A Difference Ring Theory for Symbolic Summation
aa r X i v : . [ c s . S C ] F e b A Difference Ring Theory forSymbolic Summation
Carsten Schneider
Research Institute for Symbolic Computation (RISC)Johannes Kepler UniversityAltenbergerstraße 69, 4040 Linz, Austria
Abstract
A summation framework is developed that enhances Karr’s difference field approach. It cov-ers not only indefinite nested sums and products in terms of transcendental extensions, but itcan treat, e.g., nested products defined over roots of unity. The theory of the so-called R ΠΣ ∗ -extensions is supplemented by algorithms that support the construction of such difference ringsautomatically and that assist in the task to tackle symbolic summation problems. Algorithmsare presented that solve parameterized telescoping equations, and more generally parameterizedfirst-order difference equations, in the given difference ring. As a consequence, one obtains algo-rithms for the summation paradigms of telescoping and Zeilberger’s creative telescoping. Withthis difference ring theory one gets a rigorous summation machinery that has been applied tonumerous challenging problems coming, e.g., from combinatorics and particle physics. Key words: difference ring extensions, roots of unity, indefinite nested sums and products,parameterized telescoping (telescoping, creative telescoping), semi-constants, semi-invariants
1. Introduction
In his pioneering work [24,25] M. Karr introduced a very general class of differencefields, the so-called ΠΣ-fields, in which expressions in terms of indefinite nested sums andproducts can be represented. In particular, he developed an algorithm that decides con-structively if for a given expression f ( k ) represented in a ΠΣ-field F there is an expression g ( k ) represented in the field F such that the telescoping equation (anti-difference) f ( k ) = g ( k + 1) − g ( k ) (1) ⋆ Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15) and the EuropeanCommission through contract PITN-GA-2010-264564 (LHCPhenoNet).
Email address:
[email protected] (Carsten Schneider).
Preprint submitted to Elsevier 5 October 2018 olds. If such a solution exists, one obtains for an appropriately chosen a ∈ N the identity b X k = a f ( k ) = g ( b + 1) − g ( a ) . (2)His algorithms can be viewed as the discrete version of Risch’s integration algorithm;see [40,13]. In the last years the ΠΣ-field theory has been pushed forward. It is nowpossible to obtain sum representations, i.e., right hand sides in (2) with certain optimalitycriteria such as minimal nesting depth [53,56], minimal number of generators in thesummands [45] or minimal degrees in the denominators [51]. For the simplification ofproducts see [48,8]. We emphasize that exactly such refined representations give rise tomore efficient telescoping algorithms worked out in [55,59].A striking application is that Karr’s algorithm and all the enhanced versions can beused to solve the parameterized telescoping problem [41,54]: for given indefinite nestedproduct-sum expressions f ( k ) , . . . , f n ( k ) represented in F , find constants c , . . . , c n , freeof k and not all zero, and find g ( k ) represented in F such that g ( k + 1) − g ( k ) = c f ( k ) + · · · + c n f n ( k ) (3)holds. In particular, this problem covers Zeilberger’s creative telescoping paradigm [62]for a bivariate function F ( m, k ) by setting f i ( k ) = F ( m + i − , k ) with i ∈ { , . . . , n } and representing these f i ( k ) in F . Namely, if one finds such a solution, one ends up atthe recurrence g ( m, b + 1) − g ( m, a ) = c b X k = a f ( m, k ) + · · · + c n b X k = a f ( m + n − , k ) . In a nutshell, one cannot only treat indefinite summation but also definite summationproblems. In this regard, also recurrence solvers have been developed where the co-efficients of the recurrence and the inhomogeneous part can be elements from a ΠΣ-field [14,49,6]. All these algorithms generalize and enhance substantially the ( q –)hyper-geometric and holonomic toolbox [5,18,61,62,36,34,37,35,9,15,26,30] in order to rewritedefinite sums to indefinite nested sums. For details on these aspects we refer to [58].Besides all these sophisticated developments, e.g., within the summation package Sigma [52], there is one critical gap which concerns all the developed tools in the settingof difference fields: Algebraic products, like( − k = k Y i =1 ( − , ( − k +12 ) = k Y i =1 i Y j =1 ( − , ( − k +23 ) = k Y i =1 i Y j =1 j Y k =1 ( − , . . . (4)cannot be expressed in ΠΣ-fields, which are built by a tower of transcendental fieldextensions. Even worse, the objects given in (4) introduce zero-divisors, like(1 − ( − k )(1 + ( − k ) = 0 (5)which cannot be treated in a field or in an integral domain. In applications these ob-jects occur rather frequently as standalone objects or in nested sums [3,4]. It is thus afundamental challenge to include such objects in an enhanced summation theory.With the elegant theory of [60,19] one can handle such objects by several copies of theunderlying difference field, i.e., by implementing the concept of interlacing in an algebraicway. First steps to combine these techniques with ΠΣ-fields have been made in [17].2ithin the package Sigma a different approach [42] has been implemented. Summationobjects like ( − k and sums over such objects are introduced by a tower of generatorssubject to the relations such as (5). In this way one obtains a direct translation betweenthe summation objects and the generators of the corresponding difference rings. This en-hancement has been applied non-trivially, e.g., to combinatorial problems [44,39], numbertheory [50,33] or to problems from particle physics [12]; for the most recent evaluationsof Feynman integrals [11,2,1] up to 300 generators were used to model the summationobjects in difference rings. But so far, this successful and very efficient machinery of Sigma was built, at least partially, on heuristic considerations.In this article we shall develop the underlying difference ring theory and supplementit with the missing algorithmic building blocks in order to obtain a rigorous summa-tion machinery. More precisely, we will enhance the difference field theory of [24,25] toa difference ring theory by introducing besides Π-extensions (for transcendental prod-uct extensions) and Σ ∗ -extensions (for transcendental sum extensions) also R -extensionswhich enables one to represent objects such as (4). An important ingredient of this theoryis the exploration of the so-called semi-constants (resp. semi-invariants) and the formula-tion of the symbolic summation problems within these notions. In particular, we obtainalgorithms that can solve certain classes of parameterized first-order linear differenceequations. As special instances we obtain algorithms for the parameterized telescopingproblem, in particular for the summation paradigms of telescoping and creative telescop-ing. In addition, we provide an algorithmic toolbox that supports the construction of theso-called simple R ΠΣ ∗ -extensions automatically. As a special case we demonstrate, howd’Alembertian solutions [7] of a recurrence, a subclass of Liouvillian solutions [20,38], canbe represented in such R ΠΣ ∗ -extensions. In particular, we will illustrate the underlyingproblems and their solutions by discovering the following identities b X k =1 ( − k +12 ) k k X j =1 ( − j j = 12 b X j =1 ( − j +12 ) j −
14 ( − b +12 ) (cid:0) − − b + 2 b (cid:1) + ( − b +12 ) 12 (cid:0) b ( b + 2) + ( − b (cid:0) b − (cid:1)(cid:1) b X j =1 ( − j j , (6) b Y k =1 − ( ι k )1 + k = (cid:0) − ι − (cid:1) − ( − b + ιb ( b + 1) (cid:16) b − Y j =1 ι j j (cid:17) ; (7)here the imaginary unit is denoted by ι , i.e., ι = − R ΠΣ ∗ -ring extensions. Furthermore, we will work out theunderlying problems in the setting of difference rings and motivate the different challengesthat will be treated in this article. In addition, we give an overview of the main resultsand show how they can be applied for symbolic summation. In the remaining sectionsthese results will be worked out in details. In Section 3 we present the crucial propertiesof single nested R ΠΣ ∗ -extensions. Special emphasis will be put on the properties of theunderlying ring. In Section 4 we will consider a tower of such extensions and explore theset of semi-constants. In Section 5 we present algorithms that calculate the order, periodand factorial order of the generators of R -extensions. Finally, in Section 6 and Section 7we elaborate algorithms that are needed to construct R ΠΣ ∗ -extensions and that solve asa special case the (parameterized) telescoping problem. A conclusion is given in Section 8.3 . Basic definitions, the outline of the problems, and the main results In this article all rings are commutative with 1 and all rings (resp. fields) have charac-teristic 0; in particular, they contain the rational numbers Q as a subring (resp. subfield).A ring (resp. field) is called computable if there are algorithms available that can performthe standard operations (including zero recognition and deciding constructively if an el-ement is invertible). The multiplicative group of units (invertible elements) of a ring A isdenoted by A ∗ . The ideal generated by S ⊆ A is denoted by h S i . If A is a subring (resp.subfield/multiplicative subgroup) of ˜ A we also write A ≤ ˜ A . The non-negative integersare denoted by N = { , , , . . . } .In this section we will present a general framework in which our symbolic summationproblems can be formulated and tackled in the setting of difference rings. Here an indef-inite nested product-sum expression f ( k ) (like in (1) or (3)) is described in a ring (resp.field) A and the shift behaviour of such an expression is reflected by a ring automorphism(resp. field automorphism) σ : A → A , i.e., σ i ( f ) with i ∈ Z represents the expression f ( k + i ). In the following we call such a ring A (resp. field) equipped with a ring auto-morphism (resp. field automorphism) σ a difference ring (resp. difference field) [16,31]and denote it by ( A , σ ). We remark that any difference field is also a difference ring.Conversely, any difference ring ( A , σ ) with A being a field is automatically a differencefield. A difference ring (resp. field) ( A , σ ) is called computable if both, A and the function σ are computable; note that in such rings one can decide if an element is a constant, i.e.,if σ ( c ) = c . The set of constants is also denoted by const( A , σ ) = { c ∈ A | σ ( c ) = c } , andif it is clear from the context, we also write const A = const( A , σ ). It is easy to check thatconst A is a subring (resp. a subfield) of A which contains as subring (resp. subfield) therational numbers Q . Throughout this article we will take care that const A is always afield (and not just a ring), called the constant field and denoted by K .In the first subsection we introduce the class of difference rings in which we willmodel indefinite nested sums and products. They will be introduced by a tower of ringextensions, the so-called R ΠΣ ∗ -ring extensions.In Subsection 2.2 we will focus on two tasks:(1) Introduce techniques that enable one to test if the given tower of extensions is an R ΠΣ ∗ -extension; even more, derive tactics that enable one to represent sums and prod-ucts automatically in R ΠΣ ∗ -extensions.(2) Work out the underlying subproblems in order to solve two central problems of sym-bolic summation: telescoping (compare (1)) and parameterized telescoping (compare (3)).In their simplest form they can be specified as follows. Problem T for ( A , σ ) . Given a difference ring ( A , σ ) and given f ∈ A . Find , ifpossible, a g ∈ A such that the telescoping (T) equation holds: σ ( g ) − g = f. (8) Problem PT for ( A , σ ) . Given a difference ring ( A , σ ) with constant field K andgiven f , . . . , f n ∈ A . Find , if possible, c , . . . , c n ∈ K (not all c i being zero) and a g ∈ A such that the parameterized telescoping (PT) holds: σ ( g ) − g = c f + · · · + c n f n . (9)4n Subsection 2.3 we will present the main results of theoretical and algorithmic natureto handle these problems, and in Subsection 2.4 we demonstrate how the new summationtheory can be used to represent d’Alembertian solutions in R ΠΣ ∗ -extensions. R ΠΣ ∗ -extensions A difference ring ( ˜ A , ˜ σ ) is a difference ring extension of a difference ring ( A , σ ) if A ≤ ˜ A and ˜ σ | A = σ , i.e., A is a subring of ˜ A and ˜ σ ( a ) = σ ( a ) for all a ∈ A . The definition ofdifference field extensions is the same by replacing the word ring with field. In short (forthe ring and field version) we also write ( A , σ ) ≤ ( ˜ A, ˜ σ ). If it is clear from the context,we do not distinguish anymore between σ and ˜ σ .For the construction of R ΠΣ ∗ -extensions, we start with the following basic properties. Lemma 2.1.
Let A be a ring with α ∈ A ∗ and β ∈ A equipped with a ring automorphism σ : A → A . Let A [ t ] be a polynomial ring and A [ t, t ] be a ring of Laurent polynomials.(1) There is a unique automorphism σ ′ : A [ t ] → A [ t ] with σ ′ | A = σ and σ ′ ( t ) = α t + β .(2) There is a unique automorphism σ ′′ : A [ t, t ] → A [ t, t ] with σ ′′ | A = σ and σ ′′ ( t ) = α t (where σ ′′ ( t ) = α − t ). In particular, if β = 0, σ ′′ | A [ t ] = σ ′ .(3) If A is field and A ( t ) is a rational function field, there is a unique field automorphism σ ′′′ : A ( t ) → A ( t ) with σ ′′′ | A = σ and σ ′′′ ( t ) = α t + β . In particular, σ ′′′ | A [ t ] = σ ′ ;moreover, σ ′′′ | A [ t, /t ] = σ ′′ if β = 0.In summary, let ( A , σ ) be a difference ring and t be transcendental over A . Then we obtainthe uniquely determined difference ring extension ( A [ t ] , σ ) of ( A , σ ) with σ ( t ) = α t + β where α ∈ A ∗ and β ∈ A . In particular, we get the uniquely determined differencering extension ( A [ t, t ] , σ ) of ( A , σ ) with σ ( t ) = α t . Thus for β = 0, we have the chain ofextensions ( A , σ ) ≤ ( A [ t ] , σ ) ≤ ( A [ t, t ] , σ ) . Moreover, if A is a field, we obtain the uniquelydetermined difference field extension ( A ( t ) , σ ) of ( A , σ ) with σ ( t ) = α t + β . Followingthe notions of [14] each of the extensions, i.e., ( A , σ ) ≤ ( A [ t ] , σ ), ( A , σ ) ≤ ( A [ t, t ] , σ ) or( A , σ ) ≤ ( A ( t ) , σ ) are called unimonomial extensions (of polynomial, Laurent polynomialor of rational function type, respectively). Example 2.2. (0) Take the difference field ( Q , σ ) with σ ( c ) = c for all c ∈ Q .(1) Take the unimonomial field extension ( Q ( k ) , σ ) of ( Q , σ ) with σ ( k ) = k + 1: Q ( k ) isa rational function field and σ is extended from Q to Q ( k ) with σ ( k ) = k + 1.(2) Take the unimonomial ring extension ( Q ( k )[ t, t ] , σ ) of ( Q ( k ) , σ ) with σ ( t ) = ( k +1) t : Q ( k )[ t, t ] is a ring of Laurent polynomials with coefficients from Q ( k ) and theautomorphism is extended from Q ( k ) to Q ( k )[ t, t ] with σ ( t ) = ( k + 1) t .Finally, we consider those extensions where the constants remain unchanged. Definition 2.3.
Let ( A , σ ) be a difference ring. • A unimonomial ring extension ( A [ t ] , σ ) of ( A , σ ) with σ ( t ) − t ∈ A and const A [ t ] =const A is called Σ ∗ -ring extension (in short Σ ∗ -extension). • If A is a field, a unimonomial field extension ( A ( t ) , σ ) of ( A , σ ) with σ ( t ) − t ∈ A andconst A ( t ) = const A is called Σ ∗ -field extension. We restrict Karr’s Σ-field extensions to Σ ∗ -field extensions being slightly less general but covering allsums treated explicitly in Karr’s work [24]. A unimonomial ring extension ( A [ t, t ] , σ ) of ( A , σ ) with σ ( t ) t ∈ A ∗ and const A [ t, t ] =const A is called Π-ring extension (in short Π-extension). • If A is a field, a unimonomial field extension ( A ( t ) , σ ) of ( A , σ ) with σ ( t ) t ∈ A ∗ = A ( t ) \ { } and const A ( t ) = const A is called Π-field extension.The generators of a Σ ∗ -extension (in the ring or field version) and a Π-extension (in thering or field version) are called Σ ∗ -monomial and Π-monomial, respectively. Remark 2.4.
Keeping the constants unchanged is a central property to tackle the (pa-rameterized) telescoping problem. E.g., if the constants are extended, there do not existbounds on the degrees as utilized in Subsection 7.1.1. Additionally, introducing no extraconstants is the essential property to embed the derived difference rings into the ring ofsequences; this fact has been worked out, e.g., in [54] which is related to [19].
Example 2.5 (Cont. Ex. 2.2) . For ( Q , σ ) ≤ ( Q ( k ) , σ ) ≤ ( Q ( k )[ t, t ] , σ ) from Example 2.2we have that const Q ( k )[ t, t ] = const Q ( k ) = const Q = Q , which can be checked easily.Thus ( Q ( k ) , σ ) is a Σ ∗ -field extension of ( Q , σ ) and ( Q ( k )[ t, t ] , σ ) is a Π-extension of( Q ( k ) , σ ). The generator k is a Σ ∗ -monomial and the generator t is a Π-monomial.For more complicated extensions it is rather demanding to check if the constants remainunchanged. In this regard, we refer to the field-algorithms given in [24] or to our enhancedring-algorithms given below which can perform these checks automatically.For further considerations we introduce the order function ord: A → N withord( h ) = ( ∄ n > h n = 1min { n > | h n = 1 } otherwise . (10)The third type of extensions is concerned with algebraic objects like (4). Let λ ∈ N with λ >
1, take a root of unity α ∈ A ∗ with α λ = 1 and construct the unimonomialextension ( A [ y ] , σ ) of ( A , σ ) with σ ( y ) = α y . Now take the ideal I := h y λ − i andconsider the quotient ring E = A [ y ] /I . Since I is closed under σ , i.e., I is a reflexivedifference ideal [16, page 71], one can verify that σ : E → E with σ ( f + I ) = σ ( f ) + I forms a ring automorphism. In other words, ( E , σ ) is a difference ring. Moreover, thereis the natural embedding of A into E with a → a + I . By identifying a with a + I , ( E , σ )is a difference ring extension of ( A , σ ). Lemma 2.6.
Let ( A , σ ) be a difference ring and α ∈ A ∗ with α λ = 1 for some λ > A [ x ] , σ ) of ( A , σ ) with x / ∈ A subject to the relations x λ = 1 and σ ( x ) = α x . Proof.
Consider the difference ring extension ( E , σ ) of ( A , σ ) constructed above. Define x := y + I . Then σ ( x ) = α x and x λ = y λ + I = 1 + I = 1. Further, E = { P λ − i =0 a i x i | a i ∈ A } . Thus we obtain a difference ring extension as claimed in the lemma. Now supposethat there is another difference ring extension ( A [ x ′ ] , σ ′ ) of ( A , σ ) with x ′ / ∈ A subjectto the relations σ ′ ( x ′ ) = α x ′ and x ′ λ = 1. Then by the first isomorphism theorem,there is the ring isomorphism τ : E → A [ x ′ ] with τ ( P λ − i =0 f i x i ) = P λ − i =0 f i x ′ i . Since τ ( σ ( x )) = τ ( α x ) = τ ( α ) τ ( x ) = α x ′ = σ ′ ( x ′ ), it follows that τ ( σ ( f )) = σ ′ ( τ ( f )) for all f ∈ A [ x ]. Summarizing, τ is a difference ring isomorphism. ✷ The extension ( A [ x ] , σ ) of ( A , σ ) in Lemma 2.6 is called algebraic extension of order λ .6 xample 2.7. (0) Take the Σ ∗ -ext. ( Q ( k ) , σ ) of ( Q , σ ) with σ ( k ) = k + 1 from Ex. 2.5.(1) Take the algebraic extension ( Q ( k )[ x ] , σ ) of ( Q ( k ) , σ ) with σ ( x ) = − x of order 2: Q ( k )[ x ] is an algebraic ring extension of Q ( k ) subject to the relation x = 1 and σ isextended from Q ( k ) to Q ( k )[ x ] with σ ( x ) = − x . Note that x represents the expression X ( k ) = ( − k with X ( k + 1) = − X ( k ).(2) Take the algebraic extension ( Q ( k )[ x ][ y ] , σ ) of ( Q ( k )[ x ] , σ ) with σ ( y ) = − x y of or-der 2: Q ( k )[ x ][ y ] is a ring extension of Q ( k )[ x ] with y = 1 and σ is extended from Q ( k )[ x ] to Q ( k )[ x ][ y ] with σ ( y ) = − x y . Note that y represents the expression Y ( k ) =( − k +12 ) = Q kj =1 ( − j with Y ( k + 1) = − ( − k Y ( k ).As for unimonomial extensions, we restrict now to those algebraic extensions where theconstants remain unchanged. For the underlying motivation we refer to Remark 2.4. Definition 2.8.
Let λ ∈ N \ { , } . An algebraic extension ( A [ x ] , σ ) of ( A , σ ) order λ with const A [ x ] = const A is called root of unity extension (in short R -extension) of order λ . The generator x is called R -monomial. Example 2.9 (Cont. Ex. 2.7) . For ( Q , σ ) ≤ ( Q ( k ) , σ ) ≤ ( Q ( k )[ x ] , σ ) ≤ ( Q ( k )[ x ][ y ] , σ )from Example 2.7 we have that const Q ( k )[ x ][ y ] = const Q ( k )[ x ] = const Q ( k ) = Q ,which can be checked algorithmically; see Example 2.13 below. Thus ( Q ( k )[ x ] , σ ) is an R -extension of ( Q ( k ) , σ ) and ( Q ( k )[ x ][ y ] , σ ) is an R -extension of ( Q ( k )[ x ] , σ ).To this end, we define a tower of such extensions. First, we introduce the followingnotion. Let ( A , σ ) ≤ ( E , σ ) with t ∈ E . In the following A h t i denotes the polynomial ring A [ t ] if ( A [ t ] , σ ) is a Σ ∗ -extension of ( A , σ ). A h t i denotes the ring of Laurent polynomials A [ t, t ] if ( A [ t, t ] , σ ) is a Π-extension of ( A , σ ). Finally, A h t i denotes the ring A [ t ] with t / ∈ A subject to the relation t λ = 1 if ( A [ t ] , σ ) is an R -extension of ( A , σ ) of order λ . Definition 2.10.
A difference ring extension ( A h t i , σ ) of ( A , σ ) is called R ΠΣ ∗ -extensionif it is an R -extension, Π-extension or Σ ∗ -extension. Analogously, it is called R Σ ∗ -extension, R Π-extension or ΠΣ ∗ -extension if it is one of the corresponding extensions.More generally, ( G h t ih t i . . . h t e i , σ ) is a (nested) R ΠΣ ∗ -extension (resp. R Π, R Σ ∗ , ΠΣ ∗ -, R -, Π-, Σ ∗ -extension) of ( G , σ ) if it is a tower of such extensions.Similarly, if A is a field, ( A ( t ) , σ ) is called a ΠΣ ∗ -field extension if it is either a Π-fieldextension or a Σ ∗ -field extension. ( G ( t ) . . . ( t e ) , σ ) is called a ΠΣ ∗ -field extension (resp.Π-field extension, Σ ∗ -field extension) of ( G , σ ) if it is a tower of such extensions. In par-ticular, if const G = G , ( G ( t )( t ) . . . ( t e ) , σ ) is called a ΠΣ ∗ -field over G .In both, the ring and field version, t i is called R ΠΣ ∗ -monomial (resp. R Π-, R Σ ∗ -, ΠΣ ∗ -monomial) if it is a generator of a R ΠΣ ∗ -extension (resp. R Π-, R Σ ∗ -, ΠΣ ∗ -extension). Example 2.11 (Cont. Ex. 2.9) . (1) ( Q ( k ) , σ ) is a ΠΣ ∗ -field over Q .(2) ( Q ( k ) h x ih y i , σ ) is an R -extension of ( Q ( k ) , σ ).The generators with their sequential arrangement, incorporating the recursive definitionof the automorphism, are always given explicitly. In particular, any reordering of thegenerators must respect the recursive nature induced by the automorphism.7 .2. A characterization of R ΠΣ ∗ -extensions and their algorithmic construction For the construction of R ΠΣ ∗ -extensions we rely on the following result; for the proofsof part 1, part 2 and part 3 we refer to Proof 3.9, Proof 3.16 and Proof 3.22, respectively. Theorem 2.12.
