∈ F ( t, y ) . Example
Assume u = ℓδ ( f ) + a and w = ℓτ ( f ) b, where a ∈ F ( t, x, y ) and f, b ∈ F ( t, x, y ) × . If (10) holds,then a similar calculation as above yields ℓδ ( b ) = τ ( a ) − a .Hence, δ ( b ) = 0 iff τ ( a ) = a , i.e., b ∈ F ( x, y ) iff a ∈ F ( t, x ) . Example
Assume v = ℓσ ( f ) a and w = ℓτ ( f ) b, where f, a, b ∈ F ( t, x, y ) × . Applying ℓσ, ℓτ to w, v , respec-tively, we see that ℓσ ( w ) = ℓσ ◦ ℓτ ( f ) ℓσ ( b ) , ℓτ ( v ) = ℓτ ◦ ℓσ ( f ) ℓτ ( a ) . If (11) holds, then ℓσ ( b ) = ℓτ ( a ) , because ℓσ ◦ ℓτ = ℓτ ◦ ℓσ .Hence, σ ( b ) = b iff τ ( a ) = a , i.e., b ∈ F ( t, y ) iff a ∈ F ( t, x ) .
4. PREPARATION LEMMAS
To extend Lemmas 3.1, 3.2, and 3.3 to multivariate cases,we will proceed by induction on the number of variables.There arise different expressions for a rational function inour induction. Lemmas given in this section will be used toeliminate redundant indeterminates in these expressions.We define a few additive subgroups of F ( t , x , y ) to avoidcomplicated expressions. L i = (cid:8) ℓδ i ( f ) | f ∈ F ( t , x , y ) × (cid:9) , i = 1 , . . . , l,M i = ( m X j =1 ℓδ i ( g j ) x j | g j ∈ F ( t , y ) × ) , i = 1 , . . . , l. For i = 1 , . . . , l and j = 1 , . . . , m , M i,j denotes the group j − X k =1 ℓδ i ( g k ) x k + m X k = j +1 ℓδ i ( g k ) x k | g k ∈ F ( t , x j , y ) × . Moreover, we set N i = L i + M i + F ( t , y ) and N i,j = L i + M i,j + F ( t , x j , y ) . Let Z = { t , . . . , t l , x , . . . , x m , y , . . . , y n } . We will use anevaluation trick in the sequel. Let Z ′ = { z , . . . , z s } be asubset of Z . For f ∈ F ( t , x , y ) × , there exist ξ , . . . , ξ s in F s.t. f evaluated at z = ξ , . . . , z s = ξ s is a well-definedand nonzero rational function f ′ . We say that f ′ is a properevaluation of f w.r.t. Z ′ . A proper evaluation can be carriedout for finitely many rational functions as well. In addition,we say that a rational function f is free of Z ′ if it is free ofevery indeterminate in Z ′ . Remark If Z ′ ⊂ Z , f ∈ L i and t i / ∈ Z ′ , then allproper evaluations of f w.r.t. Z ′ are also in L i . In the next example, we illustrate two typical proper evalu-ations to be used later.
Example
Let f = ℓδ i ( r ) for some f, r ∈ F ( t , x , y ) × .Assume that both f ( t , ξ , y ) and r ( t , ξ , y ) are well-definedand nonzero, where ξ ∈ F m . Then f ( t , ξ , y ) is still in L i .Let g ∈ F ( t , y ) × . Then δ i ( z ) = gz has a rational solutionin F ( t , y ) × if it has a rational solution in F ( t , x , y ) × . Thiscan also be shown by a proper evaluation. The following lemma helps us merge rational expressionsinvolving logarithmic derivatives.
Lemma
Let i ∈ { , . . . , l } .(i) Let Z , Z ⊂ Z with Z ∩ Z = ∅ . If A is any sub-field of F ( t , x , y ) whose elements are free of t i and freeof Z ∪ Z , then L i + A ( t i ) = ( L i + A ( t i , Z )) ∩ ( L i + A ( t i , Z )) . (ii) If d, e ∈ { , . . . , m } with d = e , then N i = N i,d ∩ N i,e . Proof.
