A Difference Ring Theory for Symbolic Summation

A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called $R\Pi\Sigma^*$-extensions is supplemented by algorithms that support the construction of such difference rings automatically and that assist in the task to tackle symbolic summation problems. Algorithms are presented that solve parameterized telescoping equations, and more generally parameterized first-order difference equations, in the given difference ring. As a consequence, one obtains algorithms for the summation paradigms of telescoping and Zeilberger's creative telescoping. With this difference ring theory one obtains a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics.


Introduction
In his pioneering work [24,25] M. Karr introduced a very general class of difference fields, the so-called ΠΣ-fields, in which expressions in terms of indefinite nested sums and products can be represented. In particular, he developed an algorithm that decides constructively 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) holds. If such a solution exists, one obtains for an appropriately chosen a ∈ N the identity b k=a f (k) = g(k + 1) − g (1).
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 now possible to obtain sum representations, i.e., right hand sides in (2) with certain optimality criteria such as minimal nesting depth [53,56], minimal number of generators in the summands [45] or minimal degrees in the denominators [51]. For the simplification of products see [48,8]. We emphasize that exactly such refined representations give rise to more efficient telescoping algorithms worked out in [55,58]. A striking application is that Karr's algorithm and all the enhanced versions can be used to solve the parameterized telescoping problem [41,54]: for given indefinite nested product-sum expressions f 1 (k), . . . , f n (k) represented in F, find constants c 1 , . . . , c n , free of k and not all zero, and find g(k) represented in F such that g(k + 1) − g(k + 1) = c 1 f 1 (k) + · · · + c n f n (k) (3) holds. In particular, this problem covers Zeilberger's creative telescoping paradigm [62] by setting f i (k) = F (m + i − 1, k) and representing these f i (k) in F. Namely, if one finds such a solution, one ends up at the recurrence In a nutshell one cannot only treat indefinite summation but also definite summation problems. In this regard, also recurrence solvers have been developed where the coefficients of the recurrence and the inhomogeneous part can be elements from a ΠΣfield [14,49,6]. All these algorithms generalize and enhance substantially the (q-)hypergeometric and holonomic toolbox [5,18,61,62,36,34,37,35,9,15,26,30] in order to rewrite definite sums to indefinite nested sums. For details on these aspects we refer to [59]. 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 setting of difference fields: Algebraic products, like cannot be expressed in ΠΣ-fields, which are built by a tower of transcendental field extensions. Even worse, the objects given in (4) introduce zero-divisors, like which cannot be treated in a field or in an integral domain. In applications these objects occur rather frequently as standalone objects or in nested sums [3,4]. It is thus a fundamental 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 the underlying difference field, i.e., by implementing the concept of interlacing in an algebraic way. First steps to combine these techniques with ΠΣ-fields have been made in [17].
Within the package Sigma a different approach [42] has been implemented. Summation objects like (−1) k and sums over such objects are introduced by a tower of generators subject to the relations such as (5). In this way one obtains a direct translation between the summation objects and the generators of the corresponding difference rings. This enhancement has been applied non-trivially, e.g., to combinatorial problems [44,39], number theory [50,33] or to problems from particle physics [12]; for the most recent evaluations of Feynman integrals [11,2,1] up to 300 generators were used to model the summation objects 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 supplement it with the missing algorithmic building blocks in order to obtain a rigorous summation machinery. More precisely, we will enhance the difference field theory of [24,25] to a difference ring theory by introducing besides Π-extensions (for transcendental product extensions) and Σ * -extensions (for transcendental sum extensions) also R-extensions which enables one to represent objects such as (4). An important ingredient of this theory is the exploration of the so-called semi-constants (resp. semi-invariants) and the formulation of the symbolic summation problems within these notions. In particular, we obtain algorithms that can solve parameterized first-order linear difference equations. As special instance we obtain algorithms for the parameterized telescoping problem, in particular for the summation paradigms of telescoping and creative telescoping. In addition, we provide an algorithmic toolbox that supports the construction of the so-called simple RΠΣ * -extensions automatically. As special case we demonstrate, how d'Alembertian solutions [7] of a recurrence, a subclass of Liouvillian solutions [20,38], can be represented in such RΠΣ * -extensions. In particular, we will illustrate the underlying problems and their solutions by discovering the following identities here the imaginary part is denoted with ι, i.e., ι 2 = −1.
The outline is as follows. In Section 2 we will introduce the basic notations of difference rings (resp. fields) and define RΠΣ * -ring extensions. Furthermore, we will work out the underlying problems in the setting of difference rings and motivate the different challenges that will be treated in this article. In addition, we give an overview of the main results and show how they can be applied for symbolic summation. In the remaining sections these results will be worked out in details. In Section 3 we present the crucial properties of single nested RΠΣ * -extension. Special emphasis will be put on the properties of the underling ring. In Section 4 we will consider a tower of such extensions and explore the set of semi-constants. In Section 5 we present algorithms that calculate the order, period and factorial order of the generators of R-extensions. Finally, in Section 6 and Section 7 we elaborate algorithms that are needed to construct RΠΣ * -extensions and that solve as special case the (parameterized) telescoping problem. A conclusion is given in Section 8.