Let ( A , σ ) be a difference ring. Then the following holds.(1) Let ( A [ t ] , σ ) be a unimonomial ring extension of ( A , σ ) with σ ( t ) = t + β where β ∈ A such that const A is a field. Then this is a Σ ∗ -extension (i.e., const A [ t ] = const A )iff there does not exist a g ∈ A with σ ( g ) = g + β .(2) Let ( A [ t, t ] , σ ) be a unimonomial ring extension of ( A , σ ) with σ ( t ) = α t where α ∈ A ∗ . Then this is a Π-extension (i.e., const A [ t, t ] = const A ) iff there are no g ∈ A \ { } and m ∈ Z \ { } with σ ( g ) = α m g . If it is a Π-extension, ord( α ) = 0.(3) Let ( A [ t ] , σ ) be an algebraic ring extension of ( A , σ ) of order λ > σ ( t ) = α t where α ∈ A ∗ . Then this is an R -extension (i.e., const A [ t ] = const A ) iff there areno g ∈ A \ { } and m ∈ { , . . . , λ − } with σ ( g ) = α m g . If it is an R -extension,then α is primitive, i.e., ord( α ) = λ .For Karr’s celebrated field version [24,25] of this result we refer to Theorems 3.11 and 3.18below, that can be nicely embedded in the general difference ring framework. We empha-size that Theorem 2.12 facilitates algorithmic tactics to build difference ring extensionsand to verify simultaneously if they form R ΠΣ ∗ -extensions. Here we consider two cases. R Π -extensions Let ( A , σ ) be a difference ring and let α ∈ A . Then we want to decide if we canconstruct an R Π-extension ( A h t i , σ ) of ( A , σ ) with σ ( t ) = α t . First, we have to checkif α ∈ A ∗ . E.g., for the class of difference rings ( A , σ ), built by simple R ΠΣ ∗ -extensionsintroduced in Definition 2.19 below, this task will be straightforward. Next, we need theorder of α , i.e., we have to solve the following Problem O with G := A ∗ . Problem O in G . Given a group G and α ∈ G . Find ord( α ).Given λ = ord( α ), we can decide which case has to be treated. If λ = 0, only theconstruction of a Π-extension might be possible due to Theorem 2.12. Thus we constructthe unimonomial extension ( A [ t, t ] , σ ) of ( A , σ ) with σ ( t ) = α t . Otherwise, if λ >
0, weconstruct the algebraic extension ( A [ t ] , σ ) of ( A , σ ) with σ ( t ) = α t of order λ . Finally,we check if our construction is indeed a Π-extension or R -extension, i.e., if the constantsremain unchanged. Using Theorem 2.12 this test can be accomplished by solving Problem MT in ( A , σ ) . Given a difference ring ( A , σ ) and α ∈ A ∗ with λ = ord( α ). Decide if there are a g ∈ A \ { } and an m ∈ Z \ { } for the case λ = 0 (resp. m ∈ { , . . . , λ − } for the case λ >
0) such that the multiplicative version of thetelescoping equation (MT) holds: σ ( g ) = α m g. (11)More generally, if we are given a tower of algebraic and unimonomial extensions, whichmodel indefinite nested products, Problem MT can be used to check if the constructionconstitutes a nested R Π-extension. 8 xample 2.13 (Cont. Ex. 2.9) . We will verify that ( Q ( k )[ x ][ y ] , σ ) is an R -extensionof ( Q ( k ) , σ ). (1) Take α = − λ = ord( α ) = 2. We solve Problem MP by thealgorithms presented below: there are no g ∈ Q ( k ) ∗ and m ∈ { } with σ ( g ) = ( − m g .Hence by Theorem Q ( k )[ x ] , σ ) is an R -extension of ( Q ( k ) , σ ).(2) Now we solve Problem O for α = − x and get λ = ord( − x ) = 2; see Example 5.4.(2). Inaddition, solving Problem MP for α shows that there is no g ∈ Q ( k )[ x ] \ { } with σ ( g ) = − x g . Thus by Theorem 2.12.(3) ( Q ( k )[ x ][ y ] , σ ) forms an R -extension of ( Q ( k )[ x ] , σ ). Example 2.14.
We construct a ring in which the objects in (7) can be represented.(0) Take the ΠΣ ∗ -field ( K ( k ) , σ ) over K = Q ( ι ) with σ ( k ) = k + 1.(1) Take α = ι . Then solving Problem O provides λ = ord( α ) = 4. In particular solvingthe corresponding Problem MP proves that there are no g ∈ K ( k ) ∗ and m ∈ { , , } with (11). Hence by Theorem 2.12.(3) we can construct the R -extension ( K ( k )[ x ] , σ ) of( K ( k ) , σ ) with σ ( x ) = ι x . Note that the R -monomial x represents ι k .(2) Take α = x k . Solving Problem O yields λ = ord( α ) = 0 and solving Problem MPshows that there are no g ∈ K ( k )[ x ] \ { } and m ∈ Z \ { } with (11). With Theo-rem 2.12.(2) we can construct the Π-extension ( K ( k )[ x ] , σ ) ≤ ( K ( k )[ x ] h t i , σ ) with σ ( t ) = x k t ; here the Π-monomial t represents Q k − j =1 jι j . Σ ∗ -extensions In order to verify if a unimonomial extension as given in Theorem 2.12.(1) is a Σ ∗ -extension, it suffices to solve Problem T with f = β and to check if there is not atelescoping solution. We illustrate this feature by actually constructing a difference ringin which the summand f ( k ) = ( − k +12 ) k k X j =1 ( − j j (12)given on the left hand side of (6) and the additional sum k X j =1 ( − j +12 ) j (13)occurring on the right hand side of (6) can be represented. In particular, we demonstratehow identity (6) can be discovered in this difference ring. Example 2.15 (Cont. Ex. 2.9) . (0) Take the difference ring ( A , σ ) with A = Q ( k )[ x ][ y ].(1) Take f = σ ( xk ) = − xk +1 . Then solving Problem T shows that there is no g ∈ A with σ ( g ) − g = − xk +1 . Hence we can construct the Σ ∗ -extension ( A [ s ] , σ ) of ( A , σ ) with σ ( s ) = s + − xk +1 ; note that the Σ ∗ -monomial s represents P kj =1 ( − j j .(2) Take f = σ ( yk ) = − x yk +1 . Then solving Problem T shows that there is no g ∈ A [ s ] with σ ( g ) − g = − x yk +1 . Hence we can construct the Σ ∗ -extension ( A [ s ][ S ] , σ ) of ( A [ s ] , σ ) with σ ( S ) = S + − x yk +1 ; note that the Σ ∗ -monomial S represents the sum (13).(3) Take f = y k s which represents (12). Solving Problem T produces the solution g = sy (cid:0) ( k − k + 1) x − ( k − k (cid:1) + y (cid:0) (1 − k ) − x (cid:1) + S ; (14) Note: Theorem 2.12.(3) is a shortcut for “part 3 of Theorem 2.12”. The same convention will be appliedfor other references. R Σ ∗ -monomials x, y, s, S with thecorresponding summation objects. Taking a = 1 in (2) and performing the evaluation c := g (1) = 0 ∈ Q gives the identity (6).(4) Note that we succeeded in representing the sum F ( k ) = P ki =1 f ( i ) with f from (12)in the difference ring in A [ s ][ S ] with σ ( g ) − c = σ ( g ). Namely, replacing the variables in σ ( g ) with the corresponding summation objects yields the right hand side of (6). Thisis of particular interest if there are further sums defined over F ( k ) which one wants torepresent in a Σ ∗ -extension over ( A [ s ][ S ] , σ ).We remark that for the derivation of the identity (6) it is crucial to introduce the extrasum (13). Here this was accomplished manually. But, using algorithms from [53,59] incombination with the results of this article, this sum can be determined automatically. R ΠΣ ∗ -extensions As in the difference field approach [24,49,53,59], Problem T and more generally Prob-lem PT will be solved by reducing them from ( A , σ ) to smaller difference rings (i.e.,rings built by less R ΠΣ ∗ -monomials). Likewise, this reduction technique can be appliedin order to solve a special case of Problem MT that will cover all the cases needed forour difference ring constructions. However, in order to carry out these reductions, onehas to tackle generalized problems within the recursion steps.For Problem MT the following generalization is needed. Let ( A , σ ) be a difference ring,let W ⊆ A and let f = ( f , . . . , f n ) ∈ ( A ∗ ) n . Then we define the set [24] M ( f , W ) := { ( m , . . . , m n ) ∈ Z n | σ ( g ) = f m . . . f m n n g for some g ∈ W \ { }} . In the following, we want to calculate a finite representation of M ( f , A ). If A is a field,i.e., A ∗ = A \ { } , it is immediate that M ( f , A ) is a submodule of Z n over Z and there isa basis of M ( f , A ) with rank ≤ n ; see [24]. In the setting of rings, this result carries overif the set of semi-constants (also called semi-invariants [14]) of ( A , σ ) defined bysconst( A , σ ) = { c ∈ A | σ ( c ) = u c for some u ∈ A ∗ } forms a multiplicative group (excluding the 0 element). Note: if A is a field, we have thatsconst( A , σ ) \ { } = A \ { } = A ∗ . Unfortunately, for a general difference ring the setsconst( A , σ ) \ { } is only a multiplicative monoid [14]. In order to gain more flexibility,we introduce the following refinement. For a given multiplicative subgroup G of A ∗ (inshort G ≤ A ∗ ), we define the set of semi-constants (semi-invariants) of ( A , σ ) over G bysconst G ( A , σ ) = { c ∈ A | σ ( c ) = u c for some u ∈ G } . Note that sconst ( A ∗ ) ( A , σ ) = sconst( A , σ ) and sconst { } ( A , σ ) = const( A , σ ). If it is clearfrom the context, we drop σ and just write sconst G A and sconst A , respectively.Here is one of the main challenges: For all our considerations we will choose G suchthat sconst G A \ { } is a subgroup of A ∗ (in short, sconst A \ { } ≤ A ∗ ). Then with thiscareful choice of G we can summarize the above considerations with the following lemma. Lemma 2.16.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \ { } ≤ A ∗ ;let f ∈ G n . Then M ( f , A ) = M ( f , sconst G A ). In particular, M ( f , A ) is a submodule of Z n over Z , and it has a finite Z -basis with rank ≤ n .10n the light of this property, we can state Problem PMT. Problem PMT in ( A , σ ) for G . Given a difference ring ( A , σ ) with G ≤ A ∗ suchthat sconst G A \ { } ≤ A ∗ holds; given f ∈ G n . Find a Z -basis of M ( f , A ).Observe that Problem MT can be reduced to Problem PMT for a group G with sconst G A \{ } ≤ A ∗ if we restrict to the situation that α ∈ G . More precisely, assume that wehave calculated λ = ord( α ) and succeeded in solving Problem PMT, i.e., we are givena basis of M = M (( α ) , A ) ⊆ Z . If the basis is empty, there cannot be an m ∈ Z \ { } and a g ∈ A \ { } with (11). Otherwise, if the basis is not empty, the rank is 1. Moreprecisely, we obtain m > M = m Z . Hence m is the smallest positive choice suchthat there is a g ∈ A \ { } with (11). Therefore we can again decide Problem MT.For the generalization of Problems T and PT we introduce the following set. Let( A , σ ) be a difference ring with constant field K , let W ⊆ A , and let u ∈ A \ { } and f = ( f , . . . , f n ) ∈ A n . Then we define [24] V ( u, f , ( W, σ )) = { ( c , . . . , c n , g ) ∈ K n × W | σ ( g ) − u g = c f + · · · + c n f n } ;if it is clear from the context, we write V ( u, f , W ) and suppress the automorphism σ .As with Lemma 2.16 the following result will be crucial for further considerations. Lemma 2.17.
Let ( A , σ ) be a difference ring with constant field K and let G ≤ A ∗ withsconst G A \ { } ≤ A ∗ . Let W be a K -subspace of A . Then for f ∈ A n and u ∈ G we havethat V ( u, f , W ) is a K -subspace of K n × W with dim V ( u, f , W ) ≤ n + 1. Proof.
Suppose that there are m linearly independent solutions with m > n + 1, say( c i, , . . . , c i,n , g i ) with 1 ≤ i ≤ m . Then by row operations over the field K we can deriveat least two linearly independent vectors, say v = (0 , . . . , , g ) and v = (0 , . . . , , h ).Hence we have that σ ( g ) = u g and σ ( h ) = u h where g, h ∈ sconst G A \ { } ≤ A ∗ .Consequently, σ ( gh ) = gh , thus c = g/h ∈ K ∗ and therefore v = c v ; a contradictionthat the vectors are linearly independent. ✷ This result gives rise to the following problem specification.
Problem PFLDE in ( A , σ ) for G (with constant field K ). Given a differencering ( A , σ ) with constant field K and G ≤ A ∗ such that sconst G A \ { } ≤ A ∗ holds;given u ∈ G and f ∈ A n . Find a K -basis of V ( u, f , A ).In particular, if we can solve Problem PFLDE in ( A , σ ) for G , it follows with 1 ∈ G thatwe can solve Problem T and PT in ( A , σ ). Furthermore, we can solve the multiplicativeversion of telescoping: if α ∈ G , we can determine a g ∈ A \ { } , in case of existence,such that σ ( g ) = α g holds. This feature is illustrated by the following example. Note that this restriction, in particular the choice of G , is fundamental: it is the essential step tospecify the type of products that one can handle algorithmically; see Definition 2.19. Note: If λ := ord( α ) >
0, we have that λ ∈ M , i.e., the rank of M is 1. In particular, we can constructan R -extension ( A , σ ) ≤ ( A [ t ] , σ ) with σ ( t ) = α t iff λ = m > xample 2.18 (Cont. Ex. 2.14) . Given Q ( b ) = Q bk =1 − ( ι k )1+ k on the left hand side of (7),we want to rewrite it in terms of the product P ( b ) = Q b − j =1 jι j . In a preparation stepwe constructed already the R ΠΣ ∗ -extension ( K ( k )[ x ] h t i , σ ) of ( K ( k ) , σ ) with K = Q ( ι ), σ ( x ) = ι x and σ ( t ) = k x t in Example 2.14. There we can represent − ( ι k ) k +1 with u = − xk +1 and P ( k ) with t . Now we search for a g ∈ K ( k )[ x ] h t i \ { } such that σ ( g ) = u g holds.More precisely, we are interested in a basis of V = V ( u, (0) , K ( k )[ x ] h t i ). Activating ourmachinery, we get the basis { (0 , g ) , (1 , } of V with g = x ( ι + x ) k t − . For the chosen group G with u ∈ G , that we use to solve the underlying Problem PFLDE in ( K ( k )[ x ] h t i , σ ), andthe corresponding calculation steps we refer to Example 7.6 below. Since g is a solutionof σ ( g ) = u g , g ( k ) = (cid:0) ι + ( − k (cid:1) ι k k P ( k ) − is a solution of − ι k k +1 = g ( k +1) g ( k ) . Hence by thetelescoping trick we get Q bk =1 − ι k k +1 = g ( b +1) g (1) which produces (7). Suppose that we are given a difference ring ( G , σ ) which is computable and we aregiven a group G ≤ G ∗ with sconst G G \ { } ≤ G ∗ . In this article we will restrict to certainclasses of R ΠΣ ∗ -extensions ( E , σ ) of ( G , σ ) equipped with a group ˜ G with G ≤ ˜ G ≤ E ∗ and sconst ˜ G E \ { } ≤ E ∗ such that we can derive the following algorithmic machinery:(1) Problem O in ˜ G can be reduced to Problem O in G ;(2) Problem PMT in ( E , σ ) for ˜ G can be reduced to Problem PMT in ( G , σ ) for G ;(3) Problem PFLDE in ( E , σ ) for ˜ G can be reduced to Problem PFLDE in ( G , σ ) for G (see Subsection 2.3.1) or to Problem PFLDE in ( G , σ k ) for G for all k ≥ G , σ ) and a group G ≤ G ∗ inwhich we can solve Problem O in G and Problems PMT and PFLDE in ( G , σ ) for G (resp.( G , σ k ) for G for all k ≥ E , σ ) and larger group ˜ G .As it turns out, we will succeed in this task for a subclass of R ΠΣ ∗ -extensions ( G , σ ) ≤ ( E , σ ) and a properly chosen group ˜ G ≤ E ∗ that can treat all objects (among the generalclass of R ΠΣ ∗ -extensions) that the author has encountered in practical problem solvingso far. More precisely, we will restrict to simple R ΠΣ ∗ -extensions.Let ( G h t i . . . h t e i , σ ) be a R ΠΣ ∗ -extension of ( G , σ ) and let G ≤ G ∗ . Then we define G EG = { g t m . . . t m e e | h ∈ G and m i ∈ Z where m i = 0 if t i is a Σ ∗ -monomial } . (15)It is easy to see that ˜ G = G EG forms a group. More precisely, we obtain the following chainof subgroups: G ≤ G EG ≤ E ∗ . We call G EG also the product-group over G for the R ΠΣ ∗ -extension ( E , σ ) of ( G , σ ). We are now ready to define ( G –)simple R ΠΣ ∗ -extensions. Definition 2.19.
Let ( G , σ ) be a difference ring and let G ≤ G ∗ be a group. An R ΠΣ ∗ -extension ( E , σ ) of ( G , σ ) with E = G h t ih t i . . . h t e i is called G -simple if for any R Π-monomial t i we have that σ ( t i ) /t i ∈ G EG . Moreover, an R Π, R Σ ∗ , ΠΣ ∗ -, R -, Π-, andΣ ∗ -extension of ( G , σ ) is G -simple if it is a G -simple R ΠΣ ∗ -extension. We call any suchextension simple if it is G ∗ -simple. Analogously, we call an R Π, R Σ ∗ , ΠΣ ∗ -, R -, Π-, andΣ ∗ -monomial G -simple (resp. simple) if the extension is G -simple (resp. simple).In all our examples the difference rings have been built by a simple R ΠΣ ∗ -extension ( E , σ )of ( G , σ ) where ( G , σ ) is a ΠΣ ∗ -field ( K ( k ) , σ ) over K with σ ( k ) = k + 1. In particular,12he Problems PMT and PFLDE have been considered for the constructed ( E , σ ) in G =( K ( k ) ∗ ) EK ( k ) . Before we finally turn to the class of simple R ΠΣ ∗ -extensions, we presentone example which cannot be treated properly with our toolbox under consideration. Example 2.20.
Take ( Q ( k )[ t, t ] , σ ) from Example 2.5 with σ ( k ) = k + 1 and σ ( t ) =( k + 1) t . Subsequently, we will use our notation Q ( k ) h t i = Q ( k )[ t, t ]. Then we canconstruct the R -extension ( Q ( k ) h t i [ x ] , σ ) of ( Q ( k ) h t i , σ ) with σ ( x ) = − x of order 2. Inthis ring we are given the idempotent elements e = (1 − x ) / e = ( x + 1) / e = e and e = e . Finally take α = e + e t . Then observe that α · ( e + e /t ) = 1,i.e., α ∈ Q ( k ) h t i [ x ] ∗ . Note that ord( α ) = 0. Otherwise it would follow that e λ = 0with λ = ord( α ) >
0; a contradiction that e is idempotent. Consequently, T cannotbe an R -extension, and we construct the unimonomial extension ( Q ( k ) h t i [ x ][ T, T ] , σ ) of( Q ( k ) h t i [ x ] , σ ) with σ ( T ) = α T . It seems non-trivial to derive an (algorithmic) proof (ordisproof) that T is a Π-monomial, and it would be nice to see a solution to this problem.Summarizing, we aim at solving Problems PMT and PFLDE in a G -simple R ΠΣ ∗ -extension ( G , σ ) ≤ ( E , σ ) for ˜ G = G EG , and we want to solve Problem O in ˜ G . In order toaccomplish this task, we will restrict ourselves further to the following two situations. R ΠΣ ∗ -extensions In most applications R -extensions are not nested, e.g., only objects like ( − k arise.In addition, such objects do not occur in transcendental products, but only in sums, likecyclotomic sums [3] or generalized harmonic sums [4]. A formal definition of this special,but very practical oriented class of R ΠΣ ∗ -extensions is as follows. Definition 2.21. An R ΠΣ ∗ -extension ( E , σ ) of ( G , σ ) is called single-rooted if the gen-erators of the extension can be reordered to E = G h t i . . . h t r ih x i . . . h x u ih s i . . . h s v i , (16)respecting the recursive nature of the automorphism, such that the t i are Π-monomials,the x i are R -monomials with σ ( x i ) /x i ∈ G ∗ and the s i are Σ ∗ -monomials.Given this class of single-rooted and simple R ΠΣ ∗ -extension, we will show the followingtheorem in Proof 4.8. Theorem 2.22.
Let ( G , σ ) be a difference ring and let G ≤ G ∗ with sconst G G \{ } ≤ G ∗ .Let ( E , σ ) be a simple and single-rooted R ΠΣ ∗ -extension of ( G , σ ) with (16) as specifiedin Definition 2.21, and let ˜ G = G G h t i ... h t r i G . Then sconst ˜ G E \ { } ≤ E ∗ withsconst ˜ G E = { h t m . . . t m r r x n . . . x n u u | h ∈ sconst G G , m i ∈ Z and n i ∈ N } . In particular, we obtain the following reduction algorithms summarized in Theorem 2.23;for a proof of part 1 see Proof 6.7 and of part 2 see Proof 7.10.
Theorem 2.23.
Let ( G , σ ) be a computable difference ring with G ≤ G ∗ and sconst G G \{ } ≤ G ∗ . Let ( E , σ ) be a single-rooted and G -simple R ΠΣ ∗ -extension of ( G , σ ) with (16)as given in Definition 2.21, and let ˜ G = G G h t i ... h t r i G . Then the following holds. Note: If G is a field, any single-rooted R ΠΣ ∗ -extension is simple by Corollary 4.15. E , σ ) for ˜ G if it is solvable in ( G , σ ) for G .(2) Problem PFLDE is solvable in ( E , σ ) for ˜ G if Problems PFLDE and PMT aresolvable in ( G , σ ) for G and if Problem O is solvable in G .All the calculations in [44,39,50,33,11,2,1] rely precisely on this machinery. For one ofthe most important applications we refer to Subsection 2.4. R ΠΣ ∗ -extensions of a strong constant-stable difference field In the following we restrict to simple R ΠΣ ∗ -extensions where the ground domain G = F is a field. In this setting, the semi-constants form a multiplicative group. Moreprecisely, we will show the following result in Proof 4.11. Theorem 2.24.
