Solving linear difference equations with coefficients in rings with idempotent representations
aa r X i v : . [ c s . S C ] F e b RISC-Linz Report Series No. 21-04
Solving linear difference equations with coefficients in ringswith idempotent representations ∗ Jakob Ablinger
[email protected] Kepler University Linz, RISCLinz, Austria
Carsten Schneider
[email protected] Kepler University Linz, RISCLinz, Austria
ABSTRACT
We introduce a general reduction strategy that enables one to searchfor solutions of parameterized linear difference equations in differ-ence rings. Here we assume that the ring itself can be decomposedby a direct sum of integral domains (using idempotent elements)that enjoys certain technical features and that the coefficients ofthe difference equation are not degenerated. Using this mechanismwe can reduce the problem to find solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains.Utilizing existing solvers in this integral domain setting, we ob-tain a general solver where the components of the linear differenceequations and the solutions can be taken from difference rings thatare built e.g., by 𝑅 ΠΣ -extensions over ΠΣ -fields. This class of dif-ference rings contains, e.g., nested sums and products, productsover roots of unity and nested sums defined over such objects. KEYWORDS linear difference equations, differnence rings, idempotent elements
In the following we denote by ( E , 𝜎 ) a difference ring (resp. field) ,this means that E is a ring (resp. field) E equipped with a ring (resp.field) automorphism 𝜎 : E → E . We call ( E , 𝜎 ) computable if thebasic operations of E and 𝜎 are computable. We define the ring ofconstants of ( E , 𝜎 ) by K = const 𝜎 E = { 𝑐 ∈ E | 𝜎 ( 𝑐 ) = 𝑐 } . Byconstruction K will be a field, called the constant field of ( E , 𝜎 ) .Given such a difference ring ( E , 𝜎 ) with a constant field K , weare interested in the following problem: Given 𝒂 = ( 𝑎 , . . . , 𝑎 𝑚 ) ∈ E 𝑚 + and 𝒇 = ( 𝑓 , . . . , 𝑓 𝑑 ) ∈ E 𝑑 , find (if this is possible) a finiterepresentation of all solutions 𝑔 ∈ E and 𝑐 , . . . , 𝑐 𝑑 ∈ K of the parameterized linear difference equation (in short PLDE) 𝑎 𝑔 + 𝑎 𝜎 ( 𝑔 ) + · · · + 𝑎 𝑚 𝜎 𝑚 ( 𝑔 ) = 𝑐 𝑓 + · · · + 𝑐 𝑑 𝑓 𝑑 (1)with coefficients 𝒂 and parameters 𝒇 . The solution set is defined by 𝑉 = 𝑉 ( 𝒂 , 𝒇 , E ) = {( 𝑐 , . . . , 𝑐 𝑑 , 𝑔 ) ∈ K 𝑑 × E | (1) holds } which forms a K -subspace of K 𝑑 × E . We say that we can computeall solutions in ( E , 𝜎 ) of an explicitly given (1) if 𝑉 is a finite dimen-sional vector space and one can compute a basis of 𝑉 . In particular,if E is an integral domain and 𝑎 𝑎 𝑚 ≠
0, we have dim ( 𝑉 ) ≤ 𝑚 + 𝑛 by [7, Thm. XII (page 272)]. In this case we say that we can solve(in general) parameterized linear difference equations in ( A , 𝜎 ) if onecan compute a basis of 𝑉 ( 𝒂 , 𝒇 , E ) for any ≠ 𝒂 ∈ E 𝑚 + and 𝒇 ∈ E 𝑑 .The problem to solve PLDEs (so far only in a field or integraldomain E ) plays a central rule in symbolic summation and var-ious algorithms. It covers as special cases the telescoping prob-lem ( 𝒂 = ( , − ) , 𝒇 ∈ E ) for, e.g., hypergeometric products [9], ∗ Supported by the Austrian Science Foundation (FWF) grant SFB F50 (F5009-N15). the creative telescoping problem ( 𝒂 = ( , − ) with appropriatelychosen 𝒇 ∈ E 𝑑 ) for, e.g., hypergeometric products [28], or recur-rence solving ( 𝑑 =
1) for, e.g., rational or hypergeometric solu-tions [2, 16, 17]. The parameterized version is used also in holo-nomic summation [6] and generalizations of it [5]. Further detailscan found, e.g., in [26].In particular, Karr’s pioneering summation algorithm [12] estab-lished a highly general solver for first-order PLDEs in the settingof his ΠΣ -field extensions (Def. 19). In this way, the coefficients 𝑎 𝑖 , parameters 𝑓 𝑖 and the solutions 𝑔 can be given in a ΠΣ -field ( E , 𝜎 ) that is built formally by indefinite nested sums and prod-ucts. Only recently, his general first-order solver has been pushedforward in [3] to the higher-order case (including also a solver tofind all hypergeometric solutions over E ), that covers most of thesummation algorithms mentioned above as special cases.In this article we aim at further generalizations allowing in addi-tion difference rings that are built by basic 𝑅 ΠΣ -ring extensions [23,24] (Def. 15) where also products over roots of unity like (− ) 𝑛 canarise. Based on the observation that such rings can be decomposedby a direct sum of integral domains using idempotent elements(which is one of the key tools in the Galois theory of differenceequations [10, 27]), we will develop in Section 2 a general strategyto solve non-degenerated PLDEs in idempotent difference rings(Def. 1). Inspired by [15, 18] we separate the potential solutions intheir different components (Thm. 9) and try to combine them ac-cordingly to the full solution (Thm. 14). Utilizing this machinery,we will invoke in Section 3 the general ΠΣ -field solver [3] (and vari-ants of it) implemented within the summation package Sigma [21]to derive various new algorithms (see Theorems 25 and 31) in orderto solve non-degenerated PLDEs in basic 𝑅 ΠΣ -rings defined over ΠΣ -field-extensions. As a special case, the ground field can be, e.g.,the mixed multibasic difference field [4] introduced in Remark 26.After a concrete example in Section 4 we conclude with Section 5. akob Ablinger and Carsten Schneider It will be convenient to denote by 𝑠 mod 𝜆 with 𝑠 ∈ Z the uniquevalue 𝑙 ∈ { , . . . , 𝜆 − } with 𝜆 | 𝑠 − 𝑙 . Definition 1.
Let ( E , 𝜎 ) be a difference ring and let 𝑒 𝑠 ∈ E with0 ≤ 𝑠 < 𝜆 be elements such that • they are idempotent (i.e., 𝑒 𝑠 = 𝑒 𝑠 ), • pairwise orthogonal (i.e., 𝑒 𝑠 𝑒 𝑡 = 𝑠 ≠ 𝑡 ), • and 𝜎 ( 𝑒 𝑠 ) = 𝑒 𝑠 + 𝜆 . If ( E , 𝜎 ) can be decomposed in the form E = 𝑒 E ⊕ 𝑒 E ⊕ · · · ⊕ 𝑒 𝜆 − E (2)such that 𝑒 𝑖 E forms an computable integral domain, then ( E , 𝜎 ) iscalled an idempotent difference ring of order 𝜆 .Note that, if ( E , 𝜎 ) is an idempotent difference ring of order 𝜆 then ( 𝑒 𝑠 E , 𝜎 𝜆 ) is a difference ring and 𝜎 is a difference ring isomor-phism between ( 𝑒 𝑠 E , 𝜎 𝜆 ) and ( 𝑒 𝑠 + 𝜆 E , 𝜎 𝜆 ) . Lemma 2.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 andlet 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 ∈ E , then applying 𝜎 means that the component 𝑒 𝑠 𝑔 𝑠 is moved cyclically to ( 𝑒 𝑠 + 𝜆 E , 𝜎 𝜆 ) Proof.
Fix 𝑠 with 0 ≤ 𝑠 < 𝜆 , since 𝑔 𝑠 ∈ E we can write 𝑔 𝑠 = Í 𝜆 − 𝑖 = 𝑒 𝑖 ℎ 𝑖 for some ℎ 𝑖 ∈ E . Now applying 𝜎 to 𝑒 𝑠 𝑔 𝑠 gives: 𝜎 ( 𝑒 𝑠 𝑔 𝑠 ) = 𝜎 𝑒 𝑠 𝜆 − Õ 𝑖 = 𝑒 𝑖 ℎ 𝑖 ! = 𝜎 ( 𝑒 𝑠 ) 𝜆 − Õ 𝑖 = 𝜎 ( 𝑒 𝑖 ) 𝜎 ( ℎ 𝑖 ) = 𝑒 𝑠 + 𝜆 𝜆 − Õ 𝑖 = 𝑒 𝑖 + 𝜆 𝜎 ( ℎ 𝑖 ) = 𝑒 𝑠 + 𝜆 𝜎 ( ℎ 𝑠 ) . Since 𝜎 ( ℎ 𝑠 ) ∈ E we have that 𝜎 ( 𝑒 𝑠 𝑔 𝑠 ) ∈ 𝑒 𝑠 + 𝜆 E . (cid:3) For an idempotent difference ring ( E , 𝜎 ) of order 𝜆 , with idem-potent elements 𝑒 𝑠 ∈ E with 0 ≤ 𝑠 < 𝜆 the structure given byLemma 2 can be illustrated as follows: E = 𝑒 E 𝜎 $ $ ⊕ 𝑒 E 𝜎 ⊕ . . . 𝜎 $ $ ⊕ 𝑒 𝜆 − E 𝜎 $ $ ⊕ 𝑒 𝜆 − E 𝜎 k k . The following lemma is immediate.