To prove the first assertion, note that L i + A ( t i )is a subset of ( L i + A ( t i , Z )) ∩ ( L i + A ( t i , Z )). Assumethat a is in ( L i + A ( t i , Z )) ∩ ( L i + A ( t i , Z )). Then thereexist a ∈ A ( t i , Z ) and a ∈ A ( t i , Z ) s.t. a ≡ a mod L i and a ≡ a mod L i . Hence, a − a ∈ L i . Let Z ′ = Z \ { t i } , and a ′ be a properevaluation of a w.r.t. Z ′ . Then a − a ′ is a proper evaluationof a − a w.r.t. Z ′ , because a is free of Z ′ . Thus, a − a ′ belongs to L i by Remark 4.1. Since a ′ is in A ( t i ), a isin L i + A ( t i ), and so is a .To prove the second assertion, assume i = 1, d = 1and e = m . Note that N ⊂ N , ∩ N ,m , because M iscontained in ( M , + F ( t , x , y )) ∩ ( M ,m + F ( t , x m , y )). Itremains to show N , ∩ N ,m ⊂ N . Let a ∈ N , ∩ N ,m . Then a = ℓδ ( f ) + (cid:16)P m − j =2 ℓδ ( g j ) x j (cid:17) + ℓδ ( g m ) x m + r (14)= ℓδ ( ˜ f ) + ℓδ (˜ g ) x + (cid:16)P m − j =2 ℓδ (˜ g j ) x j (cid:17) + ˜ r, (15)where f, ˜ f ∈ F ( t , x , y ), g j , r ∈ F ( t , x , y ), ˜ g j , ˜ r ∈ F ( t , x m , y )and f ˜ f g j ˜ g j = 0. For all j with 1 ≤ j ≤ m , let P j be thepolynomial part of a w.r.t. x j . Then deg x j P j ≤ j with 1 ≤ j ≤ m − x m P m ≤ Claim . Let b j denote the coefficient of x j in P j . Then thereexists s j ∈ F ( t , y ) s.t. b j = ℓδ ( s j ) for all j with 1 ≤ j ≤ m . Proof of Claim.
By (14) and Remark 2.2, b is the coefficientof x in the polynomial part of r w.r.t. x . So b is in F ( t , y ).By (15) and the same remark, b = ℓδ (˜ g ). Let s be aproper evaluation of ˜ g w.r.t. x m . Then b = ℓδ ( s ) as b is free of x m . By the same argument, b m = ℓδ ( s m ) forsome s m in F ( t , x ). By (14) and (15), b j = ℓδ ( g j ) = ℓδ (˜ g j )for all j with 2 ≤ j ≤ m −
1. Let s j be a proper evaluationof ˜ g j w.r.t. x m . Then ℓδ j ( g j ) = ℓδ j ( s j ), because g j is freeof x m . Hence, b j = ℓδ ( s j ). The claim holds.Set b = P mj =1 b j x j . Then a − b is in L + F ( t , x , y )and L + F ( t , x m , y ) by (14), (15) and the claim. Thus, a − b is in L + F ( t , y ) by the first assertion (setting Z = { x } , Z = { x m } , and A = F ( t , . . . , t l , y )). By the claim, b isin M . Thus, a is in L + M + F ( t , y ).We define a few multiplicative subgroups in F ( t , x , y ) × .Let G j = { ℓσ j ( f ) | f ∈ F ( t , x , y ) × } for j = 1 , . . . , m . Sim-ilarly, let H k = { ℓτ k ( f ) | f ∈ F ( t , x , y ) × } for k = 1 , . . . , n . Remark If Z ′ ⊂ Z , f ∈ H k and y k / ∈ Z ′ , then all properevaluations of f w.r.t. Z ′ are in H k . The same holds for G j . The next lemma helps us merge rational expressions in-volving shift or q -shift quotients. Lemma
Let j ∈ { , . . . , m } , k ∈ { , . . . , n } . Assumethat Z and Z are disjoint subsets of Z .i) If A is any subfield of F ( t , x , y ) whose elements arefree of x j and free of Z ∪ Z , then G j A ( x j ) × = (cid:0) G j A ( x j , Z ) × (cid:1) ∩ (cid:0) G j A ( x j , Z ) × (cid:1) . (ii) If A is any subfield of F ( t , x , y ) whose elements arefree of y k and free of Z ∪ Z , then H k A ( y k ) × = (cid:0) H k A ( y k , Z ) × (cid:1) ∩ (cid:0) H k A ( y k , Z ) × (cid:1) . (iii) If A = F ( t , y ) and B = F ( x , y ) , then G j A × B × = (cid:0) G j A × B ( Z ) × (cid:1) ∩ (cid:0) G j A × B ( Z ) × (cid:1) . Proof.