Let ( E , σ ) be a simple R ΠΣ ∗ -extension of a difference field ( F , σ ) andconsider its product-group ˜ G = ( F ∗ ) EF . Then sconst ˜ G E \ { } ≤ E ∗ .For a solution of Problems PMT and PFLDE we require in addition that ( F , σ ) is strongconstant-stable. Definition 2.25.
A difference ring ( A , σ ) with constant field K is called constant-stableif for all k > A , σ k ) = K . It is called strong constant-stable if it isconstant-stable and any root of unity of A is in K .In this setting we can treat products over roots of unity from K and, more generally,products that are built recursively over such products; for examples see (4) and forfurther (algorithmic) properties see Corollary 5.6 below. More precisely, given such atower of R ΠΣ ∗ -extensions, we can solve Problems PMT and PFLDE as follows; for theproofs, resp. the underlying algorithms, of part 1 see Proof 5.7, of part 2 see Proof 6.15and of part 3 see Proof 7.16. Theorem 2.26.
Let ( F , σ ) be a computable difference field where Problem O is solvablein (const F ) ∗ . Let ( E , σ ) be a simple R ΠΣ ∗ -extension of ( F , σ ). Then the following holds.(1) Problem O is solvable in ( F ∗ ) EF .If ( F , σ ) is in addition strong constant-stable, then(2) Problem PMT is solvable in ( E , σ ) for ( F ∗ ) EF if it is solvable in ( F , σ ) for F ∗ ;(3) Problem PFLDE is solvable in ( E , σ ) for ( F ∗ ) EF if Problem PMT is solvable in ( F , σ )for F ∗ and Problem PFLDE is solvable in ( F , σ k ) for F ∗ for all k > Instead of Problem O it suffices if know the orders of all the R -monomials in ( G , σ ) ≤ ( E , σ ). We emphasize that we will always work with the automorphism σ during the reduction process. Onlyin the base cases we might face the problem to solve instances of Problem PFLDE in ( F , σ k ) with k > σ k for some k > F , σ ) is built only by few summation objects. Then the typical phenomenonof the expression swell in symbolic summation due to σ k is prevented as much as possible. .3.3. A complete machinery: algorithms for the ground difference rings Both, Theorems 2.23 and 2.26 provide algorithms to reduce the Problems PMT andPFLDE (and thus the Problems T, PT and special cases of Problem MT) from an R ΠΣ ∗ -extension ( E , σ ) of ( G , σ ) to the ground difference ring ( G , σ ). Theorem 2.23 requiresless conditions on ( G , σ ), but considers only single–rooted R ΠΣ ∗ -extensions, whereasTheorem 2.26 requires more properties on ( G , σ ) but allows nested R -extensions whichare of the type as given in Corollary 5.6 below. Note that the algorithms for the lattercase are more demanding, in particular, one has to solve Problem PFLDE in ( G , σ k ) with k > k = 1 only.We emphasize that both theorems are applicable for a rather general class of differencefields ( G , σ ). Namely, ( G , σ ) itself can be a ΠΣ ∗ -field extension of ( H , σ ) where certainproperties in the difference field ( H , σ ) hold. Here the following remarks are in place.(a) By [24] a ΠΣ ∗ -field extension ( G , σ ) of ( H , σ ) is constant-stable if ( H , σ ) is constant-stable. In particular, if we are given a root of unity from G , it cannot depend on transcen-dental elements and is therefore from H . Thus ( G , σ ) is strong constant-stable if ( H , σ )is strong constant-stable.(b) It has been shown in [28] that one can solve Problem PMT in ( G , σ ) for G ∗ andProblem PFLDE in ( G , σ k ) for G ∗ for k > H , σ ). Among others (see Def. 1 and 2 in [28]) Problem PMT must be solvable in( H , σ ) for H ∗ and Problem PFLDE must be solvable in ( H , σ k ) for H ∗ .Summarizing, if we are given the tower of extensions( H , σ ) ΠΣ ∗ -field ext. ≤ ( G , σ ) R ΠΣ ∗ -ring ext. ≤ ( E , σ )where ( H , σ ) is strong constant-stable and the properties given in Def. 1 and 2 of [28]hold in ( H , σ ), then we can solve Problems PMT and PFLDE in ( E , σ ) for ( G ∗ ) EG .So far, the required properties have been verified and the necessary algorithms havebeen worked out for the following difference fields ( H , σ ) with constant field K .(1) K = H , i.e., ( G , σ ) is a ΠΣ ∗ -field over K ; here the constant field K can be a rationalfunction field over an algebraic number field; see [48, Theorem 3.5].(2) ( H , σ ) is a free difference field, i.e., H = K ( . . . , x − , x , x , . . . ) with σ ( x i ) = x i +1 ;here K is of the type as given in case (1). Note that in this field one can modelunspecified sequences; see [28,27].(3) ( H , σ ) can be a radical difference field representing objects like d √ k ; see [29].For simplicity, all our examples are chosen from case (1). More precisely, we always takethe ΠΣ ∗ -field ( H , σ ) = ( K ( k ) , σ ) over K ∈ { Q , Q ( ι ) } with σ ( k ) = k + 1. R ΠΣ ∗ -extensions We illustrate how an important class of d’Alembertian solutions [7], a subclass ofLiouvillian solutions [20,38], of a given linear difference operator, can be representedcompletely automatically in R ΠΣ ∗ -extensions. In order to obtain the d’Alembertian so-lutions, one starts as follows: first the linear difference operator is factored as much aspossible into linear right hand factors. This can be accomplished, e.g., with the algorithmsfrom [36,21,22] or, within the setting of ΠΣ ∗ -fields with the algorithms given in [6] whichare based on [14,42,49]. The latter machinery is available within the summation package Sigma . Then given this factored form of the operator, the d’Alembertian solutions canbe read off. They can be given by a finite number of hypergeometric expressions and15ndefinite nested sums defined over such expressions. More precisely, each solution is ofthe form k X i = λ h ( i ) i X i = λ h ( i ) · · · i r − X i r = λ r − h r ( i r ) (17)where λ i ∈ N and the hypergeometric expression h i ( k ) can be written in the form Q kj = λ i α i ( j ) with α i ( z ) being a rational function from K ( z ).Subsequently, we restrict ourselves to a field K which is a rational function field K = Q ( n , . . . , n r ) over the rational numbers. Now take the ΠΣ ∗ -field ( K ( k ) , σ ) over K with σ ( k ) = k + 1. Then the solutions, all being of the form (17), can be represented in asingle-rooted simple R ΠΣ ∗ -extension as follows.(1) In [48, Section 6] an algorithm has been presented that calculates a single-rootedsimple R Π-extension ( G , σ ) of ( K ( k ) , σ ) in which all hypergeometric expressions occurringin the d’Alembertian solutions are explicitly represented.(2) Then the challenging task is to construct a Σ ∗ -extension of ( G , σ ) and to representthere the arising sums of the d’Alembertian solutions. Given ( G , σ ) from step 1, thiscan be accomplished by applying iteratively Theorem 2.12.(1). Suppose we representedalready an inner summand in a Σ ∗ -extension ( A , σ ) of ( G , σ ) with β ∈ A . Since ( A , σ ) isa simple R ΠΣ ∗ -extension of ( K ( k ) , σ ) and ( K ( k ) , σ ) is a ΠΣ ∗ -field over K , we can solveProblem T with f = β by using the underlying algorithm of Theorem 2.23 in combinationwith the base case algorithms; see Subsection 2.3.3. If we find a g ∈ A with σ ( g ) = g + β ,we can represent the sum under consideration with g + c where c ∈ K is determined bythe boundary condition (lower summation bound) of the given sum; for further detailswe refer to Example 2.15.(4). Otherwise, we construct the Σ ∗ -extension ( A [ t ] , σ ) of ( A , σ )with σ ( t ) = t + β by Theorem 2.12.(1) and we succeeded in representing the sum underconsideration by t with the appropriate shift behaviour. Note that ( A [ t ] , σ ) is again asingle-rooted simple R ΠΣ ∗ -extension of ( K ( k ) , σ ). Proceeding iteratively, all the nestedhypergeometric sums are represented in terms of an R ΠΣ ∗ -extension over ( K ( k ) , σ ).Exactly this difference ring machinery is implemented in Sigma and has been used totackle challenging applications, like [44,39,50,33,11,2,1] mentioned already in the intro-duction. In particular, this toolbox has been combined with the algorithms worked outin [45,48,51,53,8,59] in order to find representations of d’Alembertian solutions with cer-tain optimality properties, like minimal nesting depth. For a recent summary of all thesefeatures (unfortunately, in the setting of difference fields) we refer to [57,58].
3. Single nested R ΠΣ ∗ -extensions This section delivers relevant properties of single nested R ΠΣ ∗ -extensions. The charac-terization of R ΠΣ ∗ -extensions (Theorem 2.12) will be elaborated. In addition, propertiesof the semi-constants within R ΠΣ ∗ -extensions are derived to gain further insight in thenature of R ΠΣ ∗ -extensions and to prove Theorems 2.22 and 2.24 in Section 4.We start with some general properties which will be essential throughout this article. Definition 3.1.
A ring A is called reduced if there are no non-zero nilpotent elements,i.e., for any f ∈ A \ { } and any n > f n = 0. A is called connected if 0and 1 are the only idempotent elements, i.e., for any f ∈ A \ { , } we have that f = f .16amely, we rely on the following ring properties. A polynomial P ni =0 a i x i ∈ A [ t ] withcoefficients from a ring A is invertible if and only if a ∈ A ∗ and a i with i ≥ A [ t ] ∗ = A ∗ . Besides, there is a complete characterization of invertible elementsin the ring of Laurent polynomials A [ t, t ] presented in [23, Theorem 1] (see also [32]).Based on this work we extract the following crucial result. Lemma 3.2.
Let A be a commutative ring with 1. If A is reduced, then A [ t ] ∗ = A ∗ . If A is reduced and connected, then A [ t, t ] ∗ = { u t r | u ∈ A ∗ and r ∈ Z } . Since our rings are usually not connected, Lemma 3.2 can be applied only partially.
Example 3.3.
The generators in the ring given in Example 2.20 can be reordered to Q ( k )[ x ] h t i . Since Q ( k )[ x ] has the idempotent elements e , e , it is not connected. There-fore we get relations such as (e + e t )( e + e t ) = 1 which are predicted in [23,32].Subsequently, we enumerate further definitions and properties in difference rings andfields that will be used throughout the article. Let ( A , σ ) be a difference ring. The risingfactorial (or σ -factorial) of f ∈ A ∗ to k ∈ Z is defined by f ( k,σ ) = f σ ( f ) . . . σ k − ( f ) if k >
01 if k = 0 σ − ( f − ) σ − ( f − ) . . . σ k ( f − ) if k < . If the automorphism is clear from the context, we also will write f ( k ) instead of f ( k,σ ) .We will rely on the following simple identities (compare also [25, page 307]). The proofsare omitted to the reader. Lemma 3.4.
Let ( A , σ ) be a difference ring, f, h ∈ A ∗ and n, m ∈ Z . Then:(1) ( f h ) ( n ) = f ( n ) h ( n ) .(2) f ( n + m ) = σ n ( f ( m ) ) f ( n ) .(3) f ( n m ) = ( f ( n,σ ) ) ( m,σ n ) .(4) If σ ( h ) = f h , then σ n ( h ) = f ( n ) h .(5) σ k ( f ) ∈ A ∗ and f ( n ) ∈ A ∗ .Let A h t i be a ring of (Laurent) polynomials. For f = P i f i t i ∈ A h t i we definedeg( f ) = ( max { i | f i = 0 } if f = 0 −∞ if f = 0 and ldeg( f ) = ( min { i | f i = 0 } if f = 0 ∞ if f = 0 . In addition, for a, b ∈ Z we introduce the set of truncated (Laurent) polynomials by A h t i a,b = { b X i = a f i t i | f i ∈ A } . (18)We conclude this part with the following two lemmas. Lemma 3.5.
Let ( A h t i , σ ) be a unimonomial ring extension of ( A , σ ) of (Laurent) poly-nomial type. Then for any k ∈ Z and f ∈ A h t i we have that deg( σ k ( f )) = deg( f ).17 roof. Let f = P i f i t i . If f = 0, σ k ( f ) = 0 and thus with deg(0) = −∞ the statementholds. Otherwise, let m := deg( f ) ∈ Z . Then note that σ k ( f ) = P i σ k ( f i )( σ k ( t )) i , i.e., t m is the largest possible monomial in σ k ( f ) with the coefficient h := α m ( k ) σ k ( f m ). Since σ k ( f m ) = 0 and α ( k ) ∈ A ∗ by Lemma 3.4.(5), the coefficient h is non-zero. ✷ Lemma 3.6.
Let ( F ( t ) , σ ) be a unimonomial field extension of ( F , σ ), and let p, q ∈ F [ t ] ∗ with gcd( p, q ) = 1 and k ∈ Z . Then the following holds.(1) If p | q then σ k ( p ) | σ k ( q ).(2) gcd( σ k ( p ) , σ k ( q )) = 1.(3) σ ( p/q ) p/q ∈ F if and only if σ ( p ) /p ∈ F and σ ( q ) /q ∈ F . Proof. (1) If p | q , i.e., p w = q for some w ∈ F [ t ] \ { } , then σ k ( p ) = σ k ( w ) σ k ( q ),and thus σ k ( p ) | σ k ( q ). (2) Suppose that 1 = gcd( σ k ( p ) , σ k ( q )) =: u ∈ F [ t ] \ F . Then σ − k ( u ) ∈ F [ t ] \ F . Since σ − k ( u ) | p and σ − k ( u ) | q by part 1 of the lemma, gcd( p, q ) = 1,a contradiction to the assumption. (3) The implication ⇐ is immediate. Suppose that u := σ ( p/q ) / ( p/q ) ∈ F , i.e., σ ( p ) q = u p σ ( q ). By part 2 of the lemma, σ ( p ) | p and p | σ ( p ) which implies that σ ( p ) /p ∈ F . Analogously, it follows that σ ( q ) /q ∈ F . ✷ Σ ∗ -extensions The essence of all the properties of Σ ∗ -extensions is contained in the following lemma. Lemma 3.7.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \ { } ≤ A ∗ .Let ( A [ t ] , σ ) be a unimonomial ring extension of ( A , σ ) with σ ( t ) = t + β for some β ∈ A .If there are a u ∈ G and a g ∈ A [ t ] with deg( g ) ≥ σ ( g ) − u g ) < deg( g ) − γ ∈ A with σ ( γ ) − γ = β . Proof.
Let g = P ni =0 g i t i ∈ A [ t ] with deg( g ) = n ≥ u ∈ G as stated in thelemma, and define f = σ ( g ) − u g ∈ A [ t ]. With (19) it follows that f = P n − i =0 f i t i .Thus comparing the n th and ( n − P n − i =0 f i t i = f = σ ( g ) − ug = P ni =0 σ ( g i )( t + β ) i − u P ni =0 g i t i and using ( t + β ) i = P ij =0 (cid:0) ij (cid:1) t i − j β j for 0 ≤ i ≤ n yield σ ( g n ) − ug n = 0 and σ ( g n − ) + σ ( g n ) (cid:0) n (cid:1) β − ug n − = 0 . The first equation shows that g n ∈ sconst G A \{ } ≤ A ∗ . Hence we get u = σ ( g n ) /g n . Sub-stituting u for σ ( g n ) /g n in the second equation gives σ ( g n − ) − σ ( g n ) g n g n − = − nβσ ( g n ) . Dividing this equation by − n σ ( g n ) ∈ A ∗ yields σ ( γ ) − γ = β with γ := − g n − ng n ∈ A . ✷ Lemma 3.7 leads to the following equivalent properties of Σ ∗ -extensions. Lemma 3.8.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \ { } ≤ A ∗ .Let ( A [ t ] , σ ) be a unimonomial ring extension of ( A , σ ) with σ ( t ) = t + β for some β ∈ A .Then the following statements are equivalent.(1) There is a g ∈ A [ t ] \ A and u ∈ G with σ ( g ) = u g .(2) There is a g ∈ A with σ ( g ) = g + β .(3) const A [ t ] ) const A . roof. (1) ⇒ (2): Let g ∈ A [ t ] \ A , u ∈ G with σ ( g ) = u g . Since deg( g ) ≥ σ ( g ) − u g ) < ≤ deg( g ) −
1, there is a γ ∈ A with σ ( γ ) = γ + β by Lemma 3.7.(2) ⇒ (3): Let g ∈ A with σ ( g ) = g + β . Since σ ( t ) = t + β , it follows that σ ( t − g ) = ( t − g ),i.e., t − g ∈ const A [ t ]. Since t − g / ∈ A , t − g / ∈ const A .(3) ⇒ (1): Suppose that const A ( const A [ t ] and take g ∈ const A [ t ] \ const A . Then σ ( g ) = u g with u = 1 ∈ G . Thus the lemma is proven. ✷ As a consequence we can now establish the characterization theorem of Σ ∗ -extensions. Proof 3.9. (Theorem 2.12.(1)).
For G = { } we have that sconst G A = const A = K .By assumption K is a field and thus sconst G A \ { } ≤ A ∗ . Therefore we can applyLemma 3.8 and its equivalence (2) ⇔ (3) establishes Theorem 2.12.(1). ✷ In order to rediscover the difference field version from [24,25], we specialize Lemma 3.8to difference fields by exploiting Lemma 3.6.(3).
Lemma 3.10.
Let ( F ( t ) , σ ) be a unimonomial field extension of ( F , σ ) with σ ( t ) = t + β for some β ∈ F . Then the following statements are equivalent.(1) There is a g ∈ F ( t ) \ F with σ ( g ) g ∈ F .(2) There is a g ∈ F with σ ( g ) = g + β .(3) const F ( t ) ) const F . Proof. (1) ⇒ (2): Let g ∈ F ( t ) \ F with σ ( g ) /g ∈ F . Write g = pq with p, q ∈ F [ t ] ∗ andgcd( p, q ) = 1. By Lemma 3.6, σ ( p ) /p ∈ F and σ ( q ) /q ∈ F . Since g / ∈ F , we have that p / ∈ F or q / ∈ F . Thus there is a g ′ ∈ F [ t ] with deg( g ′ ) ≥ σ ( g ′ ) − g ′ ) = deg(0) = −∞ < ≤ deg( g ′ ) −
1. Hence by Lemma 3.7 there is a γ ∈ F with σ ( γ ) = γ + β .(2) ⇒ (3) follows by Lemma 3.8. (3) ⇒ (1) is analogous to the proof of Lemma 3.8. ✷ Note that the above lemma is contained in Karr’s work by combining Theorems 2.1and 2.3 from [25]. As a consequence, we obtain the following result.
Theorem 3.11.
Let ( F ( t ) , σ ) be a unimonomial field extension of ( F , σ ) with σ ( t ) = t + β for some β ∈ F . Then this is a Σ ∗ -extension iff there is no g ∈ F with σ ( g ) = g + β .By the equivalence (3) ⇔ (1) of Lemma 3.8 we obtain the following result concerningthe semi-constants. Theorem 3.12.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ .If ( A [ t ] , σ ) is a Σ ∗ -extension of ( A , σ ), then sconst G A [ t ] = sconst G A .Furthermore, if we specialize to G = A [ t ] ∗ and assume that A is reduced, we get Theo-rem 3.14. For its proof given below we use in addition the following lemma. Lemma 3.13.
Let ( A , σ ) be a difference ring: sconst A \ { } ≤ A ∗ iff sconst A \ { } = A ∗ . Proof.
Suppose that sconst A \ { } ≤ A ∗ . If a ∈ A ∗ , then σ ( a ) ∈ A ∗ . Thus u := σ ( a ) a ∈ A ∗ .With σ ( a ) = u a it follows that a ∈ sconst A \ { } . Hence sconst A \ { } ⊇ A ∗ and withsconst A \ { } ≤ A ∗ we have sconst A \ { } ⊆ A ∗ . The other implication is immediate. ✷ Theorem 3.14.
Let ( A [ t ] , σ ) be a Σ ∗ -extension of ( A , σ ) where A is reduced and sconst A \{ } ≤ A ∗ . Then sconst A [ t ] \ { } = sconst A \ { } = A ∗ .19 roof. By Lemma 3.13 it follows that sconst A \ { } = A ∗ . Since A is reduced, A [ t ] ∗ = A ∗ by Lemma 3.2 and thus sconst A [ t ] = sconst A [ t ] ∗ A [ t ] = sconst A ∗ A [ t ]. Now take G = A ∗ and apply Theorem 3.12. Hence sconst A ∗ A [ t ] = sconst A ∗ A = sconst A . ✷ Π -extensions Analogously to Lemma 3.7 we obtain by coefficient comparison the following lemma.
Lemma 3.15.
Let ( A [ t, t ] , σ ) be a unimonomial ring extension of ( A , σ ) with α = σ ( t ) t ∈ A ∗ ; let u ∈ A and g = P ni =0 g i t i ∈ A [ t, t ]. If σ ( g ) = u g, then σ ( g i ) = u α − i g i for all i .Now we are in the position to obtain the characterization theorem of Π-extensions. Proof 3.16. (Theorem 2.12.(2)). “ ⇐ ”: Let m ∈ Z \ { } and g ∈ A \ { } with σ ( g ) = α m g . Since σ ( t m ) = α m t m , it follows that σ ( g/t m ) = g/t m , i.e., g/t m ∈ const A [ t, t ].Clearly g/t m / ∈ A which implies that g/t m / ∈ const A .“ ⇒ ”: Let g = P i g i t i ∈ A [ t, t ] \ A such that σ ( g ) = g . Thus g m = 0 for some m = 0. ByLemma 3.15 we have that σ ( g m ) = α − m g m .Suppose that t is a Π-monomial, but ord( α ) = n >
0. Then σ ( t n ) = α n t n = t n , which isa contradiction to the first part of the statement. ✷ Requiring in addition that the semi-constants form a group, this result can be sharpened.
Theorem 3.17.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ .Let ( A [ t, t ] , σ ) be a unimonomial extension of ( A , σ ) with σ ( t ) = α t for some α ∈ G .Then this is a Π-extension iff there are no g ∈ sconst G A \{ } and m > σ ( g ) = α m g . Proof. ⇒ : Suppose that t is not a Π-monomial. Then we can take g ∈ A \ { } and m ∈ Z \ { } with σ ( g ) = α m g . Hence g ∈ sconst G A \ { } ≤ A ∗ . Thus if m <
0, we get σ (˜ g ) = α − m ˜ g with ˜ g = g ∈ A ∗ . The other direction is immediate by Theorem 2.12.(2). ✷ Together with Lemma 3.6 we rediscover Karr’s field version; see [25, Theorem 2.2]
Theorem 3.18.
Let ( F ( t ) , σ ) be a unimonomial field extension of ( F , σ ) with α = σ ( t ) t ∈ F ∗ . Then this is a Π-extension iff there are no g ∈ F ∗ and m > σ ( g ) = α m g . Proof.