Lemma 3.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 and let 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 ∈ E and 𝑗 ∈ N then 𝜎 𝑗 ( 𝑔 ) = 𝜆 − Õ 𝑠 = 𝑒 𝑠 + 𝑗 mod 𝜆 𝜎 𝑗 ( 𝑔 𝑠 ) = 𝜆 − Õ 𝑠 = 𝑒 𝑠 𝜎 𝑗 ( 𝑔 𝑠 − 𝑗 mod 𝜆 ) . (3) Definition 4.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 with idempotent elements 𝑒 𝑠 ∈ E with 0 ≤ 𝑠 < 𝜆 . Then 𝜋 : E → E with 𝜋 ( 𝑔 ) ↦→ 𝑔 where 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 is called a projection .In this article we will always consider the projection on the firstcomponent, however each projection to an arbitrary componentwould do the job. The following lemma summarizes several prop-erties of the projection. A difference isomorphism 𝜏 : A → A between two difference rings ( A 𝑖 , 𝜎 𝑖 ) with 𝑖 = , is a ring isomorphism with 𝜏 ( 𝜎 ( 𝑓 )) = 𝜎 ( 𝜏 ( 𝑓 )) for all 𝑓 ∈ A . Lemma 5.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 with idempotent elements 𝑒 𝑠 ∈ E with ≤ 𝑠 < 𝜆 and let 𝜋 : E → E be a projection. For 𝑔, ℎ ∈ E we have 𝜋 ( 𝑔 + ℎ ) = 𝜋 ( 𝑔 ) + 𝜋 ( ℎ ) and 𝜋 ( 𝑔 · ℎ ) = 𝜋 ( 𝑔 ) · 𝜋 ( ℎ ) . (4) In addition, for 𝑗 ∈ N and ≤ 𝑠 < 𝜆 we have 𝜋 ( 𝜎 𝑗 ( 𝑒 𝑠 )) = ( if 𝑠 + 𝑗 = ( mod 𝜆 ) if 𝑠 + 𝑗 ≠ ( mod 𝜆 ) , (5) and for 𝑗 ∈ N and 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 we have 𝜋 ( 𝑔 ) = 𝑒 𝑔 and 𝜋 ( 𝜎 𝑗 ( 𝑔 )) = 𝜎 𝑗 ( 𝑔 − 𝑗 mod 𝜆 ) (6) Proof.
Let 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 ∈ E and ℎ = Í 𝜆 − 𝑠 = 𝑒 𝑠 ℎ 𝑠 ∈ E then 𝑔 + ℎ = Í 𝜆 − 𝑠 = 𝑒 𝑠 ( 𝑔 𝑠 + ℎ 𝑠 ) ∈ E and hence 𝜋 ( 𝑔 + ℎ ) = 𝑔 + ℎ = 𝜋 ( 𝑔 ) + 𝜋 ( ℎ ) . Similarly, since 𝑔 · ℎ = Í 𝜆 − 𝑠 = 𝑒 𝑠 ( 𝑔 𝑠 · ℎ 𝑠 ) ∈ E we have 𝜋 ( 𝑔 · ℎ ) = 𝑔 · ℎ = 𝜋 ( 𝑔 ) · 𝜋 ( ℎ ) . For 𝑗 ∈ N , ≤ 𝑠 < 𝜆 we have that 𝜎 𝑗 ( 𝑒 𝑠 ) = 𝑒 𝑠 + 𝑗 mod 𝜆 , hence 𝜋 ( 𝜎 𝑗 ( 𝑒 𝑠 )) = 𝜋 ( 𝑒 𝑠 + 𝑗 mod 𝜆 ) which clearly evalu-ates to 1 if 𝑠 + 𝑗 = ( mod 𝜆 ) and to 0 if 𝑠 + 𝑗 ≠ ( mod 𝜆 ) . Finally,from Lemma 3 we know that 𝜎 𝑗 ( 𝑔 ) = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝜎 𝑗 ( 𝑔 𝑠 − 𝑗 mod 𝜆 ) , hence 𝜋 ( 𝜎 𝑗 ( 𝑔 )) = Í 𝜆 − 𝑠 = 𝜋 ( 𝑒 𝑠 ) 𝜋 ( 𝜎 𝑗 ( 𝑔 𝑠 − 𝑗 mod 𝜆 )) = 𝜋 ( 𝜎 𝑗 ( 𝑔 − 𝑗 mod 𝜆 )) . Since 𝜋 ( 𝜎 𝑗 ( 𝑔 − 𝑗 mod 𝜆 )) = 𝑒 𝜎 𝑗 ( 𝑔 − 𝑗 mod 𝜆 ) = 𝜎 𝑗 ( 𝑒 − 𝑗 mod 𝜆 𝑔 − 𝑗 mod 𝜆 ) wehave that 𝜎 𝑗 ( 𝑔 ) = 𝜎 𝑗 ( 𝑔 − 𝑗 mod 𝜆 ) . (cid:3) Definition 6.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 and let 𝜋 : E → E be a projection. For 𝒂 = ( 𝑎 , 𝑎 , . . . , 𝑎 𝑚 ) ∈ E 𝑚 + we define the ( 𝑚 + ) 𝜆 − 𝑚 × ( 𝑚 + ) 𝜆 shift projection matrixby 𝑀 𝜎,𝜋 ( 𝒂 ) : = © « 𝜋 ( 𝑝 ) 𝜋 ( 𝑝 ) · · · 𝜋 ( 𝑝𝑚 ) · · · 𝜋 ( 𝜎 ( 𝑝 )) · · · 𝜋 ( 𝜎 ( 𝑝𝑚 − )) 𝜋 ( 𝜎 ( 𝑝𝑚 )) · · · ... ... · · · 𝜋 ( 𝜎𝑘 ( 𝑝 )) · · · 𝜋 ( 𝜎𝑘 ( 𝑝𝑚 )) ª®®®®®®¬ , where 𝑘 : = ( 𝑚 + ) 𝜆 − 𝑚 − . Definition 7.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 and let 𝜋 : E → E be a projection. A vector 𝒂 = ( 𝑎 , 𝑎 , . . . , 𝑎 𝑚 ) ∈ E 𝑚 + is called non-degenerate if the shift projection matrix 𝑀 𝜎,𝜋 ( 𝒂 ) has full rank, i.e., the rows are linearly independent. Likewise, alinear difference operator Í 𝑚𝑖 = 𝑎 𝑖 𝜎 𝑖 ∈ E [ 𝜎 ] with 𝑎 𝑖 ∈ E is called non-degenerate if 𝒂 is non-degenerate .Note, that for instance a linear difference operator 𝐿 = Í 𝑚𝑖 = 𝑎 𝑖 𝜎 𝑖 ∈ E [ 𝜎 ] that is a multiple of an idempotent element 𝑒 𝑖 i.e., 𝑒 𝑖 | 𝑎 𝑖 forall 0 ≤ 𝑖 ≤ 𝑚 is not non-degenerate, since for such an operatorthe shift projection matrix would contain a zero row. Similarly, 𝐿 for which all coefficients vanish for a certain component is as welldegenerate, since for such an operator the shift projection matrixwould contain 𝑚 + non-degenerate . Lemma 8.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 and let 𝜋 : E → E be a projection. A linear difference operator 𝐿 : = Í 𝑚𝑖 = 𝑝 𝑖 𝜎 𝑖 ∈ E [ 𝜎 ] , with 𝑎 𝑖 ∈ E , is non-degenerate if either 𝑎 𝑚 or 𝑎 is a unit in E . olving linear difference equations with coefficients in rings with idempotent representations Given a non-degenerate linear difference operator, the followingtheorem shows, that it is possible to define non-zero linear differ-ence operators for each component. It is inspired by [15, 18].
Theorem 9.
Let ( E , 𝜎 ) be an idempotent difference ring of order 𝜆 with idempotent elements 𝑒 𝑠 ∈ E with ≤ 𝑠 < 𝜆 , let 𝜋 : E → E be a projection and let 𝒂 = ( 𝑎 , . . . , 𝑎 𝑚 ) ∈ E 𝑚 + with 𝑎 𝑚 ≠ benon-degenerated. Consider the linear difference equation 𝑚 Õ 𝑖 = 𝑎 𝑖 𝜎 𝑖 ( 𝑔 ) = 𝜑. (7) with 𝜑 ∈ E , which is satisfied by 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 ∈ E and let 𝑘 ∈ N with ≤ 𝑘 < 𝜆 . Then there exist 𝑏 𝑘,𝑖 ∈ 𝑒 𝑘 E , not all zero, and 𝜑 𝑘 ∈ 𝑒 𝑘 E such that 𝑚 Õ 𝑖 = 𝑏 𝑘,𝑖 ( 𝜎 𝜆 ) 𝑖 ( 𝑔 𝑘 ) = 𝜑 𝑘 . (8) If ( E , 𝜎 ) is computable, then the 𝑏 𝑘,𝑖 and 𝜑 𝑘 can be computed. Proof.
From (7) we can deduce for 𝑗 ∈ N that 𝜎 𝑗 𝑚 Õ 𝑖 = 𝑎 𝑖 𝜎 𝑖 ( 𝑔 ) ! = 𝜎 𝑗 ( 𝜑 ) (9)or equivalently 𝑚 Õ 𝑖 = 𝜎 𝑗 ( 𝑎 𝑖 ) (cid:16) 𝜎 𝑖 + 𝑗 ( 𝑒 ) 𝜎 𝑖 + 𝑗 ( 𝑔 ) + · · ·· · · + 𝜎 𝑖 + 𝑗 ( 𝑒 𝜆 − ) 𝜎 𝑖 + 𝑗 ( 𝑔 𝜆 − ) (cid:17) = 𝜎 𝑗 ( 𝜑 ) . Applying the projection 𝜋 and using Lemma 5 yields 𝑚 Õ 𝑖 = 𝜋 ( 𝜎 𝑗 ( 𝑎 𝑖 )) 𝜋 ( 𝜎 𝑖 + 𝑗 ( 𝑔 −( 𝑖 + 𝑗 ) mod 𝜆 )) = 𝜋 (cid:16) 𝜎 𝑗 ( 𝜑 ) (cid:17) , since for 1 ≤ 𝑙 < 𝜆 , 𝜋 ( 𝜎 𝑖 + 𝑗 ( 𝑒 𝑙 )) = ( 𝑙 = −( 𝑖 + 𝑗 ) ( mod 𝜆 ) 𝑙 ≠ −( 𝑖 + 𝑗 ) ( mod 𝜆 ) . Now, by Lemma 3 and Lemma 5 we find 𝑚 Õ 𝑖 = 𝜋 ( 𝜎 𝑗 ( 𝑎 𝑖 )) 𝜎 𝑖 + 𝑗 ( 𝑔 −( 𝑖 + 𝑗 ) mod 𝜆 ) = 𝜋 (cid:16) 𝜎 𝑗 ( 𝜑 ) (cid:17) . (10)Now, plugging in 𝑗 = , , , . . . , ( 𝑚 + ) 𝜆 − 𝑚 − 𝑀 𝜎,𝜋 ( 𝒂 ) · © « 𝜎 ( 𝑔 𝜆 ) 𝜎 ( 𝑔 − 𝜆 ) 𝜎 ( 𝑔 − 𝜆 ) ...𝜎 𝜈 ( 𝑔 − 𝑘 mod 𝜆 ) ª®®®®®®¬ = © « 𝜋 ( 𝜎 ( 𝜑 )) 𝜋 ( 𝜎 ( 𝜑 )) 𝜋 ( 𝜎 ( 𝜑 )) ...𝜋 ( 𝜎 𝜈 ( 𝜑 )) ª®®®®®®¬ , (11)where 𝜈 : = ( 𝑚 + ) 𝜆 − 𝑚 −
1. Since 𝒂 is non-degenerate and hence 𝑀 𝜎,𝜋 ( 𝒂 ) has full rank, we can solve this system in terms of 𝑚 vari-ables. Finally, we can plug this solution into (8). Since this leadsto a linear system of at most 𝑚 + 𝑚 + 𝑏 𝑘,𝑖 and 𝜑 𝑘 of (8). In particular, if E is computable, the 𝑏 𝑘,𝑖 and 𝜑 𝑘 can be computed. (cid:3) Remark . Let ( E , 𝜎 ) be a field extension of a difference ring ( A , 𝜎 ′ ) ,i.e., A is a subring of E and 𝜎 | A = 𝜎 ′ , and suppose that the 𝒂 ∈ A 𝑚 + and 𝜙 ∈ E . Then, since we plug solutions of the linear sys-tem (11) into (8), the right-hand sides in (8) have the form 𝜑 𝑘 = 𝑠 Õ 𝑙 = 𝑓 𝑙 𝜋 ( 𝜎 𝑙 ( 𝜑 )) with 𝑓 , . . . , 𝑓 𝑠 ∈ A for some 𝑠 ∈ N . Example 11.