The proofs of the first two assertions are similarto that of Lemma 4.1 (i). So we only outline the proof ofthe second assertion. Clearly, H k A ( y k ) × ⊂ (cid:0) H k A ( y k , Z ) × (cid:1) ∩ (cid:0) H k A ( y k , Z ) × (cid:1) . For an element a ∈ (cid:0) H k A ( y k , Z ) × (cid:1) ∩ (cid:0) H k A ( y k , Z ) × (cid:1) , thereexist a ∈ A ( y k , Z ) × and a ∈ A ( y k , Z ) × s.t. a ≡ a mod H k and a ≡ a mod H k . Using a proper evaluation, one sees that a is in H k A ( y k ) × .We present a detailed proof of the third assertion due tothe presence of both A and B , though the idea goes along thesame line as before. It suffices to show that the intersectionof G j A × B ( Z ) × and G j A × B ( Z ) × is a subset of G j A × B × .Assume that a is in the intersection. Then a ≡ a b mod G j and a ≡ a b mod G j (16)for some a , a in A × , b in B ( Z ) × , and b in B ( Z ) × .Let Z ′ = Z \ B , and c be a proper evaluation of a / ( a b )w.r.t. Z ′ . Then c b is a proper evaluation of a b / ( a b )w.r.t. Z ′ , as b is free of Z ′ . So c b is in G j by Remark 4.3.Since c is in A × B × , b is in G j A × B × , and so is a .The next lemma says that some compatible rational func-tions belong to a common coset. Lemma
Let v , . . . , v m , w , . . . , w n ∈ F ( t , x , y ) × . As-sume that the compatibility conditions in (4) and (5) hold.(i) If v j is in G j F ( t , y ) × F ( x , y ) × for all j with ≤ j ≤ m ,then there exists f ∈ F ( t , x , y ) s.t. each v j is in thecoset ℓσ j ( f ) F ( t , y ) × F ( x , y ) × .(ii) Let E be a subfield of F ( t , x ) . If w k ∈ H k E ( y ) × forall k with ≤ k ≤ n , then there exists f ∈ F ( t , x , y ) s.t. each w k is in the coset ℓτ k ( f ) E ( y ) × . Proof.
We are going to show the second assertion. Thefirst one can be proved in the same fashion.The second assertion clearly holds when n = 1. Assumethat n > n −
1. Then there ex-ist g ∈ F ( t , x , y ) and b , . . . , b n − ∈ E ( y ) s.t. w k = ℓτ k ( g ) b k for all k with 1 ≤ k ≤ n −
1. Assume w n = ℓτ n ( g ) a for some a ∈ F ( t , x , y ) . (17)Then the compatibility conditions in (5) imply that thefirst-order system { τ k ( z ) = ℓτ n ( b k ) z | k = 1 , . . . , n − } hasa solution a in F ( t , x , y ) × . It follows from the hypothe-sis b k ∈ E ( y ) for all k with 1 ≤ k ≤ n − a ′ in E ( y ) × . Thus, a = c a ′ for some constant c w.r.t. τ , . . . , τ n − . Consequently, c be-longs to F ( t , x , y n ). On one hand, (17) leads to w n = ℓτ n ( g ) ca ′ . (18) On the other hand, w n ∈ H n E ( y ) × implies c = ℓτ n ( s ) r forsome r in E ( y ) and s in F ( t , x , y ). Let Z ′ = { y , . . . , y n − } ,and let s ′ and r ′ be two proper evaluations of s and r w.r.t. Z ′ at a point in F n − , respectively. Then c = ℓτ n ( s ′ ) r ′ since c is free of Z ′ . By (18), w n = ℓτ n ( s ′ g ) r ′ a ′ . Set f = s ′ g and b n = r ′ a ′ . Then w k = ℓτ k ( f ) b k for all k with 1 ≤ k ≤ n ,as s ′ is a constant w.r.t. τ , . . . , τ n − .
5. A STRUCTURE THEOREM
In this section, we extend Lemmas 3.1, 3.2 and 3.3, andthen combine these results to a structure theorem on ∆-compatible rational functions.The first proposition extends Lemma 3.1.
Proposition
Let u , . . . , u l , v , . . . , v m be ratio-nal functions in F ( t , x , y ) with v · · · v m = 0 . If the com-patibility conditions in (3) , (4) and (6) hold, then thereexist f in F ( t , x , y ) , a , . . . , a m , b , . . . , b l in F ( t , y ) ,and c , . . . , c m in F ( x , y ) s.t., for all i with ≤ i ≤ l , u i = ℓδ i ( f ) + ℓδ i ( a ) x + · · · + ℓδ i ( a m ) x m + b i , and, for all j with ≤ j ≤ m , v j = ℓσ j ( f ) a j c j . Moreover, the sequence b , . . . , b l , c , . . . , c m is compatiblew.r.t. { δ , . . . , δ l , σ , . . . , σ m } . Proof.