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 characteristic 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 perform the standard operations (including zero recognition and deciding constructively if an element is invertible). The multiplicative group of units (invertible elements) of A is denoted by A * . If A is a subring (resp. subfield/multiplicative subgroup) ofÃ we also write A ≤Ã. The non-negative integers are denoted by N = {0, 1, 2, . . . }.
In this section we will present a general framework in which the symbolic summation problems can formulated and tackled in the setting of difference rings. Here indefinite nested product-sum expression f (k) such as in (1) and (3) are 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 automorphism (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 difference field. 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}, and if it is clear from the context we also write constA = const(A, σ). It is easy to check that constA is a subring (resp. a subfield) of A which contains as subring (resp. subfield) the rational numbers Q. Throughout this article we will take care that constA is always a field (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 will model indefinite nested sums and products. They will be introduced by a tower of ring extensions, the so-called RΠΣ * -difference 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 RΠΣ * -extension; even more, derive tactics that enable one to represent sums and products automatically in RΠΣ * -extensions.
(2) Work out the underlying subproblems in order to solve two central problems of symbolic 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, if possible, a g ∈ A such that the telescoping (T) equation holds: Problem PT for (A, σ). Given a difference ring (A, σ) with constant field K and given f 1 , . . . , f n ∈ A. Find, if possible, c 1 , . . . , c n ∈ K (not all c i being zero) and a g ∈ A such that the parameterized telescoping (PT) holds: In Subsection 2.3 we will present the main results of theoretical and algorithmic nature to handle these problems, and in Subsection 2.4 we demonstrate how the new summation theory can be used to represent d'Alembertian solutions in RΠΣ * -extensions.

The definition of RΠΣ * -extensions
A difference ring (Ã,σ) is a difference ring extension of a difference ring (A, σ) if A ≤Ã andσ| A = σ, i.e., A is a subring ofÃ andσ(a) = σ(a) for all a ∈ A. The definition of difference field extensions is the same by replacing the word ring with field. In short (for the ring and field version) we also write (A, σ) ≤ (Ã,σ). If it is clear from the context, we do not distinguish anymore between σ andσ. For the construction of RΠΣ * -extensions, we start with the following basic properties. Lemma 1. Let A be a ring with α ∈ A * and β ∈ A together with a ring automorphism σ : A → A. Let A[t] be a polynomial ring and A[t, 1 t ] be the ring of Laurent polynomials. (1) There is a unique automorphism σ ′ : In summary, we obtain the 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 difference ring extension ( A is a field, we obtain the uniquely determined difference field extension (A(t), σ) of (A, σ) with σ(t) = α t + β. Following the notions of [14] each of the extensions, i.e., , σ) are called unimonomial extensions (of polynomial, Laurent polynomial or of rational function type, respectively).
Finally, we consider those extensions where the constants remain unchanged.
The generators of a Σ * -extension (in the ring or field version) and a Π-extension (in the ring or field version) are called Σ * -monomials and Π-monomials, respectively. Remark 1. Keeping the constants unchanged is a central property to tackle the parameterized telescoping problem. E.g., if the constants are extended, there do not exist bounds on the degrees as utilized in Subsection 7.1.1. Moreover, introducing no extra constants is the essential property to embed the derived difference rings into the ring of sequences; this fact has been worked out, e.g., in [54] which is related to [19].
For further considerations we introduce the order function ord: A → N with 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 unimonomial extension (A[y], σ) of (A, σ) with σ(y) = α y. Now take the ideal I := y λ − 1 and consider the quotient ring E = A[y]/I. Since I is closed under σ, i.e., I is a difference ideal, 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, there is 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. Let (A, σ) be a difference ring and α ∈ A * with α λ = 1 for some λ > 1. Then there is (up to a difference ring isomorphism) a unique difference ring extension (A[x], σ) of (A, σ) subject to the relations x λ = 1 and σ(x) = α x.
Thus we obtain a difference ring extension as claimed in the lemma. Now suppose that there is another difference ring extension (A[x ′ ], σ ′ ) of (A, σ) subject to the relations σ ′ (x ′ ) = α x ′ and x ′λ = 1. Then by the first isomorphism theorem, there is the ring isomorphism τ : The extension (A[x], σ) of (A, σ) in Lemma 2 is called algebraic extension of order λ.
is an algebraic ring extension of Q(k) subject to the relation As for unimonomial extensions, we restrict now to those algebraic extensions where the constants remain unchanged. For the underlying motivation we refer to Remark 1.
The generator x is called R-monomial.
The generators with their sequential arrangement, incorporating the recursive definition of the automorphism, are always given explicitly. In particular, any reordering of the generators must respect the recursive nature induced by the automorphism.