The direction from right to left follows by Theorem 2.12.(2) and the fact thatany Π-field extension is a Π-ring extension. Now let g ∈ F ( t ) \ F with σ ( g ) = g . Write g = p/q with p, q ∈ F ( t ) where gcd( p, q ) = 1 and q is monic. W.l.o.g. suppose thatdeg( q ) ≥ deg( p ) (otherwise take 1 /g instead of g ). By Lemma 3.6, σ ( p ) /p ∈ F and σ ( q ) /q ∈ F . (20)We consider two cases. First suppose that p ∈ F ∗ and q = t m with m >
0. Then pt m = g = σ ( g ) = σ ( p ) α m t m which implies that σ ( p ) = α m p . What remains to consider is thecase that p / ∈ F or q = t m for some m >
0. Define a := ( p if q = t m for some m > q otherwise . The following holds. 201) a ∈ F [ t ] \ F : If a = q , note that q / ∈ F by deg( p ) ≤ deg( q ) and p/q / ∈ F ; if a = p , q = t m and hence p / ∈ F by assumption.(2) u := σ ( a ) /a ∈ F ∗ by (20).(3) a = ut m for all u ∈ F ∗ and m > a could be only of this form, if q = t m for some m >
0. Hence a = p . But since gcd( p, q ) = 1, t ∤ p .By the properties (1) and (3), it follows that a = P ni = k a i t i with a k = 0 = a n where n > k ≥
0. Property (2) and Lemma 3.15 yield σ ( a k ) = uα k a k and σ ( a n ) = uα n a n whichimplies σ ( a k a n ) = α n − k a k a n . Since a k a n ∈ F ∗ and n − k >
0, the theorem is proven. ✷ Finally, we characterize the set of semi-constants for Π-extensions.
Proposition 3.19.
Let ( A , σ ) be a difference ring with G ≤ A ∗ and sconst G A \ { } ≤ A ∗ . Let ( A [ t, t ] , σ ) be Π-extension of ( A , σ ) with σ ( t ) = α t for some α ∈ G . Thensconst G A [ t, t ] = { h t m | h ∈ sconst G A and m ∈ Z } and sconst G A [ t, t ] \ { } ≤ A [ t, t ] ∗ . Proof. “ ⊆ ”: Let g ∈ sconst G A [ t, t ], i.e., g = P i g i t i ∈ A [ t, t ] with σ ( g ) = u g for some u ∈ G . By Lemma 3.15 we get σ ( g i ) α i = u g i and thus σ ( g i ) = uα i g i Now supposethat there are r, s ∈ Z with s > r and g r = 0 = g s . As uα s ∈ G , it follows that g s ∈ sconst G A \ { } ≤ A ∗ . Thus we conclude that σ ( g r g s ) = α s − r g r g s with s − r >
0; acontradiction to Theorem 2.12.(2). Hence g = h t m for some h ∈ sconst G A , m ∈ Z .“ ⊇ ”: Let g = h t m with h ∈ sconst G A , m ∈ Z . Then there is a u ∈ G with σ ( h ) = u h .Hence σ ( g ) = σ ( h ) α m t m = u α m h t m = u α m g with u α m ∈ G . Thus g ∈ sconst G A [ t, t ].Summarizing, we proved equality which implies that sconst G A [ t, t ] \ { } ≤ A [ t, t ] ∗ . ✷ So far we obtained a description of the semi-constants for a subgroup G of A ∗ . Now wewill lift this result to the group˜ G = G A h t i A = { h t m | h ∈ G and m ∈ Z } ≤ A h t i ∗ } . Theorem 3.20.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ .Let ( A [ t, t ] , σ ) be Π-extension of ( A , σ ) with σ ( t ) = α t for some α ∈ G and let ˜ G = G A h t i A .Then sconst ˜ G A [ t, t ] = sconst G A [ t, t ] = { h t m | h ∈ sconst G A and m ∈ Z } . Proof.
We show that sconst ˜ G A [ t, t ] = sconst G A [ t, t ]. Then by Proposition 3.19 the the-orem is proven. Since G ≤ ˜ G , the inclusion sconst ˜ G A [ t, t ] ⊇ sconst G A [ t, t ] is immediate.Now suppose that g = P i g i t i ∈ sconst ˜ G A [ t, t ]. Hence there are an m ∈ Z and an h ∈ G with σ ( g ) = h t m g . By coefficient comparison it follows that σ ( g i ) α i = hg i − m . If m ≥ s minimal such that g s = 0. Then σ ( g s ) α s = 0. But by the choice of s , we get g s − m = 0 and thus h g s − m = 0, a contradiction. Otherwise, if m <
0, take s maximalsuch that g s − m = 0. Then h g s − m = 0. But by the choice of s , we get σ ( g s ) α s = 0, againa contradiction. Thus m = 0 and consequently, g ∈ sconst G A [ t, t ]. ✷ We close this subsection with Theorem 3.21. It provides a description of sconst A [ t, t ]under the assumption that A is reduced and connected. This result is not applicable ifgeneral R -extensions pop up; see Example 3.3. But, it will be used for further insightssummarized in Corollary 4.6.(2), Proposition 4.14 and Corollary 4.15 below. Theorem 3.21.
Let ( A , σ ) be a difference ring being reduced and connected withsconst A \ { } = A ∗ . Let ( A [ t, t ] , σ ) be Π-extension of ( A , σ ) with σ ( t ) = α t for some α ∈ A ∗ . Then sconst A [ t, t ] = { h t m | h ∈ sconst A and m ∈ Z } . roof. Take ˜ G = ( A ∗ ) A h t i A . Then ˜ G = A [ t, t ] ∗ by Lemma 3.2. Thus sconst A [ t, t ] =sconst A [ t, /t ] ∗ A [ t, t ] = sconst ˜ G A [ t, t ] Thm. 3.20 = { h t m | h ∈ sconst A and m ∈ Z } . ✷ R -extensions We start with the proof of the characterization theorem of R -extensions. Proof 3.22. (Theorem 2.12.(3)). “ ⇐ ”: Let m ∈ { , . . . , λ − } and g ∈ A \ { } with σ ( g ) = α m g . Since σ ( t m ) = α m t m , it follows that σ ( g t λ − m ) = g t λ − m , i.e., g t λ − m ∈ const A [ t ]. Clearly g t λ − m / ∈ A which implies that g t λ − m / ∈ const A .“ ⇒ ”: Let g = P λ − i =0 g i t i ∈ A [ t ] \ A with σ ( g ) = g . Thus g r = 0 for some r ∈ { , . . . , λ − } .By coefficient comparison we get σ ( g r ) = α λ − r g r with λ − r ∈ { , . . . , λ − } .Let t be an R -monomial and let m := ord( α ) < λ . Then with g = 1 ∈ A \ { } we havethat σ ( g ) = 1 = α m α m g . A contradiction to the first statement. ✷ Finally, we work out properties for the set of semi-constants. Since the proof of thefollowing theorem is completely analogous to the proof of Proposition 3.19, it is skipped.
Proposition 3.23.
Let ( A , σ ) be a difference ring with G ≤ A ∗ and sconst G A \{ } ≤ A ∗ .Let ( A [ x ] , σ ) be an R -extension of ( A , σ ) with α = σ ( x ) x ∈ G and λ := ord( x ) = ord( α ) >
1. Then sconst G A [ x ] = { h x m | h ∈ sconst G A , ≤ m < λ } and sconst G A [ x ] \ { } ≤ A [ x ] ∗ .As in Theorem 3.20 we will lift this result from the group G ≤ A ∗ to˜ G = G A [ x ] A = { h x m | h ∈ G and m ∈ { , . . . , λ − }} ≤ A [ x ] ∗ } . We remark that there is the following subtlety. We have to assume that A [ x ] is reducedin order to prove the result below. In order to take care of this extra property, furtherinvestigations will be necessary in Subsection 4.1. Theorem 3.24.
Let ( A [ x ] , σ ) be an R -extension of ( A , σ ) and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ . If A [ x ] is reduced, then sconst ˜ G A [ x ] \ { } ≤ A [ x ] ∗ for ˜ G = G A [ x ] A . Proof.
Let α := σ ( x ) x ∈ A ∗ and n = ord( α ) = ord( x ). Let g ∈ sconst ˜ G A [ x ] \ { } , i.e., σ ( g ) = u x m g with u ∈ G and 0 ≤ m < n . Since x m n = 1, σ ( g n ) = u n g n with u n ∈ G .First suppose that v := g n ∈ A . Since A [ x ] is reduced, v = 0 and thus v ∈ sconst G A \{ } ≤ A ∗ , i.e., g ( g n − /v ) = 1. Hence g is invertible, i.e., g ∈ A [ x ] ∗ .Otherwise, suppose that v := g n / ∈ A . Define a := u n ∈ G . We consider two sub-cases. Suppose that there are a k > w ∈ A \ { } with σ ( w ) = a k w . Then w ∈ sconst G A \ { } ≤ A ∗ . Hence σ (( g n ) k /w ) = ( g n ) k /w , i.e, c := ( g n ) k /w ∈ K , and since A [ x ] is reduced, c = 0. Thus (as above) g ( g k n − /w/c ) = 1 and therefore g ∈ K [ x ] ∗ .Finally, suppose that there are no k > w ∈ A \ { } with σ ( w ) = a k g . Hence byTheorem 3.17 there is the Π-extension ( A [ t, t ] , σ ) of ( A , σ ) with σ ( t ) = a t ( a ∈ G ≤ A ∗ ).Let v = g n = P n − i =0 v i x i ∈ A [ x ] \ A . Then σ ( v ) = a v and thus by coefficient comparisonit follows that σ ( v i ) = a α n − i v i for some v i ∈ A \ { } with 1 ≤ i < n . Hence σ ( v i t ) = α n − i v i t , and thus σ ( v ni t n ) = v ni t n . Since A is reduced, we have v ni = 0, and consequently v ni t n ∈ const A [ t, t ] \ A , a contradiction that t is a Π-monomial. Thus this case can beexcluded. Summarizing, any element in sconst ˜ G A [ x ] \ { } is from A [ x ] ∗ . ✷ . Nested R ΠΣ ∗ -extensions and simple R ΠΣ ∗ -extensions We explore the set of semi-constants. First we deal with nested R -extensions in Sub-section 4.1 and with nested ΠΣ ∗ -extensions in Subsection 4.2. Finally, we obtain Theo-rems 2.22 and 2.24 for nested R ΠΣ ∗ -extensions in Subsection 4.3. In addition, we workout further structural properties of (simple) R ΠΣ ∗ -extensions. R -extensions We derive a first result of the semi-constants by applying iteratively Proposition 3.23.
Proposition 4.1.
Let ( A , σ ) be a difference ring with G ≤ A ∗ and sconst G A \ { } ≤ A ∗ .Let ( E , σ ) with E = A h x i . . . h x e i be an R -extension of ( A , σ ) with σ ( x i ) x i ∈ G and n i =ord( x i ). Then sconst G E = { h x m . . . x m e | h ∈ sconst G A and 0 ≤ m i < n i for 1 ≤ i ≤ e } and sconst G E \ { } ≤ E ∗ .In order to treat nested R -extensions, we proceed as follows. Let ( A h x i . . . h x e i , σ ) bean R -extension of ( A , σ ) with λ i = ord( x i ) and σ ( x i ) = α i x i . Moreover, take the poly-nomial ring R = A [ y , . . . , y e ] and define α ′ i = α | x → y ,...,x i − → y i − . Then we obtain theautomorphism σ ′ : R → R by σ ′ | A = σ and σ ( y i ) = α i y i , i.e., ( R, σ ′ ) is a difference ringextension of ( A , σ ). Thus by iterative application of the construction used for Lemma 2.6it follows that A h x i . . . h x e i is isomorphic to R/I where I is the ideal I = h y λ − , . . . , y λ e e − i (21)in R . In particular, we obtain the automorphism σ ′′ : R/I → R/I defined by σ ′′ ( f + I ) = σ ′ ( f ) + I and it follows that the difference ring ( A h x i . . . h x e i , σ ) is isomorphic to( R/I, σ ′′ ); here f ∈ A h x i . . . h x e i is mapped to f ′ + I with f ′ = f | x → y ,...,x e → y e .Take G = ( F ∗ ) EF ≤ E ∗ . In order to show that sconst G E \ { } ≤ E ∗ holds as claimed inCorollary 4.3 below, we use Gr¨obner bases theory. Lemma 4.2.
Let λ i ∈ N \ { } . Then the zero-dimensional ideal I given in (21) in thepolynomial ring R = F [ y , . . . , y e ] is radical. Proof.
The ideal I is zero-dimensional. Since F has characteristic 0, it is perfect. Wetherefore apply Seidenberg’s criterion (algorithm) given in [10, Thm. 8.22]. Define f i = y λ i i −
1. Then for each i (1 ≤ i ≤ e ) we have that f i ∈ R ∩ F [ y i ] and gcd( f i , ddy i f i ) =gcd( y λ i i − , λ i y λ i − ) = 1. Thus [10, Thm. 8.22] implies that h f , . . . , f e i is radical. ✷ Corollary 4.3.
Let ( E , σ ) be an R -extension of a difference field ( F , σ ) and let G = ( F ∗ ) EF .Then E is reduced and sconst G E \ { } ≤ E ∗ . Proof.
The difference ring ( E , σ ) with E = F h x i . . . h x r i is isomorphic to ( R/I, σ ′′ ) asdefined above with (21) where A = F . Suppose that E is not reduced. Then there are an f ∈ E \ { } and an n > f n = 0. Hence there is an h ∈ R with h + I = I and( h + I ) n = h n + I = I . This implies that h / ∈ I and h n ∈ I . Therefore I is not radical,a contradiction to Lemma 4.2. Hence E is reduced. Thus we can apply Theorem 3.24iteratively and it follows that sconst G E \ { } ≤ E ∗ . ✷ .2. Nested ΠΣ ∗ -extensions In Corollary 4.6 we will characterize the set of semi-constants within ΠΣ ∗ -extensions.Part 1 will deal with the general case. Part 2 assumes in addition that the ground ringis reduced and connected. In this setting, we rely on the following two lemmas. Lemma 4.4.
Let ( A h t i , σ ) be a ΠΣ ∗ -extension of ( A , σ ). If A is reduced, A h t i is reduced.If A is reduced and connected, A h t i is reduced and connected. Proof.
Let t be a Π-monomial. Moreover, let A be reduced. Now take f = P i f i t i ∈ A h t i = A [ t, t ] with f = 0 and f n = 0 for some n >
0. Since A is reduced, f / ∈ A . Let m ∈ Z be maximal such that f m = 0. Then the coefficient of t n m in f n is f nm . Hence f nm = 0 and thus f m is a nilpotent element in A , a contradiction.Now let A be reduced and connected and take f = P i f i t i ∈ A h t i = A [ t, t ] with f = f and f / ∈ { , } . Since A is connected, f / ∈ A . Let m be maximal such that f m = 0. If m >
0, then the coefficient of t m in f is f m and thus with f = f we have that f m = 0;a contradiction that A is reduced. Otherwise, if m = 0, we take ¯ m minimal with f ¯ m = 0.Note that ¯ m < f / ∈ A . As above, it follows that f m = 0, again a contradiction.Summarizing, if A is reduced (and connected), A [ t, t ] is reduced (and connected). For aΣ ∗ -monomial t , the same implications hold since A h t i = A [ t ] ≤ A [ t, t ]. ✷ If A is reduced, the shift behaviour of Π-monomials does not depend on Σ ∗ -monomials. Lemma 4.5.
Let ( E , σ ) be a ΠΣ ∗ -ring extension of ( A , σ ) where A is reduced. Thenthe generators can be reordered such that we get the form E = A h t i . . . h t p ih s i . . . h s e i where the t i are Π-monomials and the s i are Σ ∗ -monomials. Proof.
Let E = A h t i . . . h t e i . By iterative application of Lemma 4.4 it follows that E is reduced. Let t i be a Π-monomial where α = σ ( t i ) /t i ∈ A h t i . . . h t i − i depends ona Σ ∗ -monomial t j with j < i . Then we can reorder the generators such that we get H = A h t i . . . h t j − ih t j +1 i . . . h t i − i ; here we forget σ and argue purely in the given ring.In particular, α ∈ H h t j i = H [ t j ] \ H . Since α is invertible, α ∈ H by Lemma 3.2; acontradiction. Summarizing, for all Π-monomials t j we have that σ ( t j ) /t j is free of Σ ∗ -monomials. Thus we can shuffle all Π-monomials to the left and all Σ ∗ -monomials to theright and obtain again a ΠΣ ∗ -extension. ✷ Corollary 4.6.
Let ( E , σ ) be a ΠΣ ∗ -extension of ( A , σ ) with E = A h t ih t i . . . h t e i .(1) Let G ≤ A ∗ with sconst G A \ { } ≤ A ∗ and let ˜ G = G EA . If ( E , σ ) is a G -simpleΠΣ ∗ -extension of ( A , σ ), then sconst ˜ G E \ { } ≤ E ∗ wheresconst ˜ G E = { h t m . . . t m e e | h ∈ sconst G A and m i ∈ Z where m i = 0 if t i is a Σ ∗ -monomial } . (2) If A is reduced and connected and sconst A \ { } = A ∗ , thensconst E \ { } = { h t m . . . t m e e | h ∈ A ∗ and m i ∈ Z where m i = 0 if t i is a Σ ∗ -monomial } = E ∗ . (22)(3) If A is a field then we have that (22). Proof.
The first part is proven by induction on the number e of extensions. If e =0, nothing has to be shown. Now suppose that the first part holds and consider oneextra ˜ G -simple ΠΣ ∗ -monomial t e +1 on top. Define ˜˜ G = ˜ G E h t e +1 i E = G E h t e +1 i A . If t i is a24 ∗ -monomial, ˜˜ G = ˜ G . Together with Theorem 3.12 it follows that sconst ˜˜ G E [ t e +1 ] =sconst ˜ G E [ t e +1 ] = sconst ˜ G E and sconst ˜˜ G E [ t e +1 ] \ { } ≤ E ∗ ≤ E [ t e +1 ] ∗ . If t i is a Π-monomial, we have σ ( t e +1 ) /t e +1 ∈ ˜ G . Hence Theorem 3.20 yields sconst ˜˜ G E [ t e +1 , t e +1 ] = { h t me +1 | m ∈ Z and h ∈ sconst ˜ G E } and thus by the induction assumption we have thatsconst ˜˜ G E [ t e +1 , t e +1 ] = { h t m . . . t m e +1 e +1 | h ∈ sconst G A and m i ∈ Z where m i = 0 if t i is a Σ ∗ -monomial } and thus sconst ˜˜ G E [ t e +1 , t e +1 ] \ { } ≤ E [ t e +1 , t e +1 ] ∗ . This completes the induction step.Similarly, the first equality of part 2 follows by Theorems 3.14 and 3.21. The secondequality follows by Lemmas 3.2 and 4.4. Since any field is connected and reduced andsconst A \ { } = A ∗ by Lemma 3.13, part 3 follows by part 2. ✷ Restricting to Σ ∗ -extensions, the above result simplifies as follows. Corollary 4.7.
Let ( E , σ ) be a Σ ∗ -extension of ( A , σ ). Then the following holds.(1) If G ≤ A ∗ with sconst G A \ { } ≤ A ∗ , then sconst G E = sconst G A .(2) If A is reduced and sconst A \ { } = A ∗ , then E is reduced and sconst E = sconst A .(3) If A is a field, then sconst E \ { } = A ∗ = A \ { } . R ΠΣ ∗ -extensions and their simple and single-rooted restrictions We turn to the set of semi-constants within nested R ΠΣ ∗ -extensions. The case ofsimple and single-rooted R ΠΣ ∗ -extensions is immediate. Proof 4.8. (Theorem 2.22).
This follows by Corollary 4.6.(1) and Proposition 4.1. ✷ Likewise, simple R ΠΣ ∗ -extension can be treated if they are built in a particular form. Theorem 4.9.
Let ( H , σ ) be an R -extension of a difference field ( F , σ ) and let ( E , σ )with E = H h t ih t i . . . h t e i be a simple ΠΣ ∗ -extension of ( H , σ ). Let G = ( F ∗ ) HF and define˜ G = G EH . Then we have sconst ˜ G E \ { } ≤ E ∗ wheresconst ˜ G E = { h t m . . . t m e e | h ∈ sconst G H , m i ∈ Z where m i = 0 if t i is a Σ ∗ -monomial } . Proof.
By Cor. 4.3, sconst G H \ { } ≤ H ∗ . Hence the result follows by Cor. 4.6.(1). ✷ Next, we show that simple R ΠΣ ∗ -extensions can be always brought to the shape asassumed in Theorem 4.9. This will finally produce a proof of Theorem 2.24. Lemma 4.10.
Let ( A , σ ) be a difference ring with a group G ≤ A ∗ and let ( E , σ ) be a G -simple R ΠΣ ∗ -extension of ( A , σ ).(1) The R ΠΣ ∗ -monomials can be reordered to the form E = A h t ih t i . . . h t e i with r, p ∈ N (0 ≤ r ≤ p ≤ e ) such that the following holds. • For all i (1 ≤ i ≤ r ), t i is an R -monomial with σ ( t i ) /t i = u i t z . . . t z i − i − for someroot of unity u i ∈ G and z i ∈ N . • For all i ( r < i ≤ p ), t i is a Π-monomial with σ ( t i ) /t i = u i t z . . . t z i − i − for some u i ∈ G and z i ∈ Z . • For all i ( p < i ≤ e ), t i is a Σ ∗ -monomial with σ ( t i ) − t i ∈ A h t ih t i . . . h t i − i . (2) For any f ∈ G EA which depends on a Π-monomial we have that ord( f ) = 0.25 roof. We show the lemma by induction on the number of R ΠΣ ∗ -monomials. Supposethat the lemma holds for e extensions. Now let E = A h t i . . . h t e i and consider the R ΠΣ ∗ -monomial t e +1 on top of E . By the induction assumption we can reorder E such that ithas the desired form (all R -monomials are on the left, all Π-monomials are in the middleand all Σ ∗ -monomials are on the right). If t e +1 is a Σ ∗ -monomial, the required shape isfulfilled. If t e +1 is an R -monomial, observe that α := σ ( t e +1 ) /t e +1 ∈ G EA . Since ord( α ) =ord( t e +1 ) > α is free of Π-monomials by the induction assumptionand (by definition) free of Σ ∗ -monomials. Thus we can shuffle t e +1 to the left (such thatall ΠΣ ∗ -monomials are to the right), and the required shape is satisfied. Similarly, if t e +1 is a Π-monomial, σ ( t e +1 ) /t e +1 ∈ G EA is free of Σ ∗ -monomials by definition and we canshuffle t e +1 to the left such that all Σ ∗ -monomials are to the right. This completes thefirst part of the lemma. Now let E = A h x i . . . h x e +1 i be in the desired ordered form.If x e +1 is a Σ ∗ -monomial, we have that G E h x e +1 i A = G EA . Thus the second part holds bythe induction assumption. If x e +1 is an R -extension, also all x i with 1 ≤ i ≤ e are R -monomials, and the second statement holds trivially. Finally, let x e +1 be a Π-monomialand take f ∈ G E h x e +1 i A . If f ∈ E and f depends on Π-monomials, we have again thatord( f ) = 0 by the induction assumption. To this end, suppose that f depends on x e +1 andwe have that ord( f ) = n >
0. Then f = u x me +1 where m = 0 and u ∈ E ∗ . Since f n = 1, u n x m ne +1 = 1 where u n = 0. Hence x e +1 is not transcendental over E , a contradiction tothe definition of a Π-monomial. Thus ord( f ) = n = 0. This completes the proof. ✷ Proof 4.11. (Theorem 2.24).