First, we consider the case in which l = 1 and m arbitrary. The proposition holds when m = 1 by Lemma 3.1.Assume that m > m . Applying the induction hypothesis to t , x , . . . , x m − and to t , x , . . . , x m , respectively, we see thatboth u ∈ N ,m and u ∈ N , . Since m > u ∈ N byLemma 4.1 (ii). Hence, u = ℓδ ( f ) + ℓδ ( a ) x + · · · + ℓδ ( a m ) x m + b for some f ∈ F ( t , x , y ) and a , . . . , a m , b ∈ F ( t , y ). As-sume that v j = ℓσ j ( f ) a j c j . Then c , . . . , c m are in F ( x , y )by the compatibility conditions in (6) (see Example 3.2).The proposition holds for l = 1 and m arbitrary.Second, we show that the proposition holds for all l and m by induction on l . It holds if l = 1 by the preceding para-graph. Assume that l > l . Applying the induction hypothesisto t , . . . , t l − , x and to t , . . . , t l , x , respectively, we have v j ∈ (cid:0) G j A × B ( Z ) × (cid:1) ∩ (cid:0) G j A × B ( Z ) × (cid:1) , where A = F ( t , y ), B = F ( x , y ), Z = { t l } , and Z = { t } . We seethat v j ∈ G j A × B × by Lemma 4.2 (iii). So v j ∈ ℓσ j ( f ) A × B × for some f in F ( t , x , y ) by Lemma 4.3 (i). Thus, v j = ℓσ j ( f ) a j c j , where a j ∈ A , c j ∈ B and j = 1 , . . . , m . Assume that, forall i with 1 ≤ i ≤ l , u i = ℓδ i ( f ) + P mj =1 ℓδ i ( a j ) x j + b i . All the b i ’s belong to F ( t , y ) by the compatibility conditionsin (6) (see Example 3.2). The sequence b , . . . , b l , c , . . . , c m is compatible because of (3), (4) and (6).The second proposition extends Lemma 3.2. Proposition
Let u , . . . , u l , w , . . . , w n be ratio-nal functions in F ( t , x , y ) with w · · · w n = 0 . Assume thatthe compatibility conditions (3) , (5) and (7) hold. Thenhere exist f in F ( t , x , y ) , a , . . . , a l in F ( t , x ) , and b , . . . , b n in F ( x , y ) s.t. u i = ℓδ i ( f ) + a i and w k = ℓτ k ( f ) b k for all i with ≤ i ≤ l and k with ≤ k ≤ n . Moreover,the sequence a , . . . , a l , b , . . . , b n is compatible w.r.t. theset { δ , . . . , δ l , τ , . . . , τ n } . The proof of this proposition goes along the same line as inthat of Proposition 5.1.The last proposition extends Lemma 3.3.
Proposition
Let v , . . . , v m , w , . . . , w n be rationalfunctions in F ( t , x , y ) × . Assume that the compatibility con-ditions in (4) , (5) and (8) hold. Then there exist a rationalfunction f in F ( t , x , y ) , a , . . . , a m in F ( t , x ) , and b , . . . , b n in F ( t , y ) s.t., for all j with ≤ j ≤ m and k with ≤ k ≤ n , v j = ℓσ j ( f ) a j and w k = ℓτ k ( f ) b k . Furthermore, the sequence a , . . . , a m , b , . . . , b n is compat-ible w.r.t. { σ , . . . , σ m , τ , . . . , τ n } . Proof.
First, we consider the case, in which m = 1 and n arbitrary. We proceed by induction on n . The propositionholds when n = 1 by Lemma 3.3. Assume that n>
1, and theproposition holds for the values lower than n . Applying theinduction hypothesis to x , y , . . . , y n − and to x , y , . . . , y n ,respectively, we get v ∈ G F ( t , x , y n ) × ∩ G F ( t , x , y ) × .Setting A = F ( t ), Z = { y n } and Z = { y } in Lemma 4.2 (i),we see that v ∈ G F ( t , x ) × , which, together with thedefinition of G F ( t , x ) × , there exist f in F ( t , x , y ) and a in F ( t , x ) s.t. v = ℓσ ( f ) a . Assume that w k = ℓτ k ( f ) b k for some b k in F ( t , x , y ) and for all k with 1 ≤ k ≤ n .By (8), σ ( b k )= b k , i.e., b k ∈ F ( t , y ) (see Example 3.4). Theproposition holds for m = 1 and n arbitrary.Second, assume that m > m and arbitrary n . Applying this in-duction hypothesis to x , . . . , x m − , y and to x , . . . , x m , y ,respectively, we have w k ∈ (cid:0) H k A ( y k , Z ) × (cid:1) ∩ (cid:0) H k A ( y k , Z ) × (cid:1) , where A = F ( t , y , . . ., y k − , y k +1 , . . ., y n ), Z and Z are equalto { x m } and { x } , respectively. Thus, w k ∈ H k A ( y k ) × by Lemma 4.2 (ii), and w k ∈ ℓτ k ( f ) A ( y k ) × for some f in F ( t , x , y ) by Lemma 4.3 (ii). Let w k = ℓτ k ( f ) b k , where b k is in A ( y k ) × , and k = 1 , . . . , n . Let a j = v j /ℓσ j ( f ) for all j with 1 ≤ j ≤ m . Then τ k ( a j ) = a j for all k with 1 ≤ k ≤ n and j with 1 ≤ j ≤ m by the compatibility conditions in (8)(see Example 3.4). Hence, all the a j ’s are in F ( t , x ). Thesequence a , . . . , a m , b , . . . , b n is compatible because of (4),(5) and (8).Now, we present a theorem describing the structure ofcompatible rational functions. Theorem
Let u , . . . , u l , v , . . . , v m , w , . . . , w n (19) be a sequence of rational functions in F ( t , x , y ) . If the se-quence is ∆ -compatible, then there exist f in F ( t , x , y ) , α , . . . , α m , β , . . . , β l in F ( t ) , λ , . . . , λ m in F ( x ) , and µ , . . . , µ n in F ( y ) s.t., for all i with ≤ i ≤ l , u i = ℓδ i ( f ) + ℓδ i ( α ) x + · · · + ℓδ i ( α m ) x m + β i , (20) for all j with ≤ j ≤ m , and, for all k with ≤ k ≤ n , v j = ℓσ j ( f ) α j λ j and w k = ℓτ k ( f ) µ k . (21) Moreover, the sequence β , . . . , β l , λ , . . . , λ m , µ , . . . , µ n is ∆ -compatible. Proof.