Consider the idempotent difference ring ( Q ( 𝑥 )[ 𝑦 ] , 𝜎 ) with 𝜎 ( 𝑥 ) = 𝑥 + 𝜎 ( 𝑦 ) = − 𝑦 and the idempotent elements 𝑒 = − 𝑦 and 𝑒 = − 𝑦 . Let 𝒂 = ( 𝑥, 𝑥, , 𝑦 ) , then the shift projec-tion matrix 𝑀 𝜎,𝜋 ( 𝒂 ) yields © « 𝑥 𝑥 − + 𝑥 + 𝑥 + 𝑥 + 𝑥 − + 𝑥 + 𝑥 + 𝑥 + 𝑥 − ª®®®®®¬ , which has full rank. If 𝑔 = 𝑒 𝑔 + 𝑒 𝑔 ∈ E is a solution of 𝑥𝑔 + 𝑥𝜎 ( 𝑔 ) + 𝜎 ( 𝑔 ) + 𝑦𝜎 ( 𝑔 ) = 𝑔 and 𝑔 : 𝑥 ( + 𝑥 )( + 𝑥 ) 𝑔 + ( + 𝑥 − 𝑥 − 𝑥 ) 𝜎 ( 𝑔 )+ ( + 𝑥 )( 𝜎 ) ( 𝑔 ) + ( + 𝑥 )( 𝜎 ) ( 𝑔 ) = ,𝑥 ( + 𝑥 ) 𝑔 + ( + 𝑥 − 𝑥 ) 𝜎 ( 𝑔 ) − ( 𝜎 ) ( 𝑔 ) + ( 𝜎 ) ( 𝑔 ) = . Note that even in the degenerated case it might be possible to usethe method stated in the proof of Theorem 9 to construct non-zerolinear difference equations for some of the components.
Example 12.
Again we consider the idempotent difference ring ( Q ( 𝑥 )[ 𝑦 ] , 𝜎 ) with 𝜎 ( 𝑥 ) = 𝑥 + 𝜎 ( 𝑦 ) = − 𝑦 and the idempotentelements 𝑒 = − 𝑦 and 𝑒 = − 𝑦 . Let 𝒂 = ( 𝑦 − , 𝑥 ( 𝑦 + ) , 𝑦 − , 𝑥 ( 𝑦 + )) , then the shift projection matrix 𝑀 𝜎,𝜋 ( 𝒂 ) yields © « − − ( + 𝑥 ) ( + 𝑥 ) − − ( + 𝑥 ) ( + 𝑥 )
00 0 0 0 − − ª®®®®®¬ , which clearly doesn’t have full rank. Still if 𝑔 = 𝑒 𝑔 + 𝑒 𝑔 ∈ E isa solution of ( 𝑦 − ) 𝑔 + 𝑥 ( 𝑦 + ) 𝜎 ( 𝑔 ) + ( 𝑦 − ) 𝜎 ( 𝑔 ) + 𝑥 ( 𝑦 + ) 𝜎 ( 𝑔 ) = 𝑔 satisfies 𝑔 + 𝜎 ( 𝑔 ) = 𝑔 .With this notion the following corollary is immediate. Corollary 13.
Let ( E , 𝜎 ) and 𝒂 ∈ E 𝑚 + be as stated in Theorem 9.Consider the PLDE (1) with 𝑓 𝑖 ∈ E and 𝑐 𝑖 ∈ K , which is satisfied by 𝑔 = Í 𝜆 − 𝑠 = 𝑒 𝑠 𝑔 𝑠 ∈ E and let 𝑘 ∈ N with ≤ 𝑘 < 𝜆 . Then there exist 𝑏 𝑘,𝑖 ∈ 𝑒 𝑘 E , not all zero, and 𝑓 𝑘,𝑗 ∈ 𝑒 𝑘 E such that 𝑚 Õ 𝑖 = 𝑏 𝑘,𝑖 ( 𝜎 𝜆 ) 𝑖 ( 𝑔 𝑘 ) = 𝑐 𝑓 𝑘, + · · · + 𝑐 𝑑 𝑓 𝑘,𝑑 . (12) In particular, if ( E , 𝜎 ) is computable, the 𝑎 𝑘,𝑖 and 𝑓 𝑘,𝑗 are computable. akob Ablinger and Carsten Schneider We are now ready to obtain a general strategy to solve PLDEsunder the assumption that one can solve PLDEs in ( 𝑒 E , 𝜎 𝜆 ) . Notethat the task to compute for 𝑓 , . . . , 𝑓 𝑑 ∈ 𝑒 𝑘 E a basis of {( 𝑐 , . . . , 𝑐 𝑑 ) ∈ K 𝑑 | 𝑐 𝑓 + · · · + 𝑐 𝑑 𝑓 𝑑 = } (13)is a special case by setting 𝑔 = Theorem 14.
Let ( E , 𝜎 ) be an idempotent difference ring with theidempotent elements 𝑒 , . . . , 𝑒 𝜆 − and constant field K , and let 𝒂 ∈ E 𝑚 + and 𝒇 ∈ E 𝑑 . If const 𝜎 𝜆 𝑒 E = 𝑒 K and 𝒂 is non-degenerated, 𝑉 ( 𝒂 , 𝒇 , E ) has a finite basis. If ( E , 𝜎 ) is computable and PLDEs in ( 𝑒 E , 𝜎 𝜆 ) can be computed, a basis of 𝑉 ( 𝒂 , 𝒇 , E ) can be computed. Proof.
We look for a basis of 𝑉 = 𝑉 ( 𝒂 , 𝒇 , E ) over K for a non-degenerated 𝒂 ∈ E 𝑚 + and 𝒇 ∈ E 𝑑 . By Corollary 13 there exist 𝑏 𝑘,𝑖 ∈ 𝑒 𝑘 E , not all zero, and 𝑓 𝑘,𝑗 ∈ 𝑒 𝑘 E with (12). Since 𝑒 𝑘 E for 0 ≤ 𝑘 < 𝜆 are integral domains, we can take a finite ba-sis {( 𝑒 𝑘 𝑐 ( 𝑘 ) 𝑗, , . . . , 𝑒 𝑘 𝑐 ( 𝑘 ) 𝑗,𝑑 , 𝑒 𝑘 𝛾 ( 𝑘 ) 𝑗 )} ≤ 𝑗 ≤ 𝛿 𝑘 ⊆ ( 𝑒 𝑘 K ) 𝑑 × ( 𝑒 𝑘 E ) with 𝑐 ( 𝑘 ) 𝑗,𝑙 ∈ K of 𝑉 𝑘 = 𝑉 (( 𝑏 𝑘, , . . . , 𝑏 𝑘,𝑚 ) , ( 𝑓 𝑘, , . . . , 𝑓 𝑘,𝑑 ) , 𝑒 𝑘 E ) over 𝑒 𝑘 K .If 𝛿 𝑘 = ≤ 𝑘 < 𝜆 it follows that 𝑉 = { } and we getthe empty basis. Otherwise, we can take a basis of 𝑊 = {( 𝑐 , . . . , 𝑐 𝑑 , 𝑒 𝑔 + · · · + 𝑒 𝜆 − 𝑔 𝜆 − ) ∈ K 𝑑 × E |( 𝑒 𝑘 𝑐 , . . . , 𝑒 𝑘 𝑐 𝑑 , 𝑒 𝑘 𝑔 𝑘 ) ∈ 𝑉 𝑘 for 0 ≤ 𝑘 < 𝜆 } . as follows. We define 𝐶 𝑘 = ( 𝑐 𝑗,𝑙 ) ≤ 𝑗 ≤ 𝛿 𝑘 , ≤ 𝑙 ≤ 𝑑 for 0 ≤ 𝑘 < 𝜆 andtake a K -basis, say {( 𝑑 𝑙, , , . . . , 𝑑 𝑙, ,𝛿 , . . . , 𝑑 𝑙,𝜆 − , , . . . , 𝑑 𝑙,𝜆 − ,𝛿 𝜆 − )} ≤ 𝑙 ≤ 𝑟 , of the K -vector space {( 𝑑 , , . . . , 𝑑 ,𝛿 , . . . , 𝑑 𝜆 − , , . . . , 𝑑 𝜆 − ,𝛿 𝜆 − ) ∈ K 𝛿 +···+ 𝛿 𝜆 − |( 𝑑 , , . . . , 𝑑 ,𝛿 ) 𝐶 = · · · = ( 𝑑 𝜆 − , , . . . , 𝑑 𝜆 − ,𝛿 𝜆 − ) 𝐶 𝜆 − } . • If 𝑟 >
0, we proceed as follows. We define for 1 ≤ 𝑙 ≤ 𝑟 theelements 𝑔 𝑙 = 𝑔 ( ) 𝑙 + · · · + 𝑔 ( 𝜆 − ) 𝑙 ∈ E with 𝑔 ( 𝑘 ) 𝑙 = 𝑑 𝑙,𝑘, 𝑒 𝑘 𝛾 ( 𝑘 ) + · · · + 𝑑 𝑙,𝑘,𝛿 𝑘 𝑒 𝑘 𝛾 ( 𝑘 ) 𝛿 𝑘 where 0 ≤ 𝑘 < 𝜆 ,and define for 1 ≤ 𝑙 ≤ 𝑟 the constants ( 𝑐 𝑙, , . . . , 𝑐 𝑙,𝑑 ) = ( 𝑑 𝑙, , , . . . , 𝑑 𝑙, ,𝛿 ) 𝐶 ∈ K 𝑑 . Then 𝐵 = {( 𝑐 𝑙, , . . . , 𝑐 𝑙,𝑑 , 𝑔 𝑙 )} ≤ 𝑙 ≤ 𝑟 forms a bases of 𝑊 . Now weplug in the found basis elements into (1) and obtain linear con-straints. Fulfilling them by combining the basis elements accord-ingly will lead finally to a basis of the solution space 𝑉 . For this fi-nal step, take 𝐶 = ( 𝑐 𝑙,𝑖 ) ≤ 𝑙 ≤ 𝑟, ≤ 𝑖 ≤ 𝑑 with 𝑐 𝑙,𝑖 ∈ K and 𝒈 = ( 𝑔 , . . . , 𝑔 𝑟 ) ∈ E 𝑟 , and define 𝒇 ′ : = 𝐶 𝒇 𝑡 − ( 𝑎 𝑚 𝜎 𝑚 ( 𝒈 ) + · · · + 𝑎 𝒈 ) ∈ E 𝑟 ;here applying 𝜎 to a vector means to apply 𝜎 to each component.Note that nonzero elements in 𝒇 ′ reflect the disagreement of theso far found basis 𝐵 to be also a basis of 𝑉 . To complete the con-struction, we compute for the vector space 𝑊 ′ = {( 𝜅 , . . . , 𝜅 𝑟 ) ∈ K 𝑟 | ( 𝜅 , . . . , 𝜅 𝑟 ) 𝒇 ′ } (14)the basis {( 𝜅 𝑖, , . . . , 𝜅 𝑖,𝑟 ) ≤ 𝑖 ≤ 𝑠 ⊆ K 𝑠 ; here one collects the compo-nents of 𝒇 ′ w.r.t. the 𝑒 𝑘 for 0 ≤ 𝑘 < 𝜆 (which is justified since 𝑒 , . . . , 𝑒 𝜆 − are linearly independent), derives the bases in the in-tegral domains 𝑒 𝑘 E for each 0 ≤ 𝑘 < 𝜆 and computes the inter-section of the corresponding vector spaces to get a basis of 𝑊 ′ .If 𝑠 = 𝑉 = { } and we get the empty basis of 𝑉 . Otherwise,take 𝐷 = ( 𝜅 𝑖,𝑗 ) ≤ 𝑖 ≤ 𝑠, ≤ 𝑗 ≤ 𝑟 and define the entries of the matrix ( 𝑐 ′ 𝑖,𝑗 ) ≤ 𝑖 ≤ 𝑠, ≤ 𝑗 ≤ 𝑑 : = 𝐷 𝐶 and the entries of the vector ( 𝑔 ′ , . . . , 𝑔 ′ 𝑠 ) : = 𝐷 ( 𝑔 , . . . , 𝑔 𝑟 ) ∈ E 𝑠 . By construction {( 𝑐 ′ 𝑖, , . . . , 𝑐 ′ 𝑖,𝑑 , 𝑔 ′ 𝑖 )} ≤ 𝑖 ≤ 𝑠 ⊆ K 𝑑 × E is a basis of 𝑉 . • If 𝑟 =