By Lemma 4.10 we can reorder the simple R ΠΣ ∗ -extension such that Theorem 4.9 is applicable. ✷ In the remaining part of this section we deliver insight into the structure of (simple) R ΠΣ ∗ -extensions. First observe that a tower of simple R ΠΣ ∗ -extensions is again simple. Lemma 4.12.
Let ( A , σ ) be a difference ring with a group G ≤ A ∗ and let ( A , σ ) ≤ ( H , σ ) ≤ ( E , σ ) be R ΠΣ ∗ -extensions. Then ( G HA ) EH = G EA . Moreover, if ( A , σ ) ≤ ( H , σ ) is G -simple and ( H , σ ) ≤ ( E , σ ) is G HA -simple, then ( A , σ ) ≤ ( E , σ ) is G -simple.Further, the reordering as described in Lemma 4.10 is also possible if one relaxes thecondition that the R ΠΣ ∗ -extension is simple but requires that the ground ring is a field. Lemma 4.13.
Let ( E , σ ) be a R ΠΣ ∗ -ring extension of a difference field ( F , σ ). Then( E , σ ) can be reordered to the form E = F h x i . . . h x r ih t i . . . h t p ih s i . . . h s e i where the x i are R -monomials, the t i are Π-monomials and the s i are Σ ∗ -monomials. Proof.
First we try to shuffle all R -extensions to the front. Suppose that this fails atthe first time. Then there are an R -extension ( H , σ ) of ( F , σ ), a ΠΣ ∗ -extension ( G , σ ) of( H , σ ) with G = H h y i . . . h y l i and an R -extension ( G h x i , σ ) of ( G , σ ) with α = σ ( x ) /x inwhich y l occurs. Note that H is reduced by Corollary 4.3, and G is reduced by iterativeapplication of Lemma 4.4. Write α = P i f i y il . Let m = 0 such that f m = 0 and suchthat | m | ≥ m < y l is a Π-monomial).By the choice of m , we have that the coefficient of y m nl in α n is f nm . Hence with α n = 1it follows that f nm = 0, a contradiction to the assumption that G is reduced. Thereforewe can shuffle all R -monomials to the left and all ΠΣ ∗ -monomials to the right. Since thenested R -extension is reduced by Corollary 4.3, we can apply Lemma 4.5 to reorder theΠΣ ∗ -monomials further as claimed in the statement. ✷
26y definition any (nested) Σ ∗ -extension is a also a simple Σ ∗ -extension. If the groundring is reduced and connected, we obtain the following stronger result. Proposition 4.14.
Let ( G , σ ) be a difference ring where G is reduced and connectedand where sconst G \ { } = G ∗ . Then a ΠΣ ∗ -extension ( E , σ ) of ( G , σ ) is simple. Proof.
Let E = G h t i . . . h t e i . By Lemma 4.5 we may suppose that the generators areordered such that the t . . . , t p are Π-monomials and the t p +1 . . . , t e are Σ ∗ -monomials. ByCorollary 4.6.(2) we have that σ ( t i ) t i ∈ G h t i . . . h t i − i ∗ = ( G ∗ ) G h t i ... h t i − i G with 1 ≤ i ≤ p .Thus the Π-monomials t i are G ∗ -simple. Moreover, the Σ ∗ -monomials t i on top are all G ∗ -simple by definition. Summarizing ( F , σ ) ≤ ( E , σ ) is simple. ✷ In other words, for a reduced and connected difference ring ( A , σ ) (e.g., if A is a field)the notions of ΠΣ ∗ -ring extension and simple ΠΣ ∗ -ring extension are equivalent. Thesituation becomes rather different if the ring is, e.g., not connected; see Example 2.20.But, for single-rooted R ΠΣ ∗ -extensions over a difference field, the situation is again tame. Corollary 4.15.
A single-rooted R ΠΣ ∗ -extension ( E , σ ) of a field ( G , σ ) is simple. Proof.
By definition the R ΠΣ ∗ -extension can be reordered to the form (2.21). Since G is a field, sconst G \ { } = G ∗ . By Proposition 4.14 the Π-extension ( G h t i . . . h t r i , σ ) of( G , σ ) is simple. Since σ ( x i ) x i ∈ G ∗ for 1 ≤ i ≤ u , the R -monomials x i are G ∗ -simple. Sincealso the Σ ∗ -monomials s i are G ∗ -simple, we conclude that ( G , σ ) ≤ ( E , σ ) is simple. ✷
5. The algorithmic machinery I: order, period, factorial order
An important ingredient for the development of our summation algorithms is theknowledge of the order (see its definition in (10) and the corresponding Problem O), theperiod and the factorial order. In ( A , σ ) we define the period of h ∈ A ∗ byper( h ) = ( ∄ n > σ n ( h ) = h min { n > | σ n ( h ) = h } otherwise;and the factorial order of h byford( h ) = ( ∄ n > h ( n ) = 1min { n > | h ( n ) = 1 } otherwise . Using the properties of the automorphism σ and Lemma 3.4 it is easy to see that the Z -modules generated by ord( h ), per( h ) and ford( h ) are h ord( h ) i = ord( h ) Z = { k ∈ Z | h k = 1 } , h per( h ) i = per( h ) Z = { k ∈ Z | σ k ( h ) = h } , and h ford( h ) i = ford( h ) Z = { k ∈ Z | h ( k ) = 1 } , respectively. In addition, the following basic properties hold. Lemma 5.1.
Let ( A , σ ) be a difference ring with α, h ∈ A ∗ . Then the following holds.(1) If α ∈ (const A ) ∗ , then per( α ) = 1 and ford( α ) = ord( α ).(2) If σ ( h ) = α h , then per( h ) = ford( α ).(3) If ord( α ) > α ) >
0, then per( α ) | ford( α ) | per( α ) ord( α ) andford( α ) = min( i per( α ) | ≤ i ≤ ord( α ) and α ( i per( α )) = 1) > . (23)27 roof. (1) Since σ ( α ) = α , per( α ) = 1. Since α ( n ) = α n for n ≥
0, ford( α ) = ord( α ).(2) By Lemma 3.4.(4) we have that σ n ( h ) = h iff α ( n ) = 1. Hence per( h ) = ford( α ).(3) Take p = per( α ) > v = ord( α ) >
0. Then we have that α ( p v ) = α σ ( α ) . . . σ p v − ( α ) = ( α σ ( α ) . . . σ p − ( α )) v = α v σ ( α v ) . . . σ p − ( α v ) = 1 . Consequently, we can choose n = ord( α ) per( α ) to obtain α ( n ) = 1. In particular, forany i ≥ α ( i ) = 1 we have that 1 = σ (1)1 = σ ( α ( i ) ) α ( i ) = σ i ( α ) α . Hence per( α ) | i . Thus thesmallest λ with α ( λ ) = 1 is given by (23). In particular, per( α ) | ford( α ) | ord( α ) per( α ). ✷ We will present methods to calculate the order, period and factorial order for theelements of ( A ∗ ) EA of a simple R -extension ( E , σ ) ≥ ( G , σ ) by recursion. First, we assumethat the orders of the R -monomials in ( E , σ ) ≥ ( G , σ ) are already computed and showhow the orders of the elements of ( A ∗ ) EA can be determined. Lemma 5.2.
Let ( E , σ ) with E = A h x i . . . h x e i be an R -extension of ( A , σ ) and define α := u x z . . . x z e e ∈ ( A ∗ ) EA (24)with u ∈ A ∗ and z i ∈ N . Then ord( α ) > u ) >
0. If ord( u ) >
0, thenord( α ) = lcm(ord( u ) , ord( x )gcd(ord( x ) ,z ) , . . . , ord( x e )gcd(ord( x e ) ,z e ) ) . (25) Proof. If e = 0, the lemma holds. Now let n := ord( α ) >
0. Suppose that 1 =( x z . . . x z e e ) n = x n z . . . x n z e e . Let i be maximal such that ord( x i ) ∤ z i n . Then there is an s with 0 < s < ord( x i ) with x ord x i − si = u n x z . . . x z i − i − ∈ A h x i . . . h x i − i which contra-dicts to the construction that x ord( x i ) i = 1 is the defining relation of the R -monomial. Thus( x z . . . x z e e ) n = 1 and u n = 1, i.e., ord( u ) > x z . . . x z e e ) >
0. In particular,ord( α ) = lcm(ord( u ) , ord( x z . . . x z e e )) . By similar arguments we can show that ( x z ) n = · · · = ( x z e e ) n = 1 and consequently ord( x z . . . x z e e ) = lcm(ord( x z ) , . . . , ord( x z e e ) . Sincealso ord( x z i i ) = ord( x i )gcd(ord( x i ) ,z i ) holds, the identity (25) is proven.Conversely, suppose that ord( u ) >
0. Then the value of the right right hand side of (25)is positive. Denote it by n . Then one can check that α n = 1. Therefore ord( α ) > ✷ In the next lemma we set the stage to calculate the period and factorial order.
Lemma 5.3.
Let ( E , σ ) with E = A h x i . . . h x e i be an R -extension of ( A , σ ) where wehave per( x i ) > ≤ i ≤ e . Let α ∈ ( A ∗ ) EA as in (24) with z , . . . , z e ∈ N and u ∈ A ∗ .(1) Then per( α ) > u ) >
0. If per( u ) >
0, thenper( α ) = min(1 ≤ j ≤ µ | σ j ( α ) = α and j | µ ) (26)with µ = lcm(per( u ) , per( x i ) , . . . , per( x i k )) where { i , . . . , i k } = { i : ord( x i ) ∤ z i } .(2) We have that ford( α ) > u ) > u ) , ord( u ) >
0, then ford( α ) > < per( α ) | ford( α ) | per( α ) ord( α ).(4) If the values ord( x i ) and per( x i ) for 1 ≤ i ≤ e and the values per( u ) > u ) > α ) and ford( α ) can be calculated. Proof. (1) Suppose that per( u ) >
0. Then µ >
0. In particular, it follows that σ µ ( α ) = α .Consequently, per( α ) > α ) | µ . Hence we have (26). Conversely, suppose thatper( α ) >
0. Then with ν := lcm(per( α ) , per( x ) , . . . , per( x e )) > u x z . . . x z e e =28 = σ ν ( α ) = σ ν ( u ) x z . . . x z e e . Thus σ ν ( u ) = u , and consequently ord( u ) > x i ) and per( x i ) >
0, it follows that ford( x i ) > ≤ i ≤ e . If ford( u ) >
0, take ν = lcm(ford( u ) , ford( x ) , . . . , ford( x e )) >
0. ByLemma 3.4.(1), α ( ν ) = 1 and hence ford( α ) >
0. Conversely, if ford( α ) >
0, take ν ′ = lcm(ford( α ) , ford( x ) , . . . , ford( x e )) >
0. Then again by Lemma 3.4.(1): 1 = α ( ν ′ ) =( u x z . . . x z e e ) ( ν ′ ) = u ( ν ′ ) . Thus ford( u ) > α ) >
0. And with ord( u ) > α ) > α ) | ford( α ) | ord( α ) per( α ). In particular, ford( α ) > u ) and the values per( x i ) are given, µ from part 1 can be computed. In partic-ular, if ord( u ) and ord( x i ) are given explicitly, ord( α ) can be calculated by Lemma 5.2.Thus per( α ) can be determined by (26) and then ford( α ) can be computed by (23). ✷ Example 5.4. (1) Take α = u = − ∈ Q . We get ord( α ) = 2. In addition, per( −
1) = 1.Moreover, 1 = per( − | ford( − | per( −
1) ord( −
1) = 2. Hence (26) yields ford( −
1) = 2.(2) Consider the R -extension ( Q [ x ] , σ ) of ( Q , σ ) with σ ( x ) = − x and ord( x ) = 2,and take α = − x . We get ord( α ) = lcm(ord( − , ord( x )) = 2 by (25). With µ =lcm(per( − , per( x )) = 2 we get per( α ) = 2 by using (26). Furthermore, we get 2 =per( α ) | ford( α ) | per( α ) ord( α ) = 4. Hence with (26) we get ford( α ) = 4.(3) Consider the R -extension ( K ( k )[ x ] , σ ) of ( K ( k ) , σ ) with σ ( x ) = ι x and ord( x ) = 4from Example 2.14. We have per( x ) = 4. Take α = x . We obtain the following bounds4 = per( α ) | ford( α ) | per( α ) ord( α ) = 16. Thus with (26) we determine ford( α ) = 8.Combining the two lemmas from above we arrive at the following result. Proposition 5.5.
Let ( A h x i . . . h x e i , σ ) be a simple R -extension of ( A , σ ) such that for1 ≤ i ≤ e we have that σ ( x i ) /x i = u i x m i, . . . x m i,i − i − with u i ∈ A ∗ and m i,j ∈ N . Thenthe following holds.(1) ord( u i ) > ≤ i ≤ e . In particular, if the values ord( u i ) are given explicitly(are computable), then the values ord( x i ) are computable.(2) If per( u i ) > ≤ i ≤ e , then per( x i ) > ≤ i ≤ e . In particular, if thevalues of ord( u i ) and per( u i ) for 1 ≤ i ≤ e are given explicitly (are computed), thevalues per( x i ) for all 1 ≤ i ≤ e are computable. Proof. (1) By iterative application of Lemma 5.3 it follows that ord( u i ) > ≤ i ≤ e . Moreover, suppose that ord( u i ) is given for 1 ≤ i ≤ e . Furthermore, assumethat the values ord( x i ) for 1 ≤ i ≤ s with s < e are already determined. Then define α = σ ( x s ) /x s . By (25) we obtain ord( α ) and thus ord( x s ) = ord( α ) by Theorem 2.12.(3).This completes the induction step.(2) Suppose that per( u i ) > ≤ i ≤ e . In addition, suppose that we have shownalready that d i = per( x i ) > ≤ i < s with s ≤ e . Define α = σ ( x s ) /x s . ByLemma 5.3 we have per( α ) > α ) >
0. By Lemma 5.1.(2) it follows thatper( x s ) = ford( α ) >
0. If the values ord( u i ) are given explicitly, we can compute ord( α )by part 1. If per( u s ) is given explicitly and d , . . . , d s − are given (are already computed),per( α ) can be computed with Lemma 5.1.(3). Hence ford( α ) can be calculated with (23).Thus we get ord( x s ) = ford( α ) by Lemma 5.1.(2) which completes the induction step. ✷ If we restrict to the case that the ground domain is a field F and all roots of unity of F are constants, we end up at the following properties of R -extensions.29 orollary 5.6. Let ( E , σ ) with E = F h x i . . . h x e i be a simple R -extension of a differencefield ( F , σ ) with constant field K such that all roots of unity in F are constants (e.g., if( F , σ ) is strong constant-stable). Then the following holds.(1) For 1 ≤ i ≤ e we have that σ ( x i ) /x i = u i x m i, . . . x m i,i − i − (27)for some root of unity u i ∈ K ∗ with ord( u i ) | ord( x i ) and m i,j ∈ N .(2) ( K h x i . . . h x e i , σ ) is a simple R -extension of ( K , σ ).(3) Let α = u x z . . . x z e e ∈ ( K ∗ ) K h x i ... h x e i K with z , . . . , z e ∈ N and u ∈ K ∗ . Thenord( u ) > ⇔ ord( α ) > ⇔ per( α ) > ⇔ ford( α ) > . (4) If ( K , σ ) is computable and Problem O is solvable in K ∗ then the values of ord( α ),per( α ) and ford( α ) are computable for all α ∈ ( K ∗ ) K h x i ... h x e i K .(5) Problem O is solvable in ( F ∗ ) EF if it is solvable in K ∗ and ( F , σ ) is computable. Proof. (1) By definition we have that (27) with m i ∈ N and u i ∈ F ∗ . By Lemma 5.2 itfollows that ord( u i ) > u i ) | ord( x i ). In particular, u i ∈ K ∗ since all roots ofunity from F are constants by assumption.(2) It is immediate that ( H , σ ) with H = K h x i . . . h x e i forms a difference ring. Sinceconst E = const F = K , ( H , σ ) is a simple R -extension of ( K , σ ).(3) By part 1 we get u i ∈ K ∗ and ord( u i ) > ≤ i ≤ e . In particular, per( u i ) = 1.With Proposition 5.5 we get per( x i ) >
0, and by Lemma 5.1.(1) we obtain per( u ) = 1and ford( u ) = ord( u ). Thus the equivalences follow by Lemmas 5.2 and 5.3 (parts 1,2).(4) Since u i ∈ K ∗ , the values of ord( u i ) > K ∗ . Thus by Proposition 5.5 the orders and periods of the x i can be computed. Let α := u x z . . . x z e e with u ∈ K ∗ and z i ∈ N . Then by Lemma 5.2 and the computation oford( u ) the order of α can be computed. Moreover, since per( u ) = 1 and ord( u ) = ford( u )are given, we can invoke Lemma 5.3 to calculate the period and factorial order of α .(5) Let α be given as in (24) with u ∈ F ∗ and m i ∈ N . By Lemma 5.2 ord( α ) > u ) >
0. By assumption, ord( u ) > u ∈ K ∗ . Thus, if u / ∈ K , ord( α ) = 0.Otherwise, if u ∈ K ∗ , we can apply part 4. ✷ Finally, we are in the position to prove Theorem 2.26.(1).
Proof 5.7. (Theorem 2.26.(1)).
Let ( E , σ ) be a simple R ΠΣ ∗ -extension of ( F , σ )where ( F , σ ) is computable and where any root of unity of F is from K = const F . Reorderit to the shape as given in Lemma 4.10. In particular, the R -extension ( F h t i . . . h t r i , σ )of ( F , σ ) has the shape as given in Corollary 5.6.(1). Let f ∈ ( F ∗ ) EF . Suppose first that f depends on a Π-monomial t i . Now assume that ord( f ) = n >
0, and let i be maximalsuch that a Π-monomial depends on f . Then f = v t mi with v ∈ F h t i . . . h t i − i ∗ and m ∈ Z \ { } . Hence 1 = f n = v n t m ni and thus t i is not algebraically independent over F h t i . . . h t i − i ; a contradiction. Consequently, if f depends on Π-monomials, ord( f ) = 0.Otherwise, f = u t m . . . t m r r with u ∈ F ∗ and m i ∈ N where the t i are all R -monomials.Therefore the value ord( f ) can be computed by Corollary 5.6.(5). ✷ . The algorithmic machinery II: Problem PMT We aim at proving Theorems 2.23.(1) and 2.26.(2), i.e., providing recursive algorithmsthat reduce Problem PMT from a given R ΠΣ ∗ -extension to its ground ring (resp. field).For this reduction we assume that for the given ground ring ( G , σ ) and given group G ≤ G ∗ we have that sconst G G \ { } ≤ G ∗ . This property guarantees that for any f ∈ G n a Z -basis of M ( f , G ) with rank ≤ n exists; see Lemma 2.16. In particular, we relyon the fact that there are algorithms available that solve Problem PMT in ( G , σ ) for G .For concrete classes of difference fields ( G , σ ) with these algorithmic properties we referto Subsection 2.3.3. ΠΣ ∗ -extensions First, we treat the reduction for ΠΣ ∗ -extensions. More precisely, we will obtain Theorem 6.1.
Let ( G , σ ) be a computable difference ring with G ≤ G ∗ where sconst G G \{ } ≤ G ∗ . Let ( E , σ ) be a G -simple ΠΣ ∗ -extension of ( G , σ ). Then sconst G EG E \ { } ≤ E ∗ and Problem PMT is solvable in ( E , σ ) for G EG if it is solvable in ( G , σ ) for G .For the underlying reduction method we use the following two lemmas. Lemma 6.2.
Let ( A [ t ] , σ ) be a Σ ∗ -extension of ( A , σ ) and let H ≤ A ∗ be a group withsconst H A \ { } ≤ A ∗ . Then for f ∈ H n we have that M ( f , A [ t ]) = M ( f , A ). Proof. “ ⊆ ”: Let m = ( m , . . . , m n ) ∈ M ( f , A [ t ]) with f = ( f , . . . , f n ) ∈ H n . Thustake g ∈ A [ t ] \ { } with σ ( g ) = f m . . . f m n n g . Since g ∈ sconst H A [ t ] \ { } , we have g ∈ sconst H A by Theorem 3.12. Hence m ∈ M ( f , A ). The inclusion ⊇ is obvious. ✷ Lemma 6.3.
Let ( A h t i , σ ) be a Π-extension of ( A , σ ) and let H ≤ A ∗ with sconst H A \{ } ≤ A ∗ and α := σ ( t ) /t ∈ H . Let f = ( f , . . . , f n ) ∈ ( H A h t i A ) n with f i = h i t e i , h i ∈ H, e i ∈ Z . Then M ( f , A h t i ) = M ∩ M where M = { ( m , . . . , m n ) | ( m , . . . , m n , m n +1 ) ∈ M (( h , . . . , h n , α ) , A ) } ,M = Ann Z (( e , . . . , e n )) = { ( m . . . , m n ) ∈ Z n | m e + · · · + m n e n = 0 } . Proof. “ ⊆ ”: Let ( m , . . . , m n ) ∈ M ( f , A h t i ). Hence we can take g ∈ A h t i \ { } with σ ( g ) = f m . . . f m n n g, i.e., g ∈ sconst ˜ H A h t i \ { } with ˜ H = H A h t i A . Thus by Theorem 3.20it follows that g = ˜ g t m with m ∈ Z and ˜ g ∈ sconst ˜ H A \ { } ≤ A ∗ . Hence σ (˜ g ) = f m . . . f m n n α − m ˜ g = h m . . . h m n n α − m ˜ g t m e + ··· + m n e n . Since ˜ g = 0, we conclude that σ (˜ g ) = 0. By coefficient comparison it follows then that m e + · · · + m n e n = 0, i.e., ( m , . . . , m n ) ∈ M . Thus σ (˜ g ) = h m . . . h m n n α − m ˜ g andconsequently ( m , . . . , m n , m ) ∈ M (( h , . . . , h n , α ) , A ), i.e., ( m , . . . , m n ) ∈ M .“ ⊇ ”: Let ( m , . . . , m n ) ∈ M ∩ M . Thus we can take ˜ g ∈ A \ { } and m ∈ Z with σ (˜ g ) = h m . . . h m n n α − m ˜ g. Moreover, we have that e m + · · · + e n m n = 0. Thus σ (˜ g t m ) =( h t e ) m . . . ( h n t e n ) m n ˜ g and therefore ( m , . . . , m n ) ∈ M ( f , A h t i ). ✷ Now we can deal with the underlying algorithm resp. proof of Theorem 6.1.31 roof 6.4. (Theorem 6.1).