By Propositions 5.2 and 5.3, w k = ℓτ k ( g ′ ) a ′ k = ℓτ k (˜ g ) ˜ a k for some g ′ , ˜ g ∈ F ( t , x , y ), a ′ k ∈ F ( x , y ), and ˜ a k ∈ F ( t , y )with 1 ≤ k ≤ n . Set Z = { t , . . . , t l } , Z = { x , . . . , x m } ,and A = F ( y , . . . , y k − , y k +1 , . . . , y n ) in Lemma 4.2 (ii).Then the lemma implies that there exist µ k in F ( y ) and g k in F ( t , x , y ) s.t. w k = ℓτ k ( g k ) µ k . Setting E = F in the sec-ond assertion of Lemma 4.3, we may further assume that allthe g k ’s are equal to a rational function, say g . Let u i = ℓδ i ( g ) + r i (1 ≤ i ≤ l ) and v j = ℓσ j ( g ) s j (1 ≤ j ≤ m ) . Then the compatibility conditions in (7) imply that the r i ’sare in F ( t , x ) (see Example 3.3). Similarly, those conditionsin (8) imply that the s j ’s are in F ( t , x ) (see Example 3.4).Furthermore, r , . . . , r l , s , . . . , s m are compatible w.r.t. theset { δ , . . . , δ l , σ , . . . , σ m } . By Proposition 5.1, we get r i = ℓδ i ( b ) + ℓδ i ( α ) x + · · · + ℓδ i ( α m ) x m + β i , and s j = ℓσ j ( b ) α j λ j for some b in F ( t , x ), α j , β i in F ( t ), λ j in F ( x ), 1 ≤ i ≤ l , and 1 ≤ j ≤ m . Note that b belongsto F ( t , x ). Setting f = gb , we get the desired form for u i ’s, v j ’s and w k ’s. The compatibility of the sequence β , . . . , β l , λ , . . . , λ m , µ , . . . , µ n follows from that of u , . . . , u l , v ,. . . , v m , w , . . . , w n .With the notation introduced in Theorem 5.4, we say thatthe sequence: f, α , . . . , α m , β , . . . , β l , λ , . . . , λ m , µ , . . . , µ n (22)is a representation of ∆-compatible rational functions givenin (19) if the equalities in (20) and (21) hold.A rational function F ( t , x , y ) is said to be nonsplit w.r.t. t if its denominator and numerator have no irreducible fac-tors in F [ t ]. Similarly, we define the notion of nonsplit-ness w.r.t. x or y . Let ≺ be a fixed monomial orderingon F [ t , x , y ]. A nonzero rational function in F ( t , x , y ) is saidto be monic w.r.t. ≺ if its denominator and numerator areboth monic w.r.t. ≺ . A representation (22) of ∆-compatiblerational functions in (19) is said to be standard w.r.t. ≺ if(i) f is nonsplit w.r.t. t , x , and y , that is, the nontrivialirreducible factors of den( f )num( f ) are neither in F [ t ],nor in F [ x ], nor in F [ y ];(ii) both f and α j are monic w.r.t. ≺ , j = 1 , , . . . , m .Assume that the sequence (22) is a representation of (19).Factor f = f f f f , where f is monic and nonsplit w.r.t. t , x and y , f is in F ( t ) , f in F ( x ) , and f in F ( y ). Set α j = c j α ′ j ,where c j ∈ F , and α ′ j is monic. Then f , α ′ , . . . , α ′ m , β + ℓδ ( f ) , . . . , β l + ℓδ l ( f ) ,ℓσ ( f ) c λ , . . . , ℓσ m ( f ) c m λ m , ℓτ ( f ) µ , . . . , ℓτ n ( f ) µ n is also a representation of (19). This proves the existenceof standard representations. Its uniqueness follows from theuniqueness of factorization of rational functions. Corollary A ∆ -compatible sequence has a uniquestandard representation w.r.t. a given monomial ordering. . ALGORITHMS AND APPLICATIONS In this section, we discuss how to compute a representa-tion of compatible rational functions, and present two ap-plications in analyzing H -solutions. Let us fix a monomialordering on F [ t , x , y ] for standard representations.Let the sequence given in (19) be ∆-compatible. We com-pute a representation of the sequence in the form of (22).First, we compute µ ( y ) , . . . , µ n ( y ) in the sequence (22).By gcd-computation, we write w k = a k b k , where a k is non-split w.r.t. y , b k is in F ( y ), and k = 1, . . . , n . By Theo-rem 5.4, w k = ℓτ k ( f ) µ k , where f is nonsplit w.r.t. y and µ k is in F ( y ). Thus, b k = c k µ k for some c k ∈ F × .To determine c k , write a k = ℓτ k ( g k ) r k , where g k and r k are in F ( t , x , y ) with r k being τ k -reduced. By the two ex-pressions of w k , c k r k = ℓτ k ( f/g k ). Since a k is nonsplitw.r.t. y and r k is τ k -reduced, g k can be chosen to be non-split w.r.t. y , and so is f/g k . Thus, f/g k is free