0, it follows that 𝑉 ⊂ { } 𝑑 × E , i.e., we only have to searchfor homogeneous solutions of (1). Using the above constructionwe get a basis of the form {( ,𝑔 ′ 𝑖 )} ≤ 𝑖 ≤ 𝑠 ∪ {( , )} of 𝑉 ( 𝒂 , ( ) , E ) .This gives the basis {( , . . . , , 𝑔 ′ 𝑖 )} ≤ 𝑖 ≤ 𝑠 ⊆ { } 𝑑 × E of 𝑉 .We observe that the construction above can be carried out explic-itly if the algorithmic assumptions hold: First, we can compute thebases of 𝑉 𝑖 ; more precisely, we move the problem with the isomor-phism 𝜎 𝜆 − 𝑖 to the zero component, solve it there and move it backwith 𝜎 𝑖 . Further, we can solve the various linear algebra problemsin K . Finally, 𝑒 𝑘 E (0 ≤ 𝑘 < 𝜆 ) are integral domains and we cancompute a basis of (14) (by assumption a basis of (13) can be com-puted). (cid:3) ( 𝑅 ) ΠΣ -EXTENSIONS We will now apply Theorem 14 to a rather general class of dif-ference rings built by basic 𝑅 ΠΣ -ring extensions [23, 24] that aredefined over ΠΣ -field extensions [12]. Before we can state Theo-rem 25 below, we will present more details on the underlying con-struction. Definition 15.
A difference ring ( E , 𝜎 ) is called an 𝑅 ΠΣ -ring ex-tension of a difference ring ( A , 𝜎 ) if A = A ≤ A ≤ · · · ≤ A 𝑒 = E is a tower of ring extensions with const 𝜎 E = const 𝜎 A where forall 1 ≤ 𝑖 ≤ 𝑒 one of the following holds: • A 𝑖 = A 𝑖 − [ 𝑡 𝑖 ] is a ring extension subject to the relation 𝑡 𝜈𝑖 = 𝜈 > 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ ( A 𝑖 − ) ∗ is a primitive 𝜈 throot of unity ( 𝑡 𝑖 is called an 𝑅 -monomial , and 𝜈 is called the order of the 𝑅 -monomial ); • A 𝑖 = A 𝑖 − [ 𝑡 𝑖 , 𝑡 − 𝑖 ] is a Laurent polynomial ring extensionwith 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ ( A 𝑖 − ) ∗ ( 𝑡 𝑖 is called a Π -monomial ); • A 𝑖 = A 𝑖 − [ 𝑡 𝑖 ] is a polynomial ring extension with 𝜎 ( 𝑡 𝑖 ) − 𝑡 𝑖 ∈ A 𝑖 − ( 𝑡 𝑖 is called an Σ -monomial ).Depending on the occurrences of the 𝑅 ΠΣ -monomials such an ex-tension is also called a 𝑅 -/ Π -/ Σ -/ 𝑅 Π -/ 𝑅 Σ -/ ΠΣ -ring extension .For convenience we use A h 𝑡 i for three different meanings: it isthe ring A [ 𝑡 ] subject to the relation 𝑡 𝜈 = 𝑡 is an 𝑅 -monomialof order 𝜈 , it is the polynomial ring A [ 𝑡 ] if 𝑡 is a Σ -monomial, or itis the Laurent polynomial ring A [ 𝑡, 𝑡 − ] if 𝑡 is a Π -monomial. Wewill restrict 𝑅 ΠΣ -ring extensions further to basic 𝑅 ΠΣ -ring exten-sions [24]. Definition 16.
Let ( E , 𝜎 ) be a 𝑅 ΠΣ -ring extension of ( A , 𝜎 ) with E = A h 𝑡 i . . . h 𝑡 𝑒 i . We define the product group by [ A ∗ ] EA : = { 𝑓 𝑡 𝑚 . . . 𝑡 𝑚 𝑒 𝑒 | 𝑓 ∈ A ∗ and 𝑚 𝑖 ∈ Z where 𝑚 𝑖 = 𝑡 𝑖 is an 𝑅 Σ -monomial } . olving linear difference equations with coefficients in rings with idempotent representations Then ( E , 𝜎 ) is called a basic 𝑅 ΠΣ -ring extension of ( A , 𝜎 ) if for all Π -monomials 𝑡 𝑖 we have 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ [ A ∗ ] A h 𝑡 i ... h 𝑡 𝑖 − i A and for all 𝑅 -monomials 𝑡 𝑖 we have 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ const 𝜎 A ∗ .In the following we seek for algorithms that solve PLDEs in abasic 𝑅 ΠΣ -ring extension ( E , 𝜎 ) of a difference field ( F , 𝜎 ) withconstant field K . By Lemma 2.22 and Proposition 2.23 in [24] itturns out that one can collect several 𝑅 -monomials to one specific 𝑅 -monomial. Thus we assume from now on that ( E , 𝜎 ) has the form E = F [ 𝑦 ]h 𝑡 i . . . h 𝑡 𝑒 i (15)where 𝑦 is an 𝑅 -monomial of order 𝜆 with 𝛼 : = 𝜎 ( 𝑦 ) 𝑦 ∈ K ∗ andwhere the 𝑡 𝑖 with 1 ≤ 𝑖 ≤ 𝑒 are ΠΣ -monomials with 𝜎 ( 𝑡 𝑖 ) = 𝛼 𝑖 𝑡 𝑖 + 𝛽 𝑖 (note that either 𝛼 𝑖 = 𝛼 𝑖 ∈ [ F ∗ ] EF with 𝛽 𝑖 = 𝑒 𝑠 = ˜ 𝑒 𝑠 ( 𝑦 ) : = 𝜆 − Î 𝑗 = 𝑗 ≠ 𝜆 − − 𝑠 ( 𝑦 − 𝛼 𝑗 ) for 0 ≤ 𝑠 < 𝜆 . Since 𝛼 is a 𝜆 th primitive root of unity, we have that˜ 𝑒 𝑠 ( 𝛼 𝜆 − − 𝑠 ) ≠
0. Thus we can define 𝑒 𝑠 = 𝑒 𝑠 ( 𝑦 ) : = ˜ 𝑒 𝑠 ( 𝑦 ) ˜ 𝑒 𝑠 ( 𝛼 𝜆 − − 𝑠 ) (16)for 0 ≤ 𝑠 < 𝜆 which fulfill precisely the properties enumeratedin Definition 1. In particular, by [24, Thm. 4.3] (compare also [27,Corollary 1.16] and [10]) it follows that ( E , 𝜎 ) is an idempotent dif-ference ring of order 𝜆 . In particular, it is constant-stable providedthat the ground field ( F , 𝜎 ) is constant-stable. Definition 17.