Let ( G , σ ) be a difference ring and let G ≤ G ∗ such thatsconst G G \ { } ≤ G ∗ holds. Suppose that Problem PMT is solvable in ( G , σ ) for G . Nowlet ( E , σ ) be a G -simple ΠΣ ∗ -extension of ( G , σ ) as in the theorem with ˜ G = G EG andlet f ∈ ˜ G n . By Corollary 4.6.(1) it follows that sconst ˜ G E \ { } ≤ E ∗ and together withLemma 2.16 it follows that M ( f , E ) = M ( f , sconst ˜ G E ) is a Z -module. The calculationof a basis of M ( f , E ) will be accomplished by recursion/induction. If E = A , nothinghas to be shown. Otherwise, let ( A , σ ) be a G -simple ΠΣ ∗ -extension of ( G , σ ) in whichwe know how one can solve Problem PMT for H = G AG , and let E = A h t i where t isa H -simple ΠΣ ∗ -monomial. We have to treat two cases. First, suppose that t is a Σ ∗ -monomial. Then it follows that ˜ G = G EG = G AG = H ≤ A ∗ and thus f ∈ H n . Hence we canactivate Lemma 6.2 and it follows that M ( f , E ) = M ( f , A ). Thus by assumption we cancompute a basis. Second, suppose that t is a H -simple Π-monomial. Then we can utilizeLemma 6.3: We calculate a basis of M by linear algebra. Furthermore, we compute abasis of M (( h , . . . , h n , α ) , A ) by the induction assumption (by recursion). Hence we canderive a basis of M and thus of M ∩ M = M ( f , A h t i ). This completes the proof. ✷ Note that the reduction presented in Lemma 6.3 is accomplished by increasing the rankof M by one. In general, the more Π-monomials are involved, the higher the rank willbe in the arising Problems PMT of the recursions.Looking closer at the reduction algorithm, we can extract the following shortcut, resp.a refined version of Theorem 2.12.(2). Corollary 6.5.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \ { } ≤ A ∗ .Let ( H , σ ) be a G -simple Π-extension of ( A , σ ) and let ( E , σ ) be a Σ ∗ -extension of ( H , σ ).Then G EA = G HA and the following holds.(1) M ( f , E ) = M ( f , H ) for any f ∈ ( G EA ) n .(2) Let α ∈ G EA . Then there is a Π-extension ( E h t i , σ ) of ( E , σ ) with σ ( t ) = α t iff thereis a Π-extension ( H h t i , σ ) of ( H , σ ) with σ ( t ) = α t . Proof.
Note that G EA ≤ H ∗ . Hence by iterative application of Lemma 6.2 part 1 is proven.Part 2 follows by part 1 and Theorem 2.12.(2). ✷ If one restricts to the special case that ( G , σ ) is a ΠΣ ∗ -field with G = G ∗ , the presentedreduction techniques boil down to the reduction presented in [24, Theorem 8]. The majorcontribution here is that Theorem 6.1 can be applied for any computable difference ring( G , σ ) with the properties given in Theorem 6.1. Subsequently, we utilize this additionalflexibility to tackle (nested) R -extensions. R -extensions and thus for R ΠΣ ∗ -extensions First, we treat the special case of single-rooted and simple R -extensions. Lemma 6.6.
Let ( A , σ ) be a difference ring and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ . Let( A [ x ] , σ ) be an R -extension of ( A , σ ) with σ ( x ) = α x where α ∈ G ; let f = ( f , . . . , f n ) ∈ G n . Then M ( f , A [ x ]) = { ( m , . . . , m n ) | ( m , . . . , m n +1 ) ∈ M (( f , . . . , f n , α ) , A ) . Proof.
Let ( m , . . . , m n ) ∈ M ( f , A [ x ]). Hence there is a g ∈ sconst G A [ x ] \ { } with σ ( g ) = f m . . . f m n n g . By Proposition 3.23 it follows that g = ˜ g x m with ˜ g ∈ A \ { } and m ∈ N . Thus σ (˜ g ) = f m . . . f m n n α − m ˜ g (28)32nd hence ( m , . . . , m n , m ) ∈ M (( f , . . . , f n , α ) , A ). Conversely, if ( m , . . . , m n , m ) ∈ M (( f , . . . , f n , /α, A ), there is a ˜ g ∈ A \ { } with (28). Therefore we conclude that σ (˜ g t m ) = f m . . . f m n n ˜ g t m which implies that ( m , . . . , m n ) ∈ M ( f , A [ x ]). ✷ As a consequence we obtain the proof of our Theorem 2.23.(1).
Proof 6.7. (Theorem 2.23.(1)).
Since Problem PMT is solvable in ( G , σ ) for G ,it follows by Theorem 6.1 that Problem PMT is solvable in ( H , σ ) for ˜ G with H = G h t i . . . h t r i and that sconst ˜ G H \ { } ≤ H ∗ . Thus by iterative applications of Lemma 6.6and Proposition 3.23 we conclude that Problem PMT is solvable in ( ¯ H , σ ) for ˜ G with¯ H = H h x i . . . h x u i and that sconst ˜ G ¯ H \{ } ≤ ¯ H ∗ . Finally, by applying again Theorem 6.1it follows that Problem PMT is solvable in ( E , σ ) for ˜ G . ✷ In order to tackle the more general case that the R -extensions are nested and that theymight occur also in Π-extensions (see the underlying algorithms for Theorem 2.26.(2) inProof 6.15 below), we require additional properties on the difference rings: they mustbe strong constant-stable; see Definition 2.25. With this extra condition the followingstructural property of the semi-constants holds. They factor into two parts: a factorwhich depends only on the R -monomials with constant coefficients and a factor which isfree of the R -monomials. Lemma 6.8.
Let ( A , σ ) be a difference ring which is constant-stable and let G ≤ A ∗ beclosed under σ where sconst G ( A , σ k ) \ { } ≤ A ∗ for any k >
0. Let ( E , σ ) be a simple R -extension of ( A , σ ) with E = A [ x ] . . . [ x e ] where we have (27) with m i,j ∈ N and u i ∈ G with per( u i ) >
0. Define r := ( lcm(ford( u ) , . . . , ford( u e ) , ford( x ) , . . . , ford( x e )) if e >
01 if e = 0 . (29)Let ˜ G = G EA with sconst ˜ G E \ { } ≤ E ∗ . Then the following holds.(1) r >
0. (2) For any g ∈ sconst ˜ G E \ { } we have that g = ˜ g h (30)with ˜ g ∈ sconst G ( A , σ r ) \ { } ≤ A ∗ and h ∈ const( K [ x , . . . , x e ] , σ r ) ∗ .(3) If σ ( g ) = v x m . . . x m e e g with v ∈ G , m i ∈ N , then σ (˜ g ) = λ v ˜ g with λ ∈ A ∗ , λ r = 1. Proof.
Let g ∈ sconst ˜ G E \ { } , i.e., σ ( g ) = v x m . . . x m e e g with v ∈ G and m i ∈ Z . Let r be given as in (29). If e = 0, i.e., r = 1, the lemma holds by taking ˜ g := g ∈ A ∗ and h = 1. Otherwise, we may suppose that e > ≤ i ≤ e . By Proposition 5.5.(1) it follows that ord( u i ) >
0. Together with theassumption that per( u i ) > u i ) > x i ) >
0. Again with ord( x i ) > x i ) > x i ) > r > r it follows that for all 1 ≤ i ≤ e we have( u i ) ( r ) = 1 , ( x i ) ( r ) = 1 and σ r ( x i ) = x i ; (31)the last equality follows by Lemma 5.3.(3). Moreover, by Lemma 3.4 we conclude that σ r ( g ) = ( v x m . . . x m e e ) ( r ) g = v ( r ) ( x ) m ( r ) . . . ( x e ) m e ( r ) g = ˜ u g u := v ( r ) . Since G is closed under σ , we have that ˜ u ∈ G . Write g = P s ∈ S g s x s where S ⊆ N e is finite, g s ∈ F ∗ and for ( s , . . . , s e ) ∈ S and x = ( x , . . . , x e ) we usethe multi-index notation x s = x s . . . x s e e . In particular, we suppose that if s , s ′ ∈ S with x s = x s ′ then s = s ′ . Then by coefficient comparison w.r.t. x i and using (31) we obtain σ r ( g i ) = ˜ u g i for any i ∈ S . Note that g i ∈ sconst G ( A , σ r ) \ { } ≤ A ∗ . Hence for any s , r ∈ S we have that σ r ( g s /g r ) = g s /g r . Thus it follows that g s /g r ∈ (const( A , σ r )) ∗ = K ∗ , i.e.,for all s ∈ S we have that g s = c s ˜ g for some c s ∈ K ∗ and ˜ g ∈ sconst G ( A , σ r ) \ { } ≤ A ∗ with σ r (˜ g ) = ˜ u ˜ g. (32)Consequently, g = ˜ g h with h = P s ∈ S c s x s . Since g ∈ A ∗ , h ∈ K [ x , . . . , x e ] ∗ . Finally,with (31) we conclude that that h ∈ const( K [ x , . . . , x e ] , σ r ) ∗ .(3) Taking s = ( s , . . . , s e ) ∈ S , it is easy to see that there is exactly one s ′ ∈ S with σ ( c s x s ˜ g ) = v x m . . . x m e e c s ′ x s ′ . This means that on both sides the same monomial x s ′ +( m ,...,m e ) in reduced form occurs.By coefficient comparison this gives σ (˜ g ) = v u − s . . . u − s e e c s ′ c s ˜ g. Thus with (31) andLemma 3.4 we get σ r (˜ g ) = v ( r ) ( c s ′ c s ) r ˜ g = ˜ u ( c s ′ c s ) r ˜ g . Hence with (32) we obtain ( c s ′ c s ) r = 1.Finally, with λ := u − s . . . u − s e e c s ′ c s we have that σ (˜ g ) = λ v ˜ g with λ r = 1 and λ ∈ A ∗ . ✷ Specializing A to a strong constant-stable difference field, the lemma reads as follows. Corollary 6.9.
Let ( F , σ ) be a difference field with K = const F which is strong constant-stable. Let ( E , σ ) be a simple R -extension of ( F , σ ) with E = F [ x ] . . . [ x e ] such that (27)holds with m i,j ∈ N , u i ∈ K ∗ , and define (29). Let ˜ G = ( F ∗ ) EF . Then: (1) r > g ∈ sconst ˜ G E \{ } we have (30) with ˜ g ∈ F ∗ and h ∈ const( K [ x , . . . , x e ] , σ r ) ∗ .(3) If σ ( g ) = v x m . . . x m e e g with v ∈ F ∗ , m i ∈ Z , then σ (˜ g ) = λ v ˜ g with λ ∈ K ∗ , λ r = 1. Proof.
Since u i ∈ K ∗ by Corollary 5.6.(1), per( u i ) = 1. Define G = F ∗ which is closedunder σ . In particular, sconst G ( F , σ k ) \ { } = F ∗ for any k >
0. In addition, sconst ˜ G E \{ } ≤ E ∗ by Corollary 4.3. Thus we can apply Lemma 6.8. The corollary follows byobserving that λ ∈ F ∗ with λ r = 1. Then by our assumption it follows that λ ∈ K ∗ . ✷ With this result we get the following reduction tactic for simple R -extensions. Lemma 6.10.
Let ( F , σ ) be a difference field with K = const F which is strong constant-stable. Let ( E , σ ) be a simple R -extension of ( F , σ ) with E = F [ x ] . . . [ x e ] where wehave (27) with m i,j ∈ N and u i ∈ K ∗ . Define r > a set { α , . . . , α s } ⊆ K ∗ of r -th roots of unity which generate multiplicatively all r -th roots ofunity of K . Let G = ( F ∗ ) EF and let f = ( f , . . . , f n ) ∈ G n with f i = ˜ f i h i where ˜ f i ∈ F ∗ and h i = x z i, . . . x z i,e e with z i,j ∈ N . Then M ( f , E ) = { ( m , . . . , m n ) | ( m , . . . , m n + s ) ∈ M ∩ M } (33) In principal, we could also take one primitive r th root of unity α . However, if α / ∈ K , we have to extendthe constant field. By efficiency reasons we prefer to stay in the original field. We remark that extendingthe constant field would not produce further relations. M = M (( ˜ f , . . . , ˜ f n , α , . . . , α s ) , F ) ,M = M (( h , . . . , h n , α , . . . , α s ) , K [ x ] . . . [ x e ]) . Proof.
Let ( m , . . . , m n ) ∈ M ( f , E ), i.e., there is a g ∈ sconst G E \ { } with σ ( g ) = f m . . . f m n n g . Hence by Corollary 6.9 it follows that g = ˜ g h with ˜ g ∈ F ∗ and h ∈ K [ x ] . . . [ x e ] ∗ . In particular, σ (˜ g ) = ˜ f m . . . ˜ f m n n λ ˜ g for λ ∈ K ∗ being an r th root of unity.Hence we can take m n +1 , . . . , m n + s ∈ N such that λ = α m n +1 . . . α m n + s s . Consequently, σ (˜ g ) = ˜ f m . . . ˜ f m n n α m n +1 . . . α m n + s s ˜ g, (34)which yields σ ( h ) = h m . . . h m n n α − m n +1 . . . α − m n + s s h. (35)Then (34) and (35) imply ( m , . . . , m n + s ) ∈ M ∩ M . Conversely, let ( m , . . . , m n ) ∈ M ∩ M . I.e., there are m i ∈ N , ˜ g ∈ F ∗ and h ∈ K [ x ] . . . [ x e ] ∗ s.t. (34) and (35) hold.Therefore σ (˜ g h ) = f m . . . f m n n ˜ g h which implies that ( m , . . . , m n ) ∈ M ( f , E ). ✷ The following remarks are in place. By Corollary 4.3 it follows that sconst G E \{ } ≤ E ∗ and thus M ( f , E ) in Lemma 6.10 has a Z -basis with rank ≤ n . In particular, we cancompute such a basis as follows. First note that both M and M given in Lemma 6.10have Z -bases with rank ≤ n + s : for M this follows since F is a field. Moreover, ifone takes H = K [ x ] . . . [ x e ] ≤ E and H = ( K ∗ ) HK , it follows by Corollary 4.3 thatsconst H H \ { } ≤ H ∗ and thus a Z -basis exists with rank ≤ n + s . Summarizing, we candetermine a Z -basis of M ( f , E ) by using (33) if bases of M and M are available. Example 6.11.
Take the ΠΣ ∗ -field ( K ( k ) , σ ) over K = Q ( ι ) with σ ( k ) = k + 1 andconsider the R -extension ( K ( k )[ x ] , σ ) of ( K ( k ) , σ ) with σ ( x ) = ι x and ord( x ) = 4 fromExample 2.9. In order to obtain a degree bound in Example 7.6 below, we need a basisof M = M ( f , K ( k )[ x ]) with f = ( kx, − xk +1 ). Here we will apply Lemma 6.10. By Ex-ample 5.4.(3) we get ford( x ) = 8. With u = 1 we determine r = 8 by (29). We define˜ f = k , ˜ f = − / ( k + 1) and h = h = x . All 8th roots of unity of K are gener-ated by α = ι . For the activation of the above lemma, we have to determine a basisof M = M (( ˜ f , ˜ f , α ) , K ( k )) = M (( k, − k +1 , ι ) , K ( k ). Here we use, e.g., the algorithmsworked out in [24] (this is the base case of our machinery, see Subsection 2.3.3) and obtainthe basis { (1 , , , (0 , , } . Moreover, we compute the basis { (1 , , , (0 , , , (0 , , } of M = M (( h , h , α ) , K [ x ]) = M (( x, x, ι ) , K [ x ]), for details see Example 6.14 below.Thus a basis of M ∩ M is { (1 , , , (0 , , } and we get the basis { (1 , } of M .By assumption (i.e., the base case in our recursion) a basis of M can be determined. Thecalculation of a Z -basis of M can be accomplished by using the following proposition. Proposition 6.12.
Let ( H , σ ) with H = K [ x ] . . . [ x e ] be a simple R -extension of ( K , σ )with a computable constant field K and given o i = ord( x i ) for 1 ≤ i ≤ e . Define G =( K ∗ ) HK and let f = ( f , . . . , f n ) ∈ G n with given λ i := ord( f i ) > ≤ i ≤ n . Then abasis of M ( f , H ) can be computed. Proof.
Define the finite sets S := { ( n , . . . , n e ) ∈ N e | ≤ n i < o i } and ˜ M := { ( m , . . . , m n ) ∈ N n | ≤ m i < λ i } . m = ( m . . . , m n ) ∈ ˜ M and check if there is a g ∈ H ∗ with σ ( g ) = f m . . . f m n n g. More precisely, we can make the Ansatz g = P i ∈ S c i x i which leadsto a linear system of equations in the c i with coefficients from K . Solving this systemgives the solution space L and we can check if the considered m from ˜ M is contained in M ( f , H ). In this way we can generate the subset M ′ = ˜ M ∩ M ( f , H ). Denote by b i ∈ K n the i th unit vector. We show thatspan( M ′ ∪ { λ b , . . . , λ n b n } ) = M ( f , H ) . (36)Namely, since M ( f , H ) is a Z -module (see the remarks above Example 6.11) and since λ i b i ∈ M ( f , H ), the left hand side is contained in the right hand side. Conversely, supposethat ( m , . . . , m n ) ∈ M ( f , H ). Then let m ′ i = m i mod λ i , i.e., 0 ≤ m ′ i < λ i with m i = m ′ i + z i λ i for some z i ∈ Z . Thus ( m , . . . , m n ) = ( m ′ , . . . , m ′ n ) + ( λ z , . . . , λ n z n ) where( m ′ , . . . , m ′ n ) ∈ ˜ M and ( λ z , . . . , λ n z n ) = z ( λ b ) + · · · + z n ( λ n b n ). Consequently,( m , . . . , m n ) is an element of the left hand side of (36). Since the number of vectors ofthe span on the left hand side is finite, we can derive a Z -basis of (36). ✷ Remark 6.13.
A basis of M ( f , H ) can be obtained more efficiently as follows. We startwith the Z -module which is given by the basis B = { λ b , . . . , λ n b n } where b i ∈ K n isthe i th unit vector. Now go through all elements from ˜ M . Take the first element m from˜ M . If it is in span( B ) (this can be easily checked), proceed to the next element. Otherwise,if it is an element from M ( f , H ) (for the check see the proof of Proposition 6.12), put it in B and transform the set again to a Z -basis. More precisely, if we compose the rows b i toa matrix, it should yield a matrix in Hermite normal form. In this way, the membershiptests for span( B ) can be carried out efficiently within the continuing calculation steps.We proceed until all elements of ˜ M are visited and update step by step B as describedabove. By construction we have that our span( B ) equals the left hand side of (36) andthus equals M ( f , H ). We remark that B consists always of n linearly independent vectors.However, the Z -span is more and more refined. Example 6.14 (Cont. Ex. 6.11) . Take the R -extension ( K [ x ] , σ ) of ( K , σ ) with K = Q ( ι ), σ ( x ) = ι x and ord( x ) = 4. We calculate a basis of M ( f , K [ x ]) with f = ( x, x, ι ) aspresented in Remark 6.13. We start with { (4 , , , (0 , , , (0 , , } whose rows form amatrix in Hermite normal form. Now we go through all elements of ˜ M , say in the order ˜ M = { (1 , , , (2 , , , (3 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (1 , , , . . . } . Since (1 , , / ∈ span( B ), we check if there is a g ∈ K [ x ] \ { } with σ ( g ) = x x ι g :this is not the case. We continue with (2 , , , , / ∈ span( B ).Now we check if there is a g ∈ K [ x ] \ { } with σ ( g ) = x x ι g . Plugging in g = g + g x + g x + g x into σ ( g ) = x g gives the constraint ( g − g ) x + ιx ( g + ιg ) + x ( − g − g ) + x ( − g − ig ) = 0 which leads to the solution g = x + ι x . Abasis of span( B ∪ { (2 , , } ) is { (2 , , , (0 , , , (0 , , } . Thus we update B to B = { (2 , , , (0 , , , (0 , , } . We have (3 , , / ∈ span( B ), but there is no g ∈ K [ x ] \ { } with σ ( g ) = x g . Similarly to (1 , , , ,
0) does not change B , and similarlyto (2 , , , ,
0) leads to the updated basis B = { (2 , , , (0 , , , (0 , , } . (0 , , B . However, for (0 , , / ∈ span( B ) we find g = x with σ ( g ) = x x ι g which yields B = { (2 , , , (0 , , , (0 , , } . We have that (0 , , , (0 , , ∈ span( B ). By arguments as in the proof of Lemma 2.16 it follows dim( L ) ≤ , , / ∈ span( B ). We find g = x + ι x with σ ( g ) = x g (as alreadyabove). Hence we update B to B = { (1 , , , (0 , , , (0 , , } (where the rows form amatrix in Hermite normal form). As it turns out, no further element from ˜ M changes B .Thus the found B is a basis of M ( f , K [ x ]). Proof 6.15. (Theorem 2.26.(2)).
By Lemma 4.10 we can reorder the generators ofthe R ΠΣ ∗ -extension such that (¯ E , σ ) is an F ∗ -simple R -extension of ( F , σ ) and ( E , σ ) isa G -simple ΠΣ ∗ -extension of (¯ E , σ ) with G = ( F ∗ ) ¯ EF . Let ¯ E = F [ x ] . . . [ x e ] with u i , α i and f ∈ G n with ˜ f i and h i as given in Lemma 6.10. By assumption we can compute abasis of M as given in Lemma 6.10. Since Problem O is solvable in K ∗ , we can compute o i = ord( x i ) and λ i = ord( u i ) by Corollary 5.6.(4). Thus we can use Proposition 6.12 tocompute a basis of M as posed in Lemma 6.10, and we get a basis of (33). Summarizing,we can solve Problem PMT in (¯ E , σ ) for G . In particular, sconst G ¯ E \ { } ≤ ¯ E ∗ byCorollary 4.3. Hence by Theorem 6.1 we can solve Problem PMT for ( E , σ ) in G E ¯ E . Since G E ¯ E = ( F ∗ ) EF by Lemma 4.12, the theorem is proven. ✷ To this end, we work out the following shortcut, resp. refined version of Theorem 2.12.(3).
Corollary 6.16.