of y k , be-cause c k r k is τ k -reduced. Accordingly, c k r k =1 and µ k = r k b k .As a byproduct, we obtain g k with ℓτ k ( f ) = ℓτ k ( g k ).Second, we compute α , . . . , α m and λ , . . . , λ m . Assumethat j is an integer with 1 ≤ j ≤ m . By gcd-computation,we write v j = s j a j b j , where s j is nonsplit w.r.t. t and x , a j is in F ( t ), and b j in F ( x ). Moreover, set a j to be monic. ByTheorem 5.4, v j = ℓσ j ( f ) α j λ j , where f is nonsplit w.r.t. t and x , α j is a monic element in F ( t ), and λ j is in F ( x ).Hence, a j = α j and b j = c j λ j for some c j ∈ F × . As inthe preceding paragraph, we write s j = ℓσ j ( g ′ j ) r j with r j being σ j -reduced. Then c j r j = ℓσ j ( f/g ′ j ). Since c j r j is σ j -reduced, c j r j = 1. Hence, λ j = r j b j . As a byproduct, wefind g ′ j with ℓσ j ( f ) = ℓσ j ( g ′ j ).Third, we compute f . Note that f is a nonzero rational so-lution of the system { σ j ( z ) = ℓσ j ( g ′ j ) z, τ k ( z ) = ℓτ k ( g k ) z } , where 1 ≤ j ≤ m , 1 ≤ k ≤ n , and g ′ j , g k are obtained in thefirst two steps. So f can be computed by several methods,e.g., the method in the proof of [14, Proposition 3].At last, we set β i = u i − ℓδ i ( f ) − P mj =1 ℓδ i ( α j ) x j , for all i with 1 ≤ i ≤ l . Using v j = ℓσ j ( f ) α j λ j and w k = ℓτ k ( f ) µ k and the compatibility conditions in (6) and (7), we see thatall the β i ’s are in F ( t ), as required. Example
Consider the case l = m = n = 1 . Let u , v and w be compatible rational functions, where u = (4 t + 2 x + y )( t + 1) + ( t + x + 1)( t + x )(2 t + y )( t + 1)( t + x )(2 t + y ) ,v = 2(2 x + 3)( x + 1)( t + 1)( t + x + 1)(5 x + y )(5 x + y + 5)( t + x ) ,w = (5 x + y )(2 t + q y )(1 + qy )(5 x + qy )(2 t + y ) . A representation of u, v, w is of the form (cid:18) (2 t + y )( t + x )5 x + y , t + 1 , , x + 3)( x + 1) , qy + 1 (cid:19) . From now on, we assume that our ground field F is al-gebraically closed. In general, ∆-extensions of F ( t , x , y )are rings. We recall that an H -solution over F ( t , x , y ) isa nonzero solution of system (1) and, given a finite num-ber of H -solutions, there is a ∆-extension of F ( t , x , y ) con-taining these H -solutions and their inverses. The ring ofconstants of this ∆-extension is equal to F by Theorem 2in [5]. We will only encounter finitely many pairwise dis-similar H -solutions. Hence, it makes sense to multiply and invert them in some ∆-extension, which will not be specifiedexplicitly if no ambiguity arises. All H -solutions we considerwill be over F ( t , x , y ). Denote by s and s the sequencesconsisting of s s H -solution is said to be a symbolic power if its certifi-cates are of the form m X j =1 x j ℓδ ( α j ) , . . . , m X j =1 x j ℓδ l ( α j ) , α , . . . , α m , n , (23)where α , . . . , α m are monic elements in F ( t ) × . It is easy toverify that such a sequence is ∆-compatible. Such a sym-bolic power is denoted α x · · · α x m m . The monicity of the α i ’sexcludes the case, in which some α i is a constant differentfrom one. By an E -solution, we mean an H -solution whosecertificates are of the form β , . . . , β l , m + n , where β , . . . , β l are in F ( t ). An E -solution is a hyperexponential functionw.r.t. the derivations, and a constant w.r.t. other operators.By a G -solution, we mean an H -solution whose certificatesare of the form l , λ , . . . , λ m , n , where λ , . . . , λ m arein F ( x ) × . A G -solution is a hypergeometric term w.r.t. theshift operators, and a constant w.r.t. other operators. Simi-larly, by a Q -solution, we mean an H -solution whose certifi-cates are of the form l , m , µ , . . . , µ n , where µ , . . . , µ n are in F ( y ) × . A Q -solution is a q -hypergeometric term w.r.t.the q -shift operators, and a constant w.r.t. other operators.The next proposition describes a multiplicative decompo-sition of H -solutions. Proposition An H -solution is a product of an el-ement in F × , a rational function in F ( t , x , y ) , a symbolicpower, an E -solution, a G -solution, and a Q -solution. Proof.