A difference ring (resp. field) ( A , 𝜎 ) is called constant-stable if const 𝜎 𝑘 A = const 𝜎 A for all 𝑘 ∈ N \ { } . Theorem 18 ([24, Thm. 4.3]) . Let ( E , 𝜎 ) be a basic 𝑅 ΠΣ -ring exten-sion of a difference field ( F , 𝜎 ) with (15) where 𝑦 is an 𝑅 -monomial oforder 𝜆 with 𝛼 = 𝜎 ( 𝑦 ) 𝑦 . Let 𝑒 , . . . , 𝑒 𝜆 − be the idempotent, pairwiseorthogonal elements defined in (16) (that sum up to one). Then: (1) We get the direct sum (2) of the rings 𝑒 𝑠 E with the multiplica-tive identities 𝑒 𝑠 . (2) We have that 𝑒 𝑠 E = 𝑒 𝑠 ˜ E with the integral domain ˜ E : = F h 𝑡 i . . . h 𝑡 𝑒 i . (17)(3) For ≤ 𝑠 < 𝜆 , ( 𝑒 𝑠 ˜ E , 𝜎 𝜆 ) is a basic ΠΣ -ring extension of ( 𝑒 𝑠 F , 𝜎 𝜆 ) . (4) 𝜎 is a difference ring isomorphism between ( 𝑒 𝑠 ˜ E , 𝜎 𝜆 ) and ( 𝑒 𝑠 + 𝜆 ˜ E , 𝜎 𝜆 ) . (5) Further, if ( F , 𝜎 ) is constant-stable, const 𝜎 𝜆 𝑒 𝑠 E = 𝑒 𝑠 const 𝜎 F . In this particular setting, the used constructions in Section 2 andin Theorem 18 can be made more precise as follows. For 𝑓 ∈ E , theprojection of the first component can be computed by 𝜋 ( 𝑓 ) : = 𝜆 − Í 𝑖 = 𝑒 𝑖 ( 𝑦 ) 𝑓 (cid:12)(cid:12)(cid:12) 𝑦 → 𝛼 𝜆 − . Furthermore, define for 𝑛 ∈ N and 𝑓 ∈ F the 𝜎 -factorial 𝑓 𝜎,𝑛 = 𝑛 − Î 𝑖 = 𝜎 𝑖 ( 𝑓 ) . Then with 𝜎 ( 𝑡 𝑖 ) = 𝛼 𝑡 𝑖 + 𝛽 𝑖 (recall that 𝛼 𝑖 = 𝛽 𝑖 =
0) we get 𝜎 𝜆 ( 𝑡 𝑖 ) = ˜ 𝛼 𝑖 𝑡 𝑖 + ˜ 𝛽 𝑖 with ˜ 𝛼 𝑖 = 𝛼 𝜎,𝑖 and ˜ 𝛽 𝑖 = Í 𝑖 − 𝑙 = 𝜎 𝑙 ( 𝛽 ) . In particular, we can define for 0 ≤ 𝑠 < 𝜆 the ring automorphism 𝜎 𝑠 : ˜ E → ˜ 𝐸 with 𝜎 𝑠 ( 𝑓 ) = 𝜎 𝜆 ( 𝑓 ) for 𝑓 ∈ F and 𝜎 𝑠 ( 𝑡 𝑖 ) = ˜ 𝛼 𝑖 𝑡 𝑖 + ( ˜ 𝛽 𝑖 | 𝑦 → 𝛼 𝜆 − − 𝑠 ) (18)for all 1 ≤ 𝑖 ≤ 𝑒 . Then ( ˜ E , 𝜎 𝑠 ) and ( 𝑒 𝑠 ˜ E , 𝜎 𝜆 ) are isomorphic withthe difference ring isomorphism 𝜏 : ˜ E → 𝑒 𝑠 ˜ E with 𝜏 ( 𝑓 ) = 𝑒 𝑠 𝑓 for 𝑓 ∈ ˜ 𝐸 ; for further details we refer to [24, page 639]. In the followingwe prefer to work with ( ˜ E , 𝜎 𝑠 ) instead of ( 𝑒 𝑠 E , 𝜎 𝜆 ) . Note that thisrepresentation is also more convenient for implementations.As observed in Theorem 18 we obtain the ΠΣ -ring extension ( ˜ 𝐸, 𝜎 𝑠 ) of ( F , 𝜎 𝜆 ) where ˜ E is an integral domain. Thus we can takethe quotient field 𝑄 ( ˜ 𝐸 ) = F ( 𝑡 ) . . . ( 𝑡 𝑒 ) and by naturally extending 𝜎 𝑠 : ˜ E → ˜ E to 𝜎 ′ 𝑠 : 𝑄 ( ˜ E ) → 𝑄 ( ˜ E ) with 𝜎 ′ 𝑠 ( 𝑎𝑏 ) = 𝜎 𝑠 ( 𝑎 ) 𝜎 𝑠 ( 𝑏 ) we geta difference field ( 𝑄 ( ˜ E ) , 𝜎 ′ 𝑠 ) ; from now on we do not distinguishanymore between 𝜎 ′ 𝑠 and 𝜎 𝑠 .Finally, we take the ring of fractions 𝑄 ( E ) = { 𝑎𝑏 | 𝑎 ∈ E , 𝑏 ∈ E ∗ } which can be written in terms of the idempotent representation 𝑄 ( E ) = 𝑒 𝑄 ( ˜ E ) ⊕ · · · ⊕ 𝑒 𝑛 − 𝑄 ( ˜ E ) . (19)In particular, we can extend the automorphism 𝜎 : E → E to 𝜎 : 𝑄 ( E ) → 𝑄 ( E ) by mapping 𝑓 = 𝑒 𝑓 + · · · + 𝑒 𝜆 − 𝑓 𝜆 − with 𝑓 𝑖 ∈ ˜ E to 𝜎 ( 𝑓 ) = 𝑒 𝜎 ( 𝑓 𝜆 − ) + 𝑒 𝜎 ( 𝑓 ) + · · · + 𝑒 𝜆 − 𝑓 𝜆 − ; compare [11, Sec. 1.3] and [10, Cor. 6.9].Summarizing, also ( 𝑄 ( E ) , 𝜎 ) is an idempotent difference ring oforder 𝜆 as introduced in Definition 1 and it seems naturally to applyTheorem 14 to this more general situation. Here we note (comparealso [24, Prop. 66]) that each component ( ˜ E , 𝜎 𝑠 ) for 0 ≤ 𝑠 < 𝜆 isactually a special case of a ΠΣ -field-extension [12, 13]. Definition 19.
A difference field ( F , 𝜎 ) is called a ΠΣ -field exten-sion of a difference field ( H , 𝜎 ) if H = H ≤ H ≤ · · · ≤ H 𝑒 = F isa tower of field extensions with const 𝜎 F = const 𝜎 H where for all1 ≤ 𝑖 ≤ 𝑒 one of the following holds: • H 𝑖 = H 𝑖 − ( 𝑡 𝑖 ) is a rational function field extension with 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ ( H 𝑖 − ) ∗ ( 𝑡 𝑖 is called a Π -field monomial ); • H 𝑖 = H 𝑖 − ( 𝑡 𝑖 ) is a rational function extension with 𝜎 ( 𝑡 𝑖 ) − 𝑡 𝑖 ∈ H 𝑖 − ( 𝑡 𝑖 is called a Σ -field monomial ).Here we will rely on the following property of ΠΣ -field extensions;the first statement has been shown in [12] for ΠΣ -fields. The sec-ond statement appears also in [25]. Proposition 20.
Let ( E , 𝜎 ) be a ΠΣ -field/ ΠΣ -ring extension of adifference field ( F , 𝜎 ) with K = const 𝜎 F . Then: (1) For 𝑘 > , ( E , 𝜎 𝑘 ) is a ΠΣ -field/ ΠΣ -ring extension of ( F , 𝜎 𝑘 ) . (2) If ( F , 𝜎 ) is constant-stable, ( E , 𝜎 ) is constant-stable. Proof. (1) Let 𝑘 > 𝑎 ∈ E \ F with with 𝜎 𝑘 ( 𝑎 ) = 𝑎 . Define ℎ = 𝑎 𝜎 ( 𝑎 ) . . . 𝜎 𝑘 − ( 𝑎 ) . By [25,Lemma 31.(3)] if follows that ℎ ∉ F . Since 𝜎 ( ℎ ) ℎ = 𝜎 𝑘 ( 𝑎 ) 𝑎 =
1, itfollows that ℎ ∈ const 𝜎 E = const 𝜎 F ⊆ F , a contradiction. Notethat for any Σ -monomial 𝑡 with 𝜎 ( 𝑡 ) = 𝑡 + 𝛽 we have 𝜎 𝑘 ( 𝑡 ) − 𝑡 = Í 𝑘 − 𝑖 = 𝜎 𝑖 ( 𝛽 ) and for any Π -monomial 𝑡 with 𝜎 ( 𝑡 ) = 𝛼 𝑡 we have 𝜎 𝑘 ( 𝑡 )/ 𝑡 = 𝑎 𝜎,𝑘 . Thus the automorphism 𝜎 𝑘 satisfies the require-ments and consequently ( E , 𝜎 𝑘 ) is a ΠΣ -field/ ΠΣ -ring extensionof ( F , 𝜎 𝑘 ) .(2) Suppose that ( F , 𝜎 ) is constant-stable and let 𝑘 >
1. Then const 𝜎 𝑘 F = akob Ablinger and Carsten Schneider const 𝜎 F . By statement (1), const 𝜎 F = const 𝜎 E and thus const 𝜎 𝑘 E = const 𝜎 𝑘 E . Hence ( E , 𝜎 ) is constant-stable. (cid:3) In this particular scenario, we can refine Theorem 14 as follows.
Proposition 21.