A characterization of RΠΣ * -extensions and their algorithmic construction
For the construction of RΠΣ * -extensions we rely on the following result; for the proofs we refer to page 18 for part 1, page 19 for part 2, and page 21 for part 3. , σ) be a unimonomial ring extension of (A, σ) with σ(t) = t+β where β ∈ A such that constA is a field. Then this is a Σ * -extension (i.e., constA[t] = constA) if there does not exist a g ∈ A with σ(g) = g + β.
For Karr's celebrated field version [24,25] of this result we refer to Theorems 6 and 10, that can be nicely embedded in the general difference ring framework. As in Karr's work, Theorem 1 facilitates algorithmic tactics to verify if a given extension is an RΠΣ *extension. Here we consider two cases.

Testing and constructing RΠ-extensions
For a unimonomial extension as given in Theorem 1.2 or an algebraic extension as given Theorem 1.3 with α = σ(t)/t one should first check if α ∈ A * . For the class of difference rings (A, σ), built by simple RΠΣ * -extensions introduced in Definition 4 below, this task will be straightforward. Next, we need the order 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, we have to test if the generator t is a Π-monomial, and if λ > 0, we have to check if the generator t is an R-monomial. Using Theorem 1 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 \ {0} and an m ∈ Z \ {0} for the case λ = 0 (resp. m ∈ {1, . . . , λ − 1} for the case λ > 0) such that the multiplicative version of the telescoping equation holds: Example 6 (Cont. Ex. 4). We verify that (Q(k)[x][y], σ) is an R-extension of (Q(k), σ).

Testing and constructing Σ * -extensions
In order to verify if a unimonomial extension as given in Theorem 1.1 is a Σ * -extension, it suffices to solve Problem T with f = β and to check if there is a telescoping solution. We illustrate this feature by actually constructing a difference ring in which the summand given on the left hand side of (6) and the additional sum occurring on the right hand side of (6) can be represented. In particular, we demonstrate how identity (6) can be discovered in this difference ring. (1) Take f = x k . Then solving Problem T shows that there is no g ∈ A with σ(g) − g = x k . Hence we can construct the Σ * -extension (A[s], σ) of (A, σ) with σ(s) = s + x k ; note that the Σ * -monomial s represents k j=1 (−1) j j in our difference ring.
(2) Take f = y k . Then solving Problem T shows that there is no g ∈ A[s] such that σ(g) − g = y k . Hence we can construct the Σ * -extension (A[s][S], σ) of (A[s], σ) with σ(S) = S + y k ; note that the Σ * -monomial S represents the sum (13).
(3) Take f = y k 2 s which represents (12). Solving Problem T produces the solution Hence this yields the solution of the telescoping equation (1) for our summand (12) by replacing the RΣ * -monomials x, y, s, S with the corresponding 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) = k i=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). This is of particular interest if there are further sums defined over F (k) which one wants to represent 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 extra sum (13). Here this was accomplished manually. But, using algorithms from [53,58] in combination with the results of this article, this sum can be determined automatically.

The underlying problems for RΠΣ * -extensions
As in the difference field approach [24,49,53,58], Problem T (and more generally Problem PT) and Problem MT will be solved by reducing these problems from (A, σ) to smaller difference rings (i.e., rings built by less RΠΣ * -monomials). However, in order to succeed in this reduction, parameterized versions of these problems have to be solved.
For Problem MT the following generalization is needed. Let (A, σ) be a difference ring, let W ⊆ A and let f = (f 1 , . . . , f n ) ∈ (A * ) n . Then we define the set In the following, we want to calculate a finite representation of M (f , A). If A is a field, i.e., A * = A \ {0}, it is immediate that M (f , A) is a submodule of Z n over Z and there is a basis of M (f , A) with rank ≤ n. In the setting of rings, this result carries over if the set of semi-constants (also called semi-invariants [14]) of (A, σ) defined by sconst(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 that sconst(A, σ) = A. Unfortunately, for a general difference ring the set sconst(A, σ) \ {0} 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 * (in short G ≤ A * ), we define the set of semi-constants (semi-invariants) of (A, σ) over G by Note that sconst (A * ) (A, σ) = sconst(A, σ) and sconst {1} (A, σ) = const(A, σ). If it is clear form the context, we drop σ and just write sconst G A and sconstA, respectively.
Here is one of the main challenges: For all our considerations we will choose G such that sconst G A \ {0} is a subgroup of A * (in short, sconstA \ {0} ≤ A * ). Then with this careful choice of G we can summarize the above considerations with the following lemma.
is a submodule of Z n over Z, and it has a finite Z-basis with rank ≤ n.
In the light of this property, we can state Problem PMT.
Observe that Problem MT can be reduced to Problem PMT. Suppose that we are given a group G with sconst G A \ {0} ≤ A * and α ∈ G. Moreover, assume that we have calculated λ = ord(α) and succeeded in solving Problem PMT, i.e., we are given a basis of M = M ((α), A) ⊆ Z 1 . If the basis