Let h be an H -solution. Then its certificates arecompatible. By Theorem 5.4, the certificates have a stan-dard representation f, α , . . . , α m , β , . . . , β l , λ , . . . , λ m ,µ , . . . , µ n . Moreover, the following three sequences: β , . . . , β l , m + n ; l , λ , . . . , λ m , n ; l , m , µ , . . . , µ n are ∆-compatible, respectively. Hence, there exist an E -solution E , a G -solution G , and a Q -solution Q s.t. theircertificates are given in the above three sequences, respec-tively. It follows from Theorem 5.4 that h and the prod-uct fα x · · · α x m m EGQ have the same certificates. So theydiffer by a multiplicative constant, which is in F .The H -solution in Example 6.1 can be decomposed as(2 t + y )( t + x )5 x + y ( t + 1) x exp( t ) (2 x + 1)! Γ q (1 + qy ) , where Γ q (1 + qy ) is a Q -solution with certificates 0 , , qy .The next proposition characterizes rational H -solutionsvia their standard representations. Proposition
Let P be a symbolic power, E an E -solution, G a G -solution and Q a Q -solution. Then PEGQ is in F ( t , x , y ) iff P ∈ F , E ∈ F ( t ) , G ∈ F ( x ) and Q ∈ F ( y ) . Proof. ( ⇐ ) Clear.( ⇒ ) Assume that f is rational and equal to PEGQ , where P , E , G , Q are a symbolic power, an E -, a G -, and a Q -solution, respectively. Suppose that the certificates of P are given in (23). Applying ℓδ i to f , i = 1 , . . . , l , we see that ℓδ i ( f ) = m X j =1 ℓδ i ( α j ) x j + ℓδ i ( E ) . omparing the polynomial parts of the left and right hand-sides of the above equality w.r.t. x j , we see that ℓδ i ( α j ) = 0by Remark 2.2 and ℓδ i ( E ) ∈ F ( t ) for all i and j . Hence, allthe α j ’s are in F , and, consequently, all the α j ’s are equalto one as they are monic. Hence, P is in F . Moreover, ℓδ i ( f ) = ℓδ i ( E ) for all i with 1 ≤ i ≤ l. Let g be a proper evaluation of f w.r.t. x and y . Then ℓδ i ( g ) = ℓδ i ( E ) for all i with 1 ≤ i ≤ l, since ℓδ i ( E ) is in F ( t ). Hence, ℓδ i ( E /g ) = 0 , ℓσ j ( E /g ) = 1 , and ℓτ k ( E /g ) = 1 , where 1 ≤ i ≤ l , 1 ≤ j ≤ m , and 1 ≤ k ≤ n . Weconclude that E = cg for some c ∈ F . So E is in F ( t ).Applying ℓσ j and ℓτ k to f leads to ℓσ j ( f ) = ℓσ j ( G ),and ℓτ k ( f ) = ℓτ k ( Q ), respectively. One can show that G is in F ( x ) and Q is in F ( y ) by similar arguments.Now, we consider how to determine whether a finite numberof H -solutions are algebraically dependent over F ( t , x , y ).Let h , · · · , h s be H -solutions. By Proposition 6.1, h i ≡ P i E i G i Q i mod F ( t , x , y ) × , i = 1 , . . . , s, (24)where P i , E i , G i , Q i are a symbolic power, an E -solution, a G -solution, and a Q -solution, respectively. Corollary
Let h , . . . , h s be H -solutions s.t. all thecongruences in (24) hold. Then they are algebraically depen-dent over F ( t , x , y ) iff there exist integers ω , . . . ω s , not allzero, s.t. P ω · · · P ω s s is in F , E ω · · · E ω s s in F ( t ) , G ω · · · G ω s s in F ( x ) and Q ω · · · Q ω s s in F ( y ) . Proof.