Let ( F , 𝜎 ) be a constant-stable difference field withconstant field K , and let ( E , 𝜎 ) with (15) be a basic 𝑅 ΠΣ -ring exten-sion with only one 𝑅 -monomial 𝑦 with 𝜎 ( 𝑦 ) 𝑦 ∈ K of order 𝜆 . Thenone can solve non-degenerated PLDEs in ( E , 𝜎 ) (resp. in ( 𝑄 ( E ) , 𝜎 ) ) if ( E , 𝜎 ) is computable and one can solve PLDEs in the ΠΣ -ring exten-sion ( ˜ E , 𝜎 ) (resp. ΠΣ -field extension ( Q ( ˜ E ) , 𝜎 ) ) of ( F , 𝜎 𝜆 ) with (17) . Proof. ( ˜ E , 𝜎 ) is a basic ΠΣ -ring extension of ( F , 𝜎 𝜆 ) by The-orem 18.(3), and thus taking the quotient field 𝑄 ( ˜ E ) , ( Q ( ˜ E ) , 𝜎 ) is a ΠΣ -field extension of ( F , 𝜎 𝜆 ) by iterative application of [24,Cor. 2.6]. Since ( F , 𝜎 ) is constant-stable, we get const 𝜎 𝑄 ( ˜ E ) = const 𝜎 ˜ E = const 𝜎 𝜆 F = const 𝜎 F = K . Finally, since we can solvePLDEs in ( ˜ E , 𝜎 ) (resp. in ( 𝑄 ( ˜ E ) , 𝜎 ) ) by assumption, we can ap-ply Theorem 14 and can compute all solutions of non-degeneratedPLDEs in ( E , 𝜎 ) (resp. in ( 𝑄 ( E ) , 𝜎 ) ). (cid:3) 𝑅 ΠΣ -ring extensionsover ΠΣ -field extensions To activate Proposition 21 we have to take an appropriate differ-ence field ( F , 𝜎 ) such that (1) it is constant-stable and such that (2)PLDEs can be solved in ( ˜ E , 𝜎 ) . As it turns out, both properties canbe fulfilled if ( F , 𝜎 ) itself is a ΠΣ -field extension of a difference field ( G , 𝜎 ) that enjoys certain algorithmic properties. In this situation,the first property can be settled using Proposition 20 from above.To deal with the second property, we will introduce the followingproblems; a certain subset of them have been introduced originallyin [14] (by analyzing Karr’s (telescoping) algorithms in [12]). Definition 22 ([3]) . A difference field ( F , 𝜎 ) with constant field K is 𝜎 -computable if ( E , 𝜎 ) is computable and the following holds.(1) One can factor multivariate polynomials over F .(2) ( F , 𝜎 𝑠 ) is torsion free for any 𝑠 ∈ Z ∗ , i.e., ∀ 𝑠, 𝑟 ∈ Z ∗ ∀ 𝑓 , 𝑔 ∈ F ∗ : 𝑓 = 𝜎 𝑠 ( 𝑔 ) 𝑔 ∧ 𝑓 𝑟 = ⇒ 𝑓 = . (3) The Π -Regularity problem is solvable: Given ( F , 𝜎 ) and 𝑓 , 𝑔 ∈ F ∗ ; find, if possible, an 𝑛 ≥ 𝑓 𝜎,𝑛 = 𝑔 .(4) The Σ -Regularity problem is solvable: Given ( F , 𝜎 ) , 𝑟 ∈ Z ∗ , 𝑓 , 𝑔 ∈ F ∗ ; find, if possible, 𝑛 ≥ 𝑓 𝜎 𝑟 , + · · · + 𝑓 𝜎 𝑟 ,𝑛 = 𝑔 .(5) The parameterized pseudo-orbit problem is solvable: Given 𝒇 = ( 𝑓 , . . . , 𝑓 𝑛 ) ∈ ( F ∗ ) 𝑑 ; compute a Z -basis of the module 𝑀 ( 𝒇 , F ) = {( 𝑧 , . . . , 𝑧 𝑑 ) ∈ Z 𝑛 | ∃ 𝑔 ∈ F ∗ 𝜎 ( 𝑔 ) 𝑔 = 𝑓 𝑧 . . . 𝑓 𝑧 𝑑 𝑑 } . (6) There is an algorithm that can compute all the hypergeomet-ric candidates for equations with coefficients in ( F , 𝜎 ) : Givena nonzero operator 𝐿 ∈ F [ 𝜎 ] ; compute a finite set 𝑆 ⊂ F such that for any 𝑟 ∈ F ∗ , if 𝜎 − 𝑟 is a right factor of 𝐿 in F [ 𝜎 ] ,then 𝑟 = 𝑢 𝜎 ( 𝑣 ) 𝑣 for some 𝑢 ∈ 𝑆 and 𝑣 ∈ F ∗ .(7) PLDEs are solvable in ( F , 𝜎 ) : Given ≠ 𝒂 ∈ F 𝑚 + , 𝒇 ∈ F 𝑑 ;compute a K -basis of 𝑉 ( 𝒂 , 𝒇 , F ) .Then using the brandnew framework summarized in [3, Thm. 10],we obtain the following result which has been implemented withinthe summation package Sigma . Theorem 23 ([3]) . Let ( E , 𝜎 ) be a (nested) ΠΣ -field extension of ( F , 𝜎 ) . If ( F , 𝜎 ) is 𝜎 -computable, then also ( E , 𝜎 ) is 𝜎 -computable. In particular, using [3, 14] (based on [12]) the properties givenin Definition 22 simplify in the special case 𝜎 = id as follows. Theorem 24.
Let K be a computable field where (1) polynomials can be factored in K [ 𝑡 , . . . , 𝑡 𝑒 ] , (2) a basis of {( 𝑧 , . . . , 𝑧 𝑑 ) ∈ Z 𝑑 | = Î 𝑑𝑖 = 𝑐 𝑧 𝑖 𝑖 } can be com-puted, (3) one can recognize if 𝑐 ∈ 𝑘 is an integer,then ( K , 𝜎 ) with const 𝜎 𝐾 = K is 𝜎 -computable. We can now state our first algorithmic framework to solve non-degenerated PLDEs in ( 𝑅 ) ΠΣ -extensions.. Theorem 25.
Let ( F , 𝜎 ) be a ΠΣ -field extension of a difference field ( G , 𝜎 ) and let ( E , 𝜎 ) be a basic 𝑅 ΠΣ -ring extension of ( F , 𝜎 ) with one 𝑅 -monomial 𝑦 with 𝜎 ( 𝑦 ) 𝑦 ∈ const 𝜎 F of order 𝜆 . Then one can solvenon-degenerated PLDEs in the quotient ring ( 𝑄 ( E ) , 𝜎 ) or in ( E , 𝜎 ) ifone of the following holds: (1) ( G , 𝜎 ) is constant-stable and ( G , 𝜎 𝜆 ) is 𝜎 -computable. (2) const 𝜎 G = G satisfies the properties in Theorem 24. (3) const 𝜎 G = G is a rat. function field over an alg. number field. Proof. (1) Since ( G , 𝜎 ) is constant-stable, it follows that ( F , 𝜎 ) is constant-stable by Proposition 20.(2). Furthermore, ( F , 𝜎 𝜆 ) is a ΠΣ -field extension of ( G , 𝜎 𝜆 ) by Proposition 20.(1) and thus ( 𝑄 ( ˜ E ) , 𝜎 ) is a ΠΣ -field extension of ( G , 𝜎 𝜆 ) . Since ( G , 𝜎 𝜆 ) is 𝜎 -computable,we conclude with Theorem 23 that also ( 𝑄 ( ˜ E ) , 𝜎 ) is 𝜎 -computable,in particular property (7) in Definition 22 holds. Hence we can ap-ply Proposition 21 and can solve all non-degenerated PLDEs in ( 𝑄 ( E ) , 𝜎 ) . Given a basis in 𝑄 ( E ) one can filter out a basis of thesubspace in E by linear algebra .(2) Since G = const 𝜎 G , 𝜎 | G = id. Thus ( G , 𝜎 ) is trivially constant-stable. In addition, if the properties of Theorem 24 are fulfilled, ( G , 𝜎 ) is 𝜎 -computable and thus we can apply part (1).(3) By [8] and [19, Thm. 3.5] if follows that the algorithms requiredin Theorem 24 are available. Thus we can apply part (2). (cid:3) Remark . Theorem 25 (Case 3) covers, e.g., the rational ( 𝑣 =
0) orthe mixed multibasic difference field ( G , 𝜎 ) with G = K ( 𝑥, 𝑥 , . . . , 𝑥 𝑣 ) where K = 𝐾 ( 𝑞 . . . , 𝑞 𝑣 ) is a rational function field ( 𝐾 itself is arational function field over an algebraic number field) and with 𝜎 | K = id, 𝜎 ( 𝑥 ) = 𝑥 + 𝜎 ( 𝑥 𝑖 ) = 𝑞 𝑖 𝑥 𝑖 for 1 ≤ 𝑖 ≤ 𝑣 . The PLDE solver summarized in Theorem 25 assumes that ( G , 𝜎 ) is 𝜎 -computable. In the following we restrict ourselves to someinteresting sub-classes of 𝑅 ΠΣ -ring extensions where the Σ - and Π -regularity problem in Definition 22 (but also the hidden shift-equivalence problem within the tower of extensions) can be avoided.As a consequence one ends up at lighter implementations wheremost of the highly recursive algorithms from [12] can be skipped.Let ( A h 𝑡 i , 𝜎 ) be a ΠΣ -ring extension of ( A , 𝜎 ) with constantfield K = const 𝜎 A . Assume in addition that A is an integral do-main and that one can solve PLDEs in ( A , 𝜎 ) . Then we can apply In Section 3.2 we will provide improved algorithms to accomplish this task directly. olving linear difference equations with coefficients in rings with idempotent representations the following tactic [20] (which is inspired by [12] and is also thebackbone strategy in [3]) to find a basis of 𝑉 = 𝑉 ( 𝒂 , 𝒇 , A h 𝑡 i) with ≠ 𝒂 = ( 𝑎 , . . . , 𝑎 𝑚 ) ∈ A h 𝑡 i 𝑚 + and 𝒇 = ( 𝑓 , . . . , 𝑓 𝑑 ) ∈ A h 𝑡 i 𝑑 .First, we bound the degree of the possible solutions: namely, wecompute 𝑎, 𝑏 ∈ Z such that for any ( 𝑐 , . . . , 𝑐 𝑑 , Í 𝑏 ′ 𝑘 = 𝑎 ′ 𝑔 𝑖 𝑡 𝑖 ) ∈ 𝑉 we have 𝑎 ≤ 𝑎 ′ and 𝑏 ′ ≤ 𝑏 ; if 𝑡 is a Σ -monomial we set 𝑎 = 𝑏 only. Then given such bounds 𝑎, 𝑏 , we make theansatz (1) with unknown 𝑐 , . . . , 𝑐 𝑑 ∈ K and 𝑔 = Í 𝑏 ′ 𝑘 = 𝑎 ′ 𝑔 𝑖 𝑡 𝑖 withunknown 𝑔 𝑎 , . . . , 𝑔 𝑏 ∈ A . By comparing coefficients in (1) w.r.t. tothe highest arising term we obtain a PLDE in ( A , 𝜎 ) which has 𝑐 , . . . , 𝑐 𝑑 and 𝑔 𝑏 ∈ A as solution. Solving this PLDE yields all pos-sible candidates for 𝑔 𝑏 . Thus plugging these choices into (1) we canproceed recursively (by degree reduction) to nail down 𝑔 𝑏 and theremaining coefficients 𝑔 𝑎 , . . . , 𝑔 𝑏 − .Due to [3, Theorem 7] it follows that one can determine 𝑏 ∈ N and 𝑎 = Σ -monomial 𝑡 if one can solve PLDEs in ( A , 𝜎 ) . Thusactivating this machinery recursively yields the following result. Proposition 27.