Let ( F , σ ) be a strong constant-stable difference field with constantfield K , and let G ≤ F ∗ with sconst G F \ { } ≤ F ∗ . Let ( H , σ ) with H = F [ x ] . . . [ x r ] bea G -simple R -extension of ( F , σ ) and let ( E , σ ) be a G HF -simple ΠΣ ∗ -extension of ( H , σ ).(1) If f ∈ G HF with ord( f ) >
0, then f ∈ ( K ∗ ∩ G ) HF .(2) M ( f , E ) = M ( f , K [ x ] . . . [ x r ]) for any f = ( f , . . . , f n ) ∈ ( G HF ) n with ord( f i ) > α ∈ G HF with ord( α ) >
0. Then there is an R -extension ( E [ t ] , σ ) of ( E , σ ) with σ ( t ) t = α iff there is an R -extension ( K [ x ] . . . [ x r ][ t ] , σ ) of ( K [ x ] . . . [ x r ] , σ ) with σ ( t ) t = α . Proof. (1) Let f ∈ G HF , i.e., f = α x m . . . x m r r where α ∈ G and m i ∈ N . With ord( f ) > α ) >
0. Since ( F , σ ) is strong constant-stable, α ∈ K ∗ . Thus α ∈ K ∗ ∩ G and hence f ∈ ( K ∗ ∩ G ) HF .(2) Let f ∈ ( G HF ) n be given as above. By part 1, f i = α i x m i, . . . x m i,r where the α i ∈ K are roots of unity and m i,j ∈ N . By Lemma 4.13 we may suppose that E = H h t i . . . h t k i [ s ] . . . [ s e ] where the t i are Π-monomials and the s i are Σ ∗ -monomials.By Corollary 6.5 we have that M ( f , E ) = M ( f , H h t i . . . h t k i ). Now let ( m , . . . , m n ) ∈ M ( f , H h t i . . . h t k i ). Then there is a g ∈ H h t i . . . h t k i \ { } with σ ( g ) = u g (37)for some u = a x µ . . . x µ r r with µ i ∈ N and with a being a root of unity from K . ByCorollary 5.6.(3) we get µ := ord( u ) >
0; in addition we have that µ ′ = ford( u ) > g = q t ν . . . t ν k k with q ∈ sconst G H \ { } and ν i ∈ Z .Since u µ = 1, it follows with (37) that σ ( g µ ) = g µ . Now suppose that g depends on t m with 1 ≤ m ≤ k being maximal. Then g µ depends also on t m which contradicts toconst H h t i . . . h t k i = const H . Consequently g = q ∈ sconst G H \ { } . By Corollary 6.9 itfollows that g = ˜ g h with h ∈ K [ x ] . . . [ x r ] ∗ and ˜ g ∈ F ∗ with σ ( h ) = λ u h where λ ∈ K ∗ is a root of unity. Recall that µ ′ = ford( u ) > µ ′′ := lcm( µ ′ , ord( λ )) > σ µ ′′ ( h ) = h and ( F , σ ) is constant-stable, it follows that h ∈ K ∗ . Therefore g ∈ K [ x ] . . . [ x r ] ∗ . Summarizing, ( m , . . . , m n ) ∈ M ( f , K [ x ] . . . [ x r ]) and we conclude that M ( f , E ) ⊆ M ( f , K [ x ] . . . [ x r ]). The other direction is immediate.(3) The third part follows by parts 1 and 2 of the corollary and Theorem 3.17. ✷ . The algorithmic machinery III: Problem PFLDE We aim at proving Theorems 2.23.(2) and 2.26.(3), i.e., providing recursive algorithmsthat reduce Problem PFLDE from a given R ΠΣ ∗ -extension to its ground ring (resp. field).If we are considering single-rooted R ΠΣ ∗ -extensions (Theorem 2.23.(2)), we rely heavilyon the fact that for a given difference ring ( G , σ ) with constant field K and given group G ≤ G ∗ we have that sconst G ( G , σ ) \ { } ≤ G ∗ . This property allows us to assume thatfor any f ∈ G n and any u ∈ G the K -vector space V = V ( u, f , ( G , σ )) has a basis withdimension ≤ n +1; see Lemma 2.17. In particular, our reduction algorithm is based on theassumption that there are algorithms available that solve Problems PFLDE and PMT in( G , σ ) for G . For general simple R ΠΣ ∗ -extensions over a strong constant-stable differencefield ( G , σ ) (Theorem 2.26.(3)) we need stronger properties: all what we stated aboveshould hold not only for ( G , σ ) but must hold for ( G , σ l ) with l ≥
1. For the currentlyexplored difference fields ( G , σ ) with these properties we refer to Subsection 2.3.3. ΠΣ ∗ -extensions In this subsection we present a reduction method for ΠΣ ∗ -extensions which can besummarized with the following theorem. Theorem 7.1.
Let ( A , σ ) be a computable difference ring and let G ≤ A ∗ with sconst G A \{ } ≤ A ∗ . Let ( A h t i , σ ) be a G -simple ΠΣ ∗ -extension of ( A , σ ).(1) If t is a Σ ∗ -monomial and Problem PFLDE is solvable in ( A , σ ) for G , then Prob-lem PFLDE is solvable in ( A h t i , σ ) for G A h t i A .(2) If t is a Π-monomial and Problems PFLDE and PMT are solvable in ( A , σ ) for G ,then Problem PFLDE is solvable in ( A h t i , σ ) for G A h t i A .In the following let ( A h t i , σ ) ≥ ( A , σ ) be a ΠΣ ∗ -extension as given in the theorem with σ ( t ) = α t + β where α ∈ G and β = 0, or α = 1 and β ∈ A . Furthermore, we define˜ G = G A h t i A and suppose that we are given a u ∈ ˜ G , i.e., u = v t m , with v ∈ G, m ∈ Z , (38)and an f = ( f , . . . , f n ) ∈ ˜ G n . By Theorem 3.20 we have that sconst ˜ G A h t i \ { } ≤ A h t i ∗ and hence by Lemma 2.17 a basis of V ( u, f , A h t i ) with dimension ≤ n + 1 exists.Subsequently, we will prove Theorem 7.1, i.e., we will work out a reduction strategythat provides a basis of V ( u, f , A h t i ) under the assumption that one can solve ProblemPFLDE in ( A , σ ) for G if t is a Σ ∗ -monomial, resp. Problems PMT and PFLDE in ( A , σ )for G if t is a Π-monomial. The two main steps of this reduction will be described in thefollowing two Subsections 7.1.1 and 7.1.2. The first essential step is to search for degree bounds: we will determine a, b ∈ Z suchthat V ( u, f , A h t i a,b ) = V ( u, f , A h t i ) (39)holds; for the definition of the truncated set of (Laurent) polynomials see (18). Fortechnical reasons we also require that the constraintmax( b, b + m ) ≥ ˜ b (40)38olds where m and ˜ b are given by (38) and˜ b = max(deg( f ) . . . , deg( f n )) . (41)The recovery of these bounds (see Lemmas 7.2 and 7.5 below) is based on generalizationsof ideas given in [24]; for further details and proofs in the setting of difference fields seealso [43,46].If t is a Σ ∗ -monomial, then A h t i = A [ t ] forms a polynomial ring, α = 1 and ˜ G = G ; inparticular we have m = 0 in (38). In this case, we can utilize the following lemma. Lemma 7.2.
Let ( A [ t ] , σ ) be a Σ ∗ -extension of ( A , σ ) and let G ≤ A ∗ such thatsconst G A \ { } ≤ A ∗ holds. Let f ∈ A [ t ] and u ∈ G . Then any solution g ∈ A [ t ] of σ ( g ) − u g = f is bounded by deg( g ) ≤ max(deg( f ) + 1 , Proof.
Suppose there is a g ∈ A [ t ] with deg( g ) > max(deg( f ) + 1 , γ ∈ A with σ ( γ ) − γ = σ ( t ) − t which contradicts to Theorem 2.12.(1). ✷ Thus we can set a = 0 and b = max(˜ b + 1 ,
0) to guarantee that (39) and (40) hold.
Example 7.3 (Cont. Ex. 2.15) . Consider the Σ ∗ -extension ( A [ S ] , σ ) of ( A , σ ) with A = Q ( k )[ x ][ y ][ s ] and σ ( S ) = S + − x yk +1 from Example 2.15.(2). As stated in Example 2.15.(3),we want to determine a g ∈ A [ S ] with σ ( g ) − g = f where f = y k s , i.e., we want tofind a basis of V (1 , f , A [ S ]) with f = ( k sy ) ∈ A [ S ] . Using Lemma 7.2 it follows thatdeg( g ) ≤
1. Consequently, V (1 , f , A [ S ]) = V (1 , f , A [ S ] ). Using our methods below (seeExample 7.8) we get the basis { (1 , g ) , (0 , } with g as given in (14).If t is a Π-monomial, then A h t i = A [ t, t ] is a ring of Laurent polynomials and β = 0.First suppose that u / ∈ A , i.e., m ∈ Z \ { } as given in (38).If f i = 0 for all i , it is easy to see that V ( u, f , A h t i ) = V ( u, f , { } ), i.e., a = 0 and b = − f i are 0, we can use the following fact; the proof is left to the reader. Lemma 7.4.
Let ( A h t i , σ ) be a Π-extension of ( A , σ ). Let v ∈ A ∗ , m ∈ Z \ { } , f = P µi = λ f i t i ∈ A h t i with λ, µ ∈ Z and g = P ˜ µi =˜ λ g i t i ∈ A h t i with ˜ λ, ˜ µ ∈ Z and g ˜ λ = 0 = g ˜ µ such that σ ( g ) − v t m g = f . Then max( λ, λ − m ) ≤ ˜ λ and ˜ µ ≤ min( µ, µ − m ).Namely, define ˜ a = min(ldeg( f ) . . . , ldeg( f n )) . Note that in this scenario we have that ˜ a, ˜ b ∈ Z ; for the definition of ˜ b see (41). Henceby setting a = ˜ a and b = ˜ b , we can conclude with Lemma 7.4 that (39) and (40) hold.What remains to consider is the case u ∈ G with m = 0. Here we utilize Lemma 7.5.
Let ( A h t i , σ ) be a Π-extension of ( A , σ ) with G ≤ A ∗ where sconst G A \{ } ≤ A ∗ and α = σ ( t ) /t ∈ G . Let u ∈ G , f = P µi = λ f i t i ∈ A h t i and g = P ˜ µi =˜ λ g i t i ∈ A h t i with g ˜ λ = 0 = g ˜ µ and σ ( g ) − u g = f. (42)If there is a ν ∈ Z with σ ( γ ) = α − ν u γ (43)for some γ ∈ sconst G A \ { } , then ν is uniquely determined and we have that min( λ, ν ) ≤ ˜ λ and ˜ µ ≤ max( µ, ν ). If there is not such a ν , we have that λ ≤ ˜ λ and ˜ µ ≤ µ .39 roof. Suppose there is a ν ∈ Z with (43) for some γ ∈ sconst G A \ { } . Take in addition,˜ ν ∈ Z such that σ (˜ γ ) = α − ˜ ν u ˜ γ holds for some ˜ γ ∈ sconst G A \ { } . Then σ ( γ/ ˜ γ ) = α ˜ ν − ν γ/ ˜ γ . Since t is a Π-monomial it follows by Theorem 2.12.(2) that ν = ˜ ν , i.e., ν is uniquely determined. Now suppose that there is an i with g i = 0 where we have i < min( λ, ν ) or i > max( µ, ν ). Then by coefficient comparison in (42) we get σ ( g i ) = u α − i g i with g i ∈ sconst G A \ { } . Consequently ν = i , a contradiction. Otherwise, suppose thatthere is not such a ν ∈ Z . Then by the same arguments it follows that ˜ λ < λ or ˜ µ > µ isnot possible, i.e., ˜ λ ≥ λ and ˜ µ ≤ µ . This completes the proof. ✷ Therefore we derive the desired bounds as follows. First, solve Problem PMT and computea basis of M (( α, u ) , A ). Then given a basis, we can decide constructively if there is a ν ∈ Z such that (43) holds. If yes, take the uniquely determined ν and we can take a = min(˜ a, ν )and b = max(˜ b, ν ) to obtain (39) and (40). Otherwise, if there is not such a ν , we can set a = min(˜ a,
0) and b = max(˜ b, − Example 7.6 (Cont. Ex. 2.18) . Take the difference field ( K ( k )[ x ] h t i , σ ) with α = σ ( t ) /t = x k defined in Example 2.18. In order to find the identity (7), we need abasis of V = V ( u, (0) , K ( k )[ x ] h t i ) with u = − xk +1 ∈ G := ( K ( k ) ∗ ) K ( k )[ x ] h t i K ( k ) ; note that G ≤ K ( k )[ x ] h t i ∗ with sconst G K ( k )[ x ] h t i \ { } ≤ K ( k )[ x ] h t i ∗ . In this setting we applyLemma 7.5. I.e., we compute a basis of M (( α, u ) , K ( k )[ x ]) = M (( k x, − x ( k +1) ) , K ( k )[ x ]). Asworked out in Example 6.11, a basis is { (1 , } . Thus we find ν = − g ∈ K ( k )[ x ] \ { } with (7.5). We conclude that V = V ( u, (0) , K ( k )[ x ] h t i − − ). Using ourmethods below (see Example 7.7) we arrive at the basis (0 , x ( ι + x ) /k/t ) , (1 , } of V .Summarizing, we obtain bounds a, b ∈ Z such that (39) and (40) hold. For Π-monomialswe rely on the extra assumption that Problem PMT is solvable in ( A , σ ) for G . The following degree reduction has been introduced in [24] in the setting of differencefields. Subsequently, we present the basic ideas in the setting of difference rings; furthertechnical details can be found in [42, Thm. 3.2.2] and [49,59].We want to determine all c , . . . , c n ∈ K = const A and g i ∈ A in g = P bi = a g i t i suchthat the following parameterized equation holds: σ ( g ) − u g = c f + · · · + c n f n . (44)If b < a , we are in the base case: g = 0 and a basis of V ( u, f , A h t i ) = V ( u, f , { } ) can bedetermined by linear algebra.Otherwise, we continue as follows. Due to (40), it follows that λ := max( b, b + m ) is thehighest possible exponent in (44). Let ˜ f i be the coefficient of the term t λ in f i . Then bycoefficient comparison w.r.t. t λ in (44) we get the following constraints:If m > − v g b = c ˜ f + · · · + c n ˜ f n ; (45)if m = 0, α b σ ( g b ) − v g b = c ˜ f + · · · + c n ˜ f n ; (46)if m < α b σ ( g b ) = c ˜ f + · · · + c n ˜ f n . (47)40or the cases m > m < K -vectorspaces { ( c , . . . , c n , g m ) | (45) holds } and { ( c , . . . , c n , g m ) | (47) holds } by linear algebra.Moreover, if m = 0, equation (46) can be written in the form σ ( g b ) − v α − b g b = c ˜ f α − b + · · · + c n ˜ f n α − b where v α − b ∈ G and ˜ f i α − b ∈ A . Thus a basis of V ( v α − b , ( ˜ f α − b , . . . , ˜ f n α − b ) , A ) (48)can be determined under our assumption that one can solve Problem PFLDE in ( A , σ )for G . Now we plug in this partial solution (i.e., the possible leading coefficient g b withthe corresponding linear combinations of the f i ), and end up at a new first- order param-eterized difference equation where the highest possible coefficient is λ −
1. In other words,we reduced the problem by degree reduction . We continue to search for the next highestcoefficient g b − . Hence we proceed recursively by updating λ → λ − b → b − λ − V ( u, f , A h t i a,b ); for further technical details we refer to [24, Thm 12] or [59, Section 3.1].Summarizing, solving various instances of Problem PFLDE with the degree reductions b → b − → · · · → a − V ( u, f , A h t i ). This concludes the proof of Theorem 7.1. Example 7.7 (Cont. Ex. 7.6) . We know that g = g − t − . Plugging in g into σ ( g )+ xk +1 =0 yields σ ( g − ) + x kk +1 g − = 0 . Therefore we look for a basis of V ( − x kk +1 , (0) , K ( k )[ x ]). Byusing the algorithms presented in Subsection 7.2 we get the basis { (1 , x ( ι + x ) /k ) , (0 , } .This finally gives the basis (0 , x ( ι + x ) /k/t ) , (1 , } of V ( − xk +1 , (0) , K ( k )[ x ] h t i ). Example 7.8 (Cont. Ex. 7.3) . We want to find a basis of V = V (1 , f , A [ S ] ) with A = Q ( k )[ x ][ y ][ s ] and f = ( y k s ). Hence we make the Ansatz ( c , g + g S ) ∈ V withthe indeterminates c ∈ Q and g , g ∈ A such that σ ( g + g S ) − ( g + g S ) = c y k s (49)holds. Doing coefficient comparison w.r.t. S yields the constraint σ ( g ) − g = c V (1 , ˜f , Q ( k )[ x ][ s ])with ˜f = (0) ∈ A . In this particular instance, the Q -basis { (1 , , (0 , } is immediateutilizing the fact that the constants are precisely Q . Summarizing, the solutions are( c , g ) ∈ Q . Consequently, our Ansatz can be refined with ( c , g + c S ) ∈ V where c ∈ Q , c (= g ) ∈ Q and g ∈ A such that σ ( g + c S ) − ( g + c S ) = c k sy holds.Bringing the c S part to the right hand side yields the new equation σ ( g ) − g = c y k s − c h (50)with h = σ ( S ) − S = − x yk +1 ∈ A . In other words, we need a basis of V (1 , h , A ) with h = ( y k s, x yk +1 ) ∈ A . Now we apply again the reduction method, but this time inthe smaller ring A without the Σ ∗ -monomial S . We skip all the details, but refer to aparticular subproblem that we will consider in Example 7.13. Finally, we get the basis { (0 , , , (1 , , (cid:0) (1 − k ) − x (cid:1) y + s (cid:0) ( k − k + 1) x − ( k − k (cid:1) y } Note that we reduced the problem to find a polynomial solution of (49) with maximal degree 1to a polynomial solution of (49) with maximal degree 0. This degree reduction has been achieved byintroducing an extra parameter c . In general, the more Σ ∗ -monomials are involved, the more parameterswill be introduced within the proposed degree reduction. V (1 , h , A ). Thus we can reconstruct the basis { (1 , g ) , (0 , } of V with g given in (14).Note that the reduction of Theorem 7.1 simplifies to Karr’s field version given in [24] ifone specializes A to a field and sets G = A ∗ = A \ { } . However, the presented versionworks not only for a field, but for any difference ring ( A , σ ) as specified in Theorem 7.1.Subsequently, we will exploit this enhancement in order to treat (nested) R -extensions. R -extensions and thus for R ΠΣ ∗ -extensions In order to treat simple and single-rooted R ΠΣ ∗ -extensions (Theorem 2.23.(2)), weutilize the following proposition. Proposition 7.9.
Let ( A , σ ) be a computable difference ring with G ≤ A ∗ and sconst G A \{ } ≤ A ∗ . Let ( A [ t ] , σ ) be an R -extension of ( A , σ ) of given order d with σ ( t ) t ∈ G .Then Problem PFLDE is solvable in ( A [ t ] , σ ) for G if it is solvable in ( A , σ ) for G Proof.
The proof follows by an adapted degree reduction presented in the proof of The-orem 7.1; see Subsection 7.1.2. Let u ∈ G and f = ( f , . . . , f n ) ∈ A [ t ] n . By definition,it follows that a solution g ∈ A [ t ] and c , . . . , c n ∈ K = const A of (44) is of the form g = P bi = a g i t i with a := 0 and b := d −
1. Thus the bounds are immediate (under theassumption that d has been determined; see Section 5). Since P d − i =0 h i t i = P d − i =0 ¯ h i t i iff h i = ¯ h i , we can activate the degree reduction as outlined in Subsection 7.1.2. Namely,by coefficient comparison of the highest term we always enter in the case (46) (note that m = 0 in (38)). By assumption we can solve Problem PFLDE in ( A , σ ) for G and thus wecan determine a basis of (48). By recursion we finally obtain a basis of V ( u, f , A [ t ]). ✷ Proof 7.10. (Theorem 2.23.(2)).
Since Problem PFLDE is solvable in ( G , σ ) for G , it follows by iterative applications of Theorem 7.1 and Corollary 4.6.(1) that Prob-lem PFLDE is solvable in ( H , σ ) for ˜ G with H = G h t i . . . h t r i and that sconst ˜ G H \ { } ≤ H ∗ . Thus by iterative applications of Propositions 7.9 and 3.23 we conclude that Prob-lem PFLDE is solvable in ( ¯ H , σ ) for ˜ G with ¯ H = H h x i . . . h x u i and that sconst ˜ G ¯ H \{ } ≤ ¯ H ∗ . Finally, by applying iteratively Theorem 7.1 and Corollary 4.6.(1) it follows thatProblem PFLDE is solvable in ( E , σ ) for ˜ G . Note that in Proposition 7.9 we have toknow the values ord( x i ) = ord( α i ) with α i ∈ G (either as input or by computing themfirst by solving instances of Problem O in G ). ✷ Finally, we present the underlying reduction method for simple R ΠΣ ∗ -extensions (The-orem 2.26.(3)) which is based on the following lemma and proposition. Lemma 7.11.
Let ( A , σ ) be a difference ring, f ∈ A , u ∈ A ∗ and λ ∈ N \ { } . Then σ ( g ) − u g = f implies that σ λ ( g ) − u ( λ ) g = λ − X j =0 u ( λ ) u ( j +1) σ j ( f ) . (51) Proof.
From σ ( g ) − u g = f we get σ j +1 ( g ) − σ j ( u ) σ j ( g ) = σ j ( f ) for all j ∈ N . Multiplyingit with u ( λ ) /u ( j +1) yields u ( λ ) u ( j +1) σ j +1 ( g ) − u ( λ ) u ( j ) σ j ( g ) = u ( λ ) u ( j +1) σ j ( f ) . Summing this equationover j from 0 to λ − ✷ roposition 7.12. Let ( A , σ ) be a constant-stable and computable difference ring withconstant field K . Let G ≤ A ∗ be closed under σ with sconst G ( A , σ l ) \ { } ≤ A ∗ for all l >
0. Let ( E , σ ) with E = A h x i . . . h x r i be a G -simple R -extension of ( A , σ ) whereord( x i ) > x i ) > ≤ i ≤ r are given and where sconst ( G EA ) E \ { } ≤ E ∗ .If Problem PFLDE is solvable in ( G , σ l ) for G for all l >
0, it is solvable in ( E , σ ) for G EA . Proof.