It follows from [15, Corollary 4.2] that h , · · · , h s are algebraically dependent over F ( t , x , y ) iff there exist in-tegers ω , . . . ω s , not all zero, s.t. h ω · · · h ω s s is in F ( t , x , y ).The corollary follows from (24) and Proposition 6.2.By the above corollary, one may determine the algebraicdependence of h , . . . , h s using the decompositions in Propo-sition 6.1. By gcd-computation, one can find all nonzero in-teger vectors ( ω , . . . ω s ) s.t. P ω · · · P ω s s is in F . Accordingto [19], one can find all nonzero integer vectors ( ω , . . . ω s )s.t. E ω · · · E ω s s ∈ F ( t ) by seeking rational number solutionsof a linear homogeneous system over F . Computing allnonzero integer vectors ( ω , . . . ω s ) s.t. G ω · · · G ω s s ∈ F ( x )reduces to the following subproblem: given c , . . . , c s ∈ F × ,compute integers ω , . . . ω s , not all zero, with c ω · · · c ω s s = 1(see [19]). Algorithms for tackling this subproblem and re-lated discussions are contained in [13, § ω , . . . ω s , not all zero, s.t. Q ω · · · Q ω s s belongs to F ( y ).The reader is referred to [6] for an extended version ofthis paper, which contains a short proof of Fact 3.1 and aproof of Proposition 5.2. A Maple implementation is beingwritten for decomposing H -solutions. We shall apply thestructure theorem to study the existence of telescopers inthe mixed cases in which any two of differential, shift and q -shift operators appear. Acknowledgments.
The authors thank Fr´ed´eric Chyzak,Bruno Salvy, Michael Singer and anonymous referees forhelpful discussions and suggestions.
7. REFERENCES [1] S. A. Abramov. When does Zeilberger’s algorithm succeed?
Adv. in Appl. Math. H -systems. J. Symbolic Comput. , 43(5):377–394,2008.[4] S. A. Abramov and M. Petkovˇsek. On the structure ofmultivariate hypergeometric terms.
Adv. in Appl. Math. ,29(3):386–411, 2002.[5] M. Bronstein, Z. Li, and M. Wu. Picard–Vessiot extensionsfor linear functional systems. In
Proc. of ISSAC ’05 )[8] W. Y. C. Chen, Q.-H. Hou, and Y.-P. Mu. Applicability ofthe q -analogue of Zeilberger’s algorithm. J. SymbolicComput. , 39(2):155–170, 2005.[9] C. Christopher. Liouvillian first integrals of second orderpolynomial differential equations.
Electron. J. DifferentialEquations , 49: 1-7 (electronic), 1999.[10] R. Feng, M. F. Singer, and M. Wu. An algorithm tocompute Liouvillian solutions of prime order lineardifference-differential equations.
J. Symbolic Comput. ,45(3):306–323, 2010.[11] I. Gel’fand, M. Graev, and V. Retakh. Generalhypergeometric systems of equations and series ofhypergeometric type.
Uspekhi Mat. Nauk (Russian), Engl.transl. in Russia Math Surveys , 47(4):3–82, 1992.[12] C. Hardouin and M. F. Singer. Differential Galois theory oflinear difference equations.
Math. Ann. , 342(2):333–377,2008.[13] M. Kauers.
Algorithms for Nonlinear Higher OrderDifference Equations . PhD thesis, RISC-Linz, Linz,Austria, 2005.[14] G. Labahn and Z. Li. Hyperexponential solutions offinite-rank ideals in orthogonal Ore rings. In
Proc. ofISSAC’04 , 213–220. ACM, New York, 2004.[15] Z. Li, M. Wu, and D. Zheng. Testing linear dependence ofhyperexponential elements.
ACM Commun. Comput.Algebra , 41(1):3–11, 2007.[16] O. Ore. Sur la forme des fonctions hyperg´eom´etriques deplusieurs variables.
J. Math. Pures Appl. , 9(4):311–326,1930.[17] G. H. Payne.
Multivariate Hypergeometric Terms . PhDthesis, Penn. State Univ., Pennsylvania, USA, 1997.[18] M. Sato. Theory of prehomogeneous vector spaces(algebraic part)– the English translation of Sato’s lecturefrom Shintani’s note.
Nagoya Math. J. , 120:1–34, 1990.[19] M.F. Singer. A note on solutions of first-order linearfunctional equations. Manuscript for discussions at theSecond NCSU-China Symbolic Computation CollaborationWorkshop, Hangzhou, March, 2007.[20] M. van der Put and M.F. Singer.
Galois Theory ofDifference Equations , volume 1666 of
Lecture Notes inMathematics . Springer-Verlag, Berlin, 1997.[21] H. Zoladek. The extended monodromy group andLiouvillian first integrals.