If one can solve PLDEs in ( A , 𝜎 ) , then one can solvePLDEs in a Σ -extension ( A h 𝑡 i . . . h 𝑡 𝑒 i , 𝜎 ) of ( A , 𝜎 ) For Π -monomials one can utilize [3, Theorem 6] to compute theabove bounds 𝑎, 𝑏 ∈ Z . If one applies this machinery recursively (asfor Σ -monomials) one ends up at the requirement that the groundring is 𝜎 -computable. In a nutshell, we rediscover the ring versionof Theorem 25 – but this time we solve it directly without comput-ing first all solutions in its quotient field.In the following we adapt slightly the proof steps of [3, Theo-rem 6] yielding the more flexible Lemma 29. For its proof, we needin addition the following result. Lemma 28.
Let ( F h 𝑡 i . . . h 𝑡 𝑒 i , 𝜎 ) be a Π -ring extension of ( F , 𝜎 ) with 𝛼 𝑖 = 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ F ∗ . Let 𝑉 = 𝑀 ( 𝛼 , . . . , 𝛼 𝑒 , 𝑢 ) for some 𝑢 ∈ F ∗ .Then 𝑉 = ∅ or 𝑉 = Z ( 𝜆 , . . . , 𝜆 𝑒 + ) for some 𝜆 𝑖 ∈ Z with 𝜆 𝑒 + > . Proof.
Suppose that 𝑉 ≠ ∅ . Suppose further that we can take ≠ ( 𝜆 , . . . , 𝜆 𝑒 , ) ∈ 𝑉 . Then we get 𝑔 ∈ F ∗ with 𝜎 ( 𝑔 ) 𝑔 = 𝛼 𝜆 . . . 𝛼 𝜆 𝑒 𝑒 ,not all 𝜆 𝑖 being zero, which is not possible by [22, Thm. 9.1]. Conse-quently, for any nonzero vector in 𝑉 we conclude that the last entrymust be nonzero. Now take 𝝀 = ( 𝜆 , . . . , 𝜆 𝑒 + ) , 𝝁 = ( 𝜇 , . . . , 𝜇 𝑒 + ) ∈ 𝑉 \ { } . Then 𝜆 𝑒 + , 𝜇 𝑒 + ≠
0. In particular, 𝒂 = 𝜇 𝑒 + 𝝀 − 𝜆 𝑒 + 𝝁 ∈ 𝑉 .Since the last entry of 𝒂 is zero, it follows that 𝒂 = . Hencetwo nonzero vectors are linearly dependent and it follows that 𝑉 = {( 𝜆 , . . . , 𝜆 𝑒 + )} Z with 𝜆 𝑒 + ≠
0. If 𝜆 𝑒 + <
0, we can choosethe alternative generator (− 𝜆 , . . . , − 𝜆 𝑒 + ) with − 𝜆 𝑒 + > (cid:3) Lemma 29.
Let ( E , 𝜎 ) with E = F h 𝑡 i . . . h 𝑡 𝑒 i be a Π -ring extensionof ( F , 𝜎 ) with 𝛼 𝑖 = 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ F ∗ . If one can solve the parameterizedpseudo problem in ( F , 𝜎 ) and can find all hypergeometric candidatesin ( F , 𝜎 ) , one can bound the degrees of the solutions w.r.t. 𝑡 𝑒 . Proof.
Let 𝒇 = ( 𝑓 , . . . , 𝑓 𝑑 ) ∈ E 𝑑 and ( 𝑎 , . . . , 𝑎 𝑚 ) ∈ E 𝑚 + with 𝑎 𝑎 𝑚 ∈ E ∗ and suppose that 𝑔 ∈ E is a solution of (1). Let 𝑙 𝑒 be the highest degree in 𝑔 w.r.t. 𝑡 𝑒 . In the following we take thelexicographic order < on 𝑀 = { 𝑡 𝑛 . . . 𝑡 𝑛 𝑒 𝑒 | 𝑛 , . . . , 𝑛 𝑒 ∈ Z } with 𝑡 < 𝑡 < · · · < 𝑡 𝑒 , and 𝑡 𝑎𝑖 < 𝑡 𝑏𝑖 iff 𝑎 < 𝑏 . Let ˜ 𝑔 = ℎ 𝑡 𝜆 . . . 𝑡 𝜆 𝑒 𝑒 be thehighest term in 𝑔 ; note that 𝜆 𝑒 = 𝑙 𝑒 . Further, let 𝜇 = 𝑡 𝜇 . . . 𝑡 𝜇 𝑒 𝑒 ∈ 𝑀 be the largest monomial of the coefficients in 𝒂 , and let ˜ 𝑎 𝑖 ∈ F for 0 ≤ 𝑖 ≤ 𝑚 be the corresponding coefficient of 𝜇 ; note that one ofthe ˜ 𝑎 𝑖 is nonzero. Take 𝐿 : = ˜ 𝑎 + ˜ 𝑎 𝜎 + · · · + ˜ 𝑎 𝑚 𝜎 𝑚 ∈ F [ 𝜎 ] .Now suppose that 𝐿 ( ˜ 𝑔 ) = 𝛼 = 𝜎 ( ℎ ) ℎ 𝛼 𝜆 . . . 𝛼 𝜆 𝑒 𝑒 ∈ F ∗ .Note that for ˜ 𝐿 = 𝜎 − 𝛼 ∈ F [ 𝜎 ] we have ˜ 𝐿 ( ˜ 𝑔 ) = 𝐿 = 𝑄 ˜ 𝐿 + 𝑅 bethe right-division of 𝐿 by ˜ 𝐿 with 𝑄 ∈ F [ 𝜎 ] and 𝑅 ∈ F . Since 0 = 𝐿 ( ˜ 𝑔 ) = 𝑄 ˜ 𝐿 ( ˜ 𝑔 )+ 𝑅 = 𝑅 , ˜ 𝐿 is a right-factor of 𝐿 . By assumption we cancompute a set 𝑆 which contains all hypergeometric candidates of˜ 𝐿 . Thus we can take 𝑢 ∈ 𝑆 with 𝜎 ( ℎ ) ℎ 𝛼 𝜆 . . . 𝛼 𝜆 𝑒 𝑒 = 𝛼 = 𝑢 𝜎 ( 𝑤 ) 𝑤 . Con-sequently, we get 𝛼 𝜆 . . . 𝛼 𝜆 𝑒 𝑒 𝑢 − = 𝜎 ( 𝑤 ′ ) 𝑤 ′ for some 𝑤 ′ ∈ F ∗ . Nowcompute a basis 𝐵 𝑢 of 𝑉 𝑢 = 𝑀 ( 𝛼 , . . . , 𝛼 𝑒 , 𝑢 − ; F ) . By Lemma 28 wecan assume that 𝐵 𝑢 = {( 𝜈 𝑢, , . . . , 𝜈 𝑢,𝑒 + )} ∈ Z 𝑒 + with 𝜈 𝑢,𝑒 + > 𝜈 𝑢,𝑒 + = 𝑙 𝑒 = 𝜆 𝑒 = 𝜈 𝑢,𝑒 .Thus to bound the leading coefficient w.r.t. 𝑡 𝑒 we proceed as fol-lows: We loop trough all 𝑢 ∈ 𝑆 and compute a basis 𝐵 𝑢 of 𝑉 𝑢 andtake 𝐶 = max { 𝜈 𝑢,𝑒 | 𝑉 𝑢 = ( 𝜈 𝑢, , . . . , 𝜈 𝑢,𝑒 , ) Z for 𝑢 ∈ 𝑆 } . Summarizing, let 𝑙 𝑒 be the highest degree in the solution 𝑔 w.r.t. 𝑡 𝑒 and let ˜ 𝑔 = ℎ𝑡 𝜆 . . . 𝑡 𝜆 𝑒 𝑒 be the highest term in 𝑔 . If ˜ 𝐿 ( ˜ 𝑔 ) =
0. then 𝑙 𝑒 ∈ 𝐶 , i.e., 𝐶 ≠ ∅ and 𝑙 𝑒 ≤ max ( 𝐶 ) . Otherwise, if 𝐶 = ∅ or ˜ 𝐿 ( ˜ 𝑦 ) ≠ 𝐿 ( ˜ 𝑦 ) = ℎ ′ 𝑡 𝜆 . . . 𝑡 𝜆 𝑒 𝑒 for some ℎ ′ ∈ F ∗ . Since ˜ 𝑦 is the largest term in our solution 𝑦 and since ˜ 𝐿 isthe contribution of the highest term in (1), it follows by coefficientcomparison in (1) that ˜ 𝐿 ( ˜ 𝑦 ) 𝑡 𝑚 . . . 𝑡 𝑚 𝑒 𝑒 = ℎ ′ 𝑡 𝑚 + 𝜆 . . . 𝑡 𝑚 𝑒 + 𝜆 𝑒 𝑒 forsome ℎ ′ ∈ F ∗ must arise in 𝑐 𝑓 + · · · + 𝑐 𝑑 𝑓 𝑑 . Thus, if 𝑑 𝑖 is thelargest exponent in 𝑓 𝑖 w.r.t. 𝑡 𝑒 , we get 𝑚 𝑒 + 𝜆 𝑒 < max ( 𝑑 , . . . , 𝑑 𝑒 ) .In conclusion, if 𝐶 = ∅ , we get 𝑙 𝑒 = 𝜆 𝑒 ≤ max ( 𝑑 , . . . , 𝑑 𝑒 ) − 𝑚 𝑒 = : 𝑏 .Otherwise, we conclude that 𝑙 𝑒 = 𝜆 𝑒 ≤ max ( max ( 𝐶 ) , 𝑏 ) . Similarly,we can bound the lowest term in 𝑔 by repeating this procedure andtaking the order < with 𝑡 < 𝑡 < · · · < 𝑡 𝑒 and 𝑡 𝑎𝑖 < 𝑡 𝑏𝑖 , iff 𝑎 > 𝑏 and replacing the max operation with the min operation, etc. (cid:3) The above results yield, in comparison to Theorem 25, the follow-ing less general but simpler (less recursive algorithms) and moreflexible (less requirements) toolbox to solve PLDEs in ΠΣ -ring ex-tensions. Theorem 30.