Let K = const A , let E = A h x i . . . h x r i , let f = ( f , . . . , f n ) ∈ E n and let u = v x m . . . x m r r ∈ G EA with v ∈ G and m i ∈ N . We will present a reduction method toobtain a basis of V ( u, f , E ). Set α := x m . . . x m r r . Then by Lemma 5.2 it follows thatord( α ) > x i ) with 1 ≤ i ≤ r . Hence wecan activate Lemma 5.3.(4) and can compute ford( α ) >
0. Now take λ = lcm(ford( α ) , per( x ) , . . . , per( x r )) . (52)Thus we have that α ( λ ) = 1 and σ λ ( x i ) = x i for all 1 ≤ i ≤ r . Finally, define w := u ( λ ) = ( α v ) ( λ ) = v ( λ ) ∈ G. (53)Now let ( c , . . . , c n , g ) ∈ V ( u, f , E ), i.e., we have that (44). Thus Lemma 7.11 yields σ λ ( g ) − w g = c ˜ f + · · · + c n ˜ f n (54)with ˜ f i = λ − X j =0 u ( λ ) u ( j +1) σ j ( f i ) . (55)Hence V ( u, f , E ) is a subset of˜ V = { ( c , . . . , c n , g ) ∈ K n × E | (54) holds } . (56)Note that ˜ V is a K -subspace of K n × E . Thus V ( u, f , E ) is a subspace of ˜ V over K . First,we show that ˜ V has a finite basis and show how one can compute it. For this task define S := { ( n , . . . , n r ) ∈ N r | ≤ n i < ord( x i ) } . Write g = P i ∈ S g i x i and ˜ f j = P i ∈ S ˜ f j, i x i in multi-index notation. Since σ λ ( x i ) = x i , it follows by coefficient comparison that for i ∈ S we have that σ λ ( g i ) − w g i = c ˜ f , i + · · · + c n ˜ f n, i . By assumption, sconst G ( A , σ λ ) \ { } ≤ A ∗ . In particular, since ( A , σ ) is constant-stable,we have that const( A , σ λ ) = K . Thus with our w ∈ G and ˜f i = ( ˜ f , i , . . . , ˜ f n, i ) ∈ A n wecan solve Problem PFLDE in ( A , σ λ ) with constant field K . Hence we get for all i ∈ S the bases for V i = V ( w, ˜f i , ( A , σ λ )) . (57)Note that by construction it follows that ˜ V from (56) is given by˜ V = { ( c , . . . , c n , X i ∈ S g i x i ) | ( c , . . . , c n , g i ) ∈ V i } . (58)Thus by linear algebra we get a basis of (58), say b , . . . , b s ∈ K n × E . Recall that V ( u, f , ( E , σ )) is a K -subspace of (58). To this end, we make the Ansatz ( c , . . . , c n , g ) = d b + · · · + d s b s for indeterminates d , . . . , d s ∈ K and plug in the generic solutioninto (44). This yields another linear system with unknowns ( d , . . . , d s ). Solving thissystem enables one to derive a basis of V ( u, f , E ). ✷ By a mild modification of the proof it suffices to take a λ such that α ( λ ) ∈ const A holds. xample 7.13 (Cont. Ex. 7.8) . In order to compute a basis of V (1 , h , A ) in Ex. 7.8,the recursive reduction enters in the following subproblem. We are given the R -extension( Q ( k )[ x ] , σ ) of ( Q ( k ) , σ ) with σ ( x ) = − x and need a basis of V ( x, f , Q ( k )[ x ]) with f =( f , f , f ) = ( ( k − x k + − k − k k , − xk , x ) = 2 andford( x ) = 4. Hence using (52) we determine λ = lcm(ford( x ) , per( x )) = 4. Using (55)with u = x yields ( ˜ f , ˜ f , ˜ f ) = ( − k +4 k +1 k ( k +2) − x ( k +1)( k +3) , − x ( k +1)( k +3) − k ( k +2) , S = { (0) , (1) } ⊆ N we get ˜f (0) = ( ˜ f , (0) , ˜ f , (0) , ˜ f , (0) ) = ( − k + 4 k + 1 k ( k + 2) , − k ( k + 2) , ˜f (1) = ( ˜ f , (1) , ˜ f , (1) , ˜ f , (1) ) = ( − k + 1)( k + 3) , − k + 1)( k + 3) , f = P ( m ) ∈ S ˜f ( m ) x m = ˜f (0) + ˜f (1) x . Following (53) we get w = 1 and we have to com-pute bases of the (57) with i ∈ S . Here we obtain the basis { (0 , , , , (1 , − , , − k/ } of V (0) = V (1 , ˜f (0) , ( Q ( k ) , σ )) and the basis { ( − , , , , (0 , , , , (0 , , , } of V (1) = V (1 , ˜f (0) , ( Q ( k ) , σ )). Therefore a basis of˜ V = { ( c , c , c , g ) ∈ Q × Q ( k )[ x ] | σ ( g ) − g = c ˜ f + c ˜ f + c ˜ f } = { ( c , c , c , X ( i ) ∈{ (0) , (1) } g i x i ) | ( c , c , c , g ( i ) ) ∈ V ( i ) } can be read off: { (1 , − / , , − k/ , (0 , , , , (0 , , , } . Since V ( x, f , Q ( k )[ x ]) is a Q -subspace of ˜ V , we plug in ( c , c , c , g ) = d (1 , − / , , − k )+ d (0 , , , d (0 , , , x )+ d (0 , , ,
1) with unknowns d , d , d , d ∈ Q into (44). Together with our given f i and u we get the linear constraint ( d − d + 2 d ) + x ( − d − d ) = 0 or equivalently the linearconstraints − d − d = 0 and ( d − d + 2 d ). This yields d = d and d = − d . Thuswe obtain the generic solution d (1 , − , , − k + x − ) + d (0 , , ,
0) of V ( x, f , Q ( k )[ x ]),i.e., the basis { (1 , − , , − k + x − ) , (0 , , , } of V ( x, f , Q ( k )[ x ]). Remark 7.14. (1) In the underlying algorithm of Proposition 7.12 we construct for all i ∈ S the solution spaces given in (57) and combine them in one stroke as proposedin (58). This approach is interesting if one wants to perform calculations in parallel.Another approach is to apply similar tactics as given in Subsection 7.1: compute a basisof one of the (57), plug in the found solutions and continue with an updated Ansatz interms of the remaining monomials. In this way, one usually shortens step by step thelength of the vectors ˜f i in (57) and ends up very soon at a trivial situation (shortcut).(2) A different approach is to consider an R -extension ( F [ x ] , σ ) of ( F , σ ) of order d as aholonomic expression [61,15,30] over a difference field. Then as worked out in [47,17], asolution g = P d − i =0 g i x i and c i ∈ const F of (9) leads to a coupled system of first-order dif-ference equations in terms of the g i that can be uncoupled explicitly. More precisely, thereis an explicitly given formula that constitutes a higher-order parameterized linear differ-ence equation in g d − and the parameters c i . Solving this difference equation in terms of g d − and the c i delivers automatically the remaining coefficients g i , i.e., the solution g of (9). Here one usually has to solve a general higher-order linear difference equation. Forfurther details on the holonomic Ansatz in the context of algebraic ring extensions (alsoon handling such objects in the basis of idempotent elements [60,19]) we refer to [17].44he advantage of the reduction technique proposed in Proposition 7.12 is that it canbe applied in one stroke for nested R -extensions. In particular, Problem PFLDE canbe always reduced again to Problem PFLDE by possibly switching to ( F , σ k ) for some k >
1. In this way, general higher-order linear difference equations can be avoided.Combining all algorithmic parts of this article we obtain the following result.
Theorem 7.15.
Let ( E , σ ) with E = F h t i . . . h t e i be a simple R ΠΣ ∗ -extension of aconstant-stable and computable field ( F , σ ). Suppose that for all R -monomials the periodsare positive, and the orders and periods of the R -monomials are given explicitly. ThenProblem PFLDE in ( E , σ ) for ( F ∗ ) EF is solvable if one of the following holds.(1) All t i are R Σ ∗ -monomials and PFLDE is solvable in ( F , σ k ) for F ∗ for all k > F , σ ) for F ∗ and Problem PFLDE is solvable in ( F , σ k )for F ∗ for all k > Proof.
Let H = ( F ∗ ) EF . Recall that by Theorem 2.24 we have that sconst H E \ { } ≤ E ∗ ,i.e., Problem PFLDE is applicable in ( E , σ ) for H . By Lemma 4.10 we can reorderthe generators of the R ΠΣ ∗ -extension such that (¯ E , σ ) is an F ∗ -simple R -extension of( F , σ ) and ( E , σ ) is a G -simple ΠΣ ∗ -extension of (¯ E , σ ) with G = ( F ∗ ) ¯ EF . Note thatthe multiplicative group F ∗ is closed under σ , sconst ( F ∗ ) ( F , σ l ) = F ∗ for all l > ( F ∗ ) ¯ E \ { } ≤ ¯ E ∗ by Corollary 4.3. Thus we can apply Proposition 7.12. HenceProblem PFLDE is solvable in (¯ E , σ ) for G . If we are in case (1), i.e., no Π-monomialsoccur, we can apply iteratively Theorem 7.1 and obtain an algorithm to solve ProblemPFLDE in ( E , σ ) for G E ¯ E = H . If we are in case (2), i.e., Π-monomials may occur, weexploit in addition our assumptions together with Theorem 2.26.(2). This shows that wecan solve Problem PMT in ( E , σ ) for H (and in any sub-difference ring by truncatingthe tower of extensions). Again the iterative application of Theorem 7.1 shows thatProblem PFLDE is solvable in ( E , σ ) for G E ¯ E = H . ✷ Proof 7.16. (Theorem 2.26.(3)).
Let ( E , σ ) be a simple R ΠΣ ∗ -extension of ( F , σ )where ( F , σ ) is computable and strong constant-stable. Then by Corollary 5.6 (parts 3and 4) the periods and orders of all R -monomials are positive and can be computed.Thus Theorem 7.15.(2) is applicable which completes the proof. ✷ We remark that in Theorem 2.26.(3) one can drop the condition that Problem PMT issolvable in ( F , σ ) for F ∗ if in the R ΠΣ ∗ -extension no Π-monomials occur, i.e., one appliespart one and not part two of Theorem 7.15.
8. Conclusion
We provided important building blocks that extend the well established difference fieldtheory to a difference ring theory. In this setting one can handle in addition objects suchas (4). We elaborated algorithms for the (multiplicative) telescoping problem (Problems Tand MT) and the (multiplicative) parameterized telescoping problem (Problems PT andPMT). In particular, Problem PT enables one to apply Zeilberger’s creative telescopingparadigm in the rather general class of simple R ΠΣ ∗ -extensions. In order to derive thesealgorithms we showed that certain semi-constants (resp. semi-invariants) of the differencerings under consideration form a multiplicative group.45urrently, the underlying engine of Theorem 2.23 with the ground field machinery ofSubsection 2.3.3 is fully implemented within the summation package Sigma . In this wayone can treat big classes of indefinite nested sums and products involving algebraic objectslike ( − k . In particular, one can treat d’Alembertian solutions of linear recurrences asworked out in Subsection 2.4. We emphasize that these algorithms are enhanced bythe refinements described in [45,48,51,53,8,59] in order to find sum representations withcertain optimality criteria, like optimal nesting depth.The machinery to handle nested R -extensions (see Theorem 2.26) is not incorporated in Sigma yet. First, further investigations will be necessary so that the new algorithms canbe merged with the difference field enhancements of
Sigma .Another challenging task is to push forward the difference ring theory and the under-lying algorithms in order to relax the requirements in Theorems 2.23 and 2.26 that the R ΠΣ ∗ -extensions are simple and/or that the ground difference ring is strong constant-stable. In this regard, we refer to the comments given in Example 2.20.In any case, the currently developed toolbox widens the class of indefinite nested sumsand products in the setting of difference rings. We are looking forward to see new kindsof applications that can be attacked with this machinery. Acknowledgement
I would like to thank Michael Singer and the anonymous referee for their helpful com-ments and suggestions to improve the presentation of this article.
Appendix: A short index M ( f , A ), 10 V ( u, f , A ), 11 A h t i , 7 A h t i a,b , 17deg, 17( A , σ ) ≤ ( ˜ A, ˜ σ ), 5 G AG , 12ford( f ), 27sconst( A , σ ), sconst A , 10 h S i , 4ldeg, 17ord( f ), 6per( f ), 27sconst G ( A , σ ), sconst G A , 10 f ( k,σ ) , f ( k ) , 17ΠΣ ∗ -field, 7constant field/ring, 4difference ring/field, 4ring/field extension, 5extension(nested) Π,Σ ∗ , R , R Π, R Σ ∗ ,ΠΣ ∗ , R ΠΣ ∗ , 7Π, 5 R , 7Σ ∗ , 5algebraic, 6simple, G -simple, 12single-rooted, 13unimonomial, 5functiondegree, 17factorial order, 27order, 6period, 27rising factorial, 17 The
Sigma package can be downloaded from ∗ , R , R Π, R Σ ∗ ,ΠΣ ∗ , R ΠΣ ∗ , 7simple, G -simple, 12ProblemFPLDE, 11MT, 8O, 8PMT, 11PT, 4 T, 4product group, 12ring (strong) constant-stable, 14connected, 16constant-stable, 14reduced, 16semi-constant, 10 References [1] J. Ablinger, A. Behring, J. Bl¨umlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round,C. Schneider, and F. Wissbrock. The 3-loop non-singlet heavy flavor contributions and anomalousdimensions for the structure function F ( x, Q ) and transversity. Nucl. Phys. B , 886:733–823, 2014.arXiv:1406.4654 [hep-ph].[2] J. Ablinger, J. Bl¨umlein, A. De Freitas A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider,and F. Wissbrock. The transition matrix element A gq ( N ) of the variable flavor number scheme at O ( α s ). Nucl. Phys. B , 882:263–288, 2014. arXiv:1402.0359 [hep-ph].[3] J. Ablinger, J. Bl¨umlein, and C. Schneider. Harmonic sums and polylogarithms generated bycyclotomic polynomials.
J. Math. Phys. , 52(10):1–52, 2011. [arXiv:1007.0375 [hep-ph]].[4] J. Ablinger, J. Bl¨umlein, and C. Schneider. Analytic and algorithmic aspects of generalized harmonicsums and polylogarithms.
J. Math. Phys. , 54(8):1–74, 2013. arXiv:1302.0378 [math-ph].[5] S. A. Abramov. On the summation of rational functions.
Zh. vychisl. mat. Fiz. , 11:1071–1074, 1971.[6] S. A. Abramov, M. Bronstein, M. Petkovˇsek, and C. Schneider.
In preparation , 2015.[7] S. A. Abramov and M. Petkovˇsek. D’Alembertian solutions of linear differential and differenceequations. In J. von zur Gathen, editor,
Proc. ISSAC’94 , pages 169–174. ACM Press, 1994.[8] S. A. Abramov and M. Petkovˇsek. Polynomial ring automorphisms, rational ( w, σ )-canonical forms,and the assignment problem.
J. Symbolic Comput. , 45(6):684–708, 2010.[9] A. Bauer and M. Petkovˇsek. Multibasic and mixed hypergeometric Gosper-type algorithms.
J. Symbolic Comput. , 28(4–5):711–736, 1999.[10] Thomas Becker and Volker Weispfenning.
Gr¨obner bases , volume 141 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1993. A computational approach to commutative algebra,In cooperation with Heinz Kredel.[11] J. Bl¨umlein, A. Hasselhuhn, S. Klein, and C. Schneider. The O ( α s n f T F C A,F ) contributions tothe gluonic massive operator matrix elements.
Nucl. Phys. B , 866:196–211, 2013. [arXiv:1205.4184[hep-ph]].[12] J. Bl¨umlein, S. Klein, C. Schneider, and F. Stan. A symbolic summation approach to Feynmanintegral calculus.
J. Symbolic Comput. , 47:1267–1289, 2012.[13] M. Bronstein.
Symbolic Integration I, Transcendental functions . Springer, Berlin-Heidelberg, 1997.[14] M. Bronstein. On solutions of linear ordinary difference equations in their coefficient field.
J. Symbolic Comput. , 29(6):841–877, 2000.[15] F. Chyzak. An extension of Zeilberger’s fast algorithm to general holonomic functions.
DiscreteMath. , 217:115–134, 2000.[16] R. M. Cohn.
Difference Algebra . Interscience Publishers, John Wiley & Sons, 1965.
17] B: Er¨ocal.
Algebraic extensions for summation in finite terms . PhD thesis, RISC, Johannes KeplerUniversity, Linz, February 2011.[18] R. W. Gosper. Decision procedures for indefinite hypergeometric summation.
Proc. Nat. Acad. Sci.U.S.A. , 75:40–42, 1978.[19] C. Hardouin and M.F. Singer. Differential Galois theory of linear difference equations.
Math. Ann. ,342(2):333–377, 2008.[20] P. A. Hendriks and M. F. Singer. Solving difference equations in finite terms.
J. Symbolic Comput. ,27(3):239–259, 1999.[21] M. van Hoeij. Finite singularities and hypergeometric solutions of linear recurrence equations.
J. Pure Appl. Algebra , 139(1-3):109–131, 1999.[22] P. Horn, W. Koepf, and T. Sprenger. m -fold hypergeometric solutions of linear recurrence equationsrevisited. Math. Comput. Sci. , 6(1):61–77, 2012.[23] G. Karpilovsky. On finite generation of unit groups of commutative group rings.
Arch. Math.(Basel) , 40(6):503–508, 1983.[24] M. Karr. Summation in finite terms.
J. ACM , 28:305–350, 1981.[25] M. Karr. Theory of summation in finite terms.
J. Symbolic Comput. , 1:303–315, 1985.[26] M. Kauers and P. Paule.
The concrete tetrahedron . Texts and Monographs in Symbolic Computation.SpringerWienNewYork, Vienna, 2011. Symbolic sums, recurrence equations, generating functions,asymptotic estimates.[27] M. Kauers and C. Schneider. Application of unspecified sequences in symbolic summation. In J.G.Dumas, editor,
Proc. ISSAC’06. , pages 177–183. ACM Press, 2006.[28] M. Kauers and C. Schneider. Indefinite summation with unspecified summands.
Discrete Math. ,306(17):2021–2140, 2006.[29] M. Kauers and C. Schneider. Symbolic summation with radical expressions. In C.W. Brown, editor,
Proc. ISSAC’07 , pages 219–226, 2007.[30] C. Koutschan. Creative telescoping for holonomic functions. In C. Schneider and J. Bl¨umlein, editors,
Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions , Textsand Monographs in Symbolic Computation, pages 171–194. Springer, 2013. arXiv:1307.4554 [cs.SC].[31] A. Levin.
Difference algebra , volume 8 of
Algebra and Applications . Springer, New York, 2008.[32] Erhard Neher. Invertible and nilpotent elements in the group algebra of a unique product group.
Acta Appl. Math. , 108(1):135–139, 2009.[33] R. Osburn and C. Schneider. Gaussian hypergeometric series and extensions of supercongruences.
Math. Comp. , 78(265):275–292, 2009.[34] P. Paule. Greatest factorial factorization and symbolic summation.
J. Symbolic Comput. , 20(3):235–268, 1995.[35] P. Paule and A. Riese. A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraicallymotivated aproach to q -hypergeometric telescoping. In M. Ismail and M. Rahman, editors, SpecialFunctions, q-Series and Related Topics , volume 14, pages 179–210. AMS, 1997.[36] M. Petkovˇsek. Hypergeometric solutions of linear recurrences with polynomial coefficients.
J. Symbolic Comput. , 14(2-3):243–264, 1992.[37] M. Petkovˇsek, H. S. Wilf, and D. Zeilberger. A = B . A. K. Peters, Wellesley, MA, 1996.[38] M. Petkovˇsek and H. Zakrajˇsek. Solving linear recurrence equations with polynomial coefficients.In C. Schneider and J. Bl¨umlein, editors, Computer Algebra in Quantum Field Theory: Integration,Summation and Special Functions , Texts and Monographs in Symbolic Computation, pages 259–284.Springer, 2013.[39] H. Prodinger, C. Schneider, and S. Wagner. Unfair permutations.
Europ. J. Comb. , 32:1282–1298,2011.
40] R. Risch. The problem of integration in finite terms.
Trans. Amer. Math. Soc. , 139:167–189, 1969.[41] C. Schneider. An Implementation of Karr’s Summation Algorithm in Mathematica.
Sem. Lothar.Combin. , S43b:1–10, 2000.[42] C. Schneider. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. KeplerUniversity, November 2001. PhD Thesis.[43] C. Schneider. A collection of denominator bounds to solve parameterized linear difference equationsin ΠΣ-extensions.
An. Univ. Timi¸soara Ser. Mat.-Inform. , 42(2):163–179, 2004. Extended versionof Proc. SYNASC’04.[44] C. Schneider. The summation package Sigma: Underlying principles and a rhombus tilingapplication.
Discrete Math. Theor. Comput. Sci. , 6:365–386, 2004.[45] C. Schneider. Symbolic summation with single-nested sum extensions. In J. Gutierrez, editor,
Proc.ISSAC’04 , pages 282–289. ACM Press, 2004.[46] C. Schneider. Degree bounds to find polynomial solutions of parameterized linear difference equationsin ΠΣ-fields.
Appl. Algebra Engrg. Comm. Comput. , 16(1):1–32, 2005.[47] C. Schneider. A new Sigma approach to multi-summation.
Adv. in Appl. Math. , 34(4):740–767,2005.[48] C. Schneider. Product representations in ΠΣ-fields.
Ann. Comb. , 9(1):75–99, 2005.[49] C. Schneider. Solving parameterized linear difference equations in terms of indefinite nested sumsand products.
J. Differ. Equations Appl. , 11(9):799–821, 2005.[50] C. Schneider. Ap´ery’s double sum is plain sailing indeed.
Electron. J. Combin. , 14, 2007.[51] C. Schneider. Simplifying sums in ΠΣ-extensions.
J. Algebra Appl. , 6(3):415–441, 2007.[52] C. Schneider. Symbolic summation assists combinatorics.
S´em. Lothar. Combin. , 56:1–36, 2007.Article B56b.[53] C. Schneider. A refined difference field theory for symbolic summation.
J. Symbolic Comput. ,43(9):611–644, 2008. [arXiv:0808.2543v1].[54] C. Schneider. Parameterized telescoping proves algebraic independence of sums.
Ann. Comb. ,14(4):533–552, 2010. [arXiv:0808.2596].[55] C. Schneider. Structural theorems for symbolic summation.
Appl. Algebra Engrg. Comm. Comput. ,21(1):1–32, 2010.[56] C. Schneider. A symbolic summation approach to find optimal nested sum representations. InA. Carey, D. Ellwood, S. Paycha, and S. Rosenberg, editors,
Motives, Quantum Field Theory, andPseudodifferential Operators , volume 12 of
Clay Mathematics Proceedings , pages 285–308. Amer.Math. Soc, 2010. arXiv:0808.2543.[57] C. Schneider. Simplifying multiple sums in difference fields. In C. Schneider and J. Bl¨umlein, editors,
Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions , Textsand Monographs in Symbolic Computation, pages 325–360. Springer, 2013. arXiv:1304.4134 [cs.SC].[58] C. Schneider. Modern summation methods for loop integrals in quantum field theory: The packagesSigma, EvaluateMultiSums and SumProduction. In
Proc. ACAT 2013 , volume 523 of
J. Phys.:Conf. Ser. , pages 1–17, 2014. arXiv:1310.0160 [cs.SC].[59] C. Schneider. Fast algorithms for refined parameterized telescoping in difference fields. InM. Weimann J. Guitierrez, J. Schicho, editor,
Computer Algebra and Polynomials, Applications ofAlgebra and Number Theory , Lecture Notes in Computer Science (LNCS), pages 157-191. Springer,2015. arXiv:1307.7887 [cs.SC].[60] M. van der Put and M.F. Singer.
Galois theory of difference equations , volume 1666 of
LectureNotes in Mathematics . Springer-Verlag, Berlin, 1997.[61] D. Zeilberger. A holonomic systems approach to special functions identities.
J. Comput. Appl.Math. , 32:321–368, 1990.[62] D. Zeilberger. The method of creative telescoping.
J. Symbolic Comput. , 11:195–204, 1991., 11:195–204, 1991.