Let ( E , 𝜎 ) with E = F h 𝑡 i . . . h 𝑡 𝑒 i be a ΠΣ -ring exten-sion of a difference field ( F , 𝜎 ) where for all Π -monomials 𝑡 𝑖 we have 𝜎 ( 𝑡 𝑖 ) 𝑡 𝑖 ∈ F ∗ . If one can solve PLDEs, the parameterized pseudo-orbitproblem and hypergeometric candidates in ( F , 𝜎 ) , then one can solvePLDEs in ( E , 𝜎 ) . Proof.
By reordering we may assume that A = F h 𝑡 i . . . h 𝑡 𝑙 i contains precisely the Π -monomials of E and that the 𝑡 𝑙 + , . . . , 𝑡 𝑒 form all Σ -monomials. By Lemma 29 we can bound the degree ofthe solutions w.r.t. 𝑡 𝑙 . By iteration (recursion) we can thus solvePLDEs in ( A , 𝜎 ) . Finally, with Prop. 27 we can solve PLDEs in ( E , 𝜎 ) . (cid:3) Combining Theorem 30 with Proposition 21 yields Theorem 31.
Theorem 31.
Let ( E , 𝜎 ) be an 𝑅 ΠΣ -ring extension of a constant-stable difference field ( F , 𝜎 ) with one 𝑅 -monomial 𝑦 with 𝜎 ( 𝑦 ) 𝑦 ∈ const 𝜎 F of order 𝜆 and where for each Π -monomial 𝑡 in the exten-sion E of F we have 𝜎 ( 𝑡 ) 𝑡 ∈ F ∗ . If one can solve PLDEs, solve the akob Ablinger and Carsten Schneider parameterized pseudo-orbit problem and can find all hypergeomet-ric candidates in ( F , 𝜎 𝜆 ) , one can solve non-degenerated PLDEs in ( E , 𝜎 ) . Using results of [4], this PLDE solver is, e.g., applicable if one spe-cializes F to the mixed multibasic case introduced in Remark 26. We will illustrate the whole machinery by solving the recurrence: h ( + 𝑛 ) ( + 𝑛 ) (cid:0)(cid:0) + 𝑛 + ( + 𝑛 ) 𝑛 Í 𝑖 = 𝑖 (cid:1) (− ) 𝑛 − ( + 𝑛 ) 𝑛 Í 𝑖 = (− ) 𝑖 𝑖 (cid:1)i 𝐺 ( 𝑛 )+ h ( + 𝑛 ) ( + 𝑛 ) (cid:0)(cid:0) + 𝑛 + ( + 𝑛 ) 𝑛 Í 𝑖 = 𝑖 (cid:1) (− ) 𝑛 − ( + 𝑛 ) 𝑛 Í 𝑖 = (− ) 𝑖 𝑖 (cid:1)i 𝐺 ( 𝑛 + )+ h ( + 𝑛 ) ( + 𝑛 ) (cid:0) (− ) 𝑛 𝑛 Í 𝑖 = 𝑖 + 𝑛 𝑛 Í 𝑖 = (− ) 𝑖 𝑖 (cid:1)i 𝐺 ( 𝑛 + ) = ( + 𝑛 ) + ( + 𝑛 ) 𝑛 Í 𝑖 = 𝑖 − ( + 𝑛 ) (− ) 𝑛 𝑛 Í 𝑖 = (− ) 𝑖 𝑖 . Internally, we represent the recurrence in the basic 𝑅 ΠΣ -ring ex-tension ( E , 𝜎 ) of ( Q ( 𝑥 ) , 𝜎 ) with E = Q ( 𝑥 )[ 𝑦 ][ 𝑠 ][ ¯ 𝑠 ] where 𝜎 ( 𝑥 ) = 𝑥 + 𝜎 ( 𝑦 ) = − 𝑦 , 𝜎 ( 𝑠 ) = 𝑠 + 𝑥 + and 𝜎 ( ¯ 𝑠 ) = ¯ 𝑠 + − 𝑦𝑥 + . Note that ( E , 𝜎 ) is an idempotent difference ring of order 2 with 𝑒 = − 𝑦 and 𝑒 = + 𝑦 . Then the recurrence turns into Í 𝑖 = 𝑎 𝑖 𝜎 𝑖 ( 𝑔 ) = 𝜑 with 𝒂 = (cid:16) ( + 𝑥 ) ( + 𝑥 ) (− ¯ 𝑠 ( + 𝑥 ) + ( + 𝑠 + 𝑥 + 𝑠𝑥 ) 𝑦 ) , ( + 𝑥 ) ( + 𝑥 )(− ¯ 𝑠 ( + 𝑥 ) + ( + 𝑥 + 𝑠 ( + 𝑥 )) 𝑦 ) , ( + 𝑥 ) ( + 𝑥 ) ( ¯ 𝑠𝑥 + 𝑠𝑦 ) (cid:17) ,𝜑 = 𝑠 ( + 𝑥 ) + ( + 𝑥 ) − 𝑠 ( + 𝑥 ) 𝑦. With Theorem 9 we compute with the package
HarmonicSums [1]for the first component the equation Í 𝑖 = 𝑏 ,𝑖 𝜎 𝑖 ( 𝑔 ) = 𝜑 with 𝒃 = ( 𝑏 𝑖, , 𝑏 𝑖, ) where 𝒃 = (cid:16) 𝑥 ( + 𝑥 + 𝑥 + 𝑥 + 𝑠 ( + 𝑥 + 𝑥 + 𝑥 ) + ¯ 𝑠 ( + 𝑥 + 𝑥 + 𝑥 𝑡 )) , − 𝑥 ( + 𝑥 + 𝑥 + 𝑥 + 𝑠 ( + 𝑥 + 𝑥 + 𝑥 ) + 𝑠 ( + 𝑥 + 𝑥 + 𝑥 )) , 𝑥 ( + 𝑥 ) ( + 𝑥 ) (cid:19) ,𝜑 = − 𝑥 ( + 𝑥 ) ( + 𝑥 ) ( + 𝑥 ) (cid:18) + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑠 ( + 𝑥 + 𝑥 + 𝑥 + 𝑥 ) + 𝑠 ( + 𝑥 + 𝑥 + 𝑥 + 𝑥 ) (cid:17) . A similar linear difference equation can be computed for the sec-ond component. Solving these equations (activating, e.g., Theo-rem 30 with
Sigma [21]) leads to the solutions 𝑡 = 𝑠 + 𝑐 + 𝑐 ( 𝑠 + ¯ 𝑠 + 𝑥 − 𝑠𝑥 − 𝑠𝑥 ) ,𝑡 = − 𝑠 + 𝑑 + 𝑑 (− 𝑠 + ¯ 𝑠 − 𝑥 + 𝑠𝑥 − 𝑠𝑥 ) , for 𝑐 , 𝑐 , 𝑑 , 𝑑 ∈ Q . Plugging 𝑔 : = 𝑒 𝑡 + 𝑒 𝑡 into Í 𝑖 = 𝑎 𝑖 𝜎 𝑖 ( 𝑔 ) = 𝜑 gives us constraints for the constants (compare Theorem 14) andwe find 𝑑 = − 𝑐 and 𝑑 = 𝑐 . These solutions can be combined tothe general solution − 𝑠𝑦 − 𝑐 𝑦 + 𝑐 ( ¯ 𝑠 − 𝑠𝑥 − 𝑠𝑦 − 𝑥𝑦 − 𝑠𝑥𝑦 ) , of Í 𝑖 = 𝑎 𝑖 𝜎 𝑖 ( 𝑔 ) = 𝜑 , i.e., {( ,𝑦 ) , ( , ¯ 𝑠 − 𝑠𝑥 − 𝑠𝑦 − 𝑥𝑦 − 𝑠𝑥𝑦 ) , ( , − 𝑠𝑦 )} is a basis of 𝑉 ( 𝒂 , ( 𝜑 ) , E ) . Finally, by reinterpreting the result interms of sums and products we find the following general solutionof the original recurrence: − 𝑛 Í 𝑖 = 𝑖 (− ) 𝑛 − 𝑐 (− ) 𝑛 + 𝑐 (cid:16) − (− ) 𝑛 𝑛 − ( + 𝑛 )(− ) 𝑛 𝑛 Í 𝑖 = 𝑖 + 𝑛 Í 𝑖 = (− ) 𝑖 𝑖 ( − 𝑛 ) (cid:17) . We have considered idempotent difference rings (heavily used inthe Galois theory of difference equations [10, 27]) and derived ageneral toolbox to solve PLDEs in this setting. More precisely, weintroduced the notion of non-degenerated linear difference opera-tors and showed that finding solutions for a given PLDE in differ-ence rings with zero-divisors can be reduced to finding solutions indifference rings that are integral domains (see Theorems 9 and 14).In the second part of this article we provided two general PLDEsolvers: Theorem 25 for the most general case which assumes thatrather strong properties hold in the ground field and Theorem 31which is less general, but where some of the complicated algorith-mic assumptions can be dropped. In both cases, the inner core (The-orem 23) is a PLDE solver for ΠΣ -field extensions that has beenelaborated in [3] and implemented within Sigma .Our notion of non-degenerated operators is motivated by ourmethod to decompose the desired solution. An interesting ques-tion is if there are equivalent (or even more flexible definitions)that are easier to verify. We also indicated that the decompositionmethod (implemented in the package
HarmonicSums ) works par-tially if the operator is degenerated. Further investigations in thisdirection, also connected to the dimension of the solution space,would be highly interesting. Finally, we are strongly motivated togeneralize our PLDE solver summarized in Theorem 31 further tomore general classes of (basic) 𝑅 ΠΣ -ring extensions. REFERENCES [1] J. Ablinger. The package HarmonicSums: Computer algebra and analytic aspectsof nested sums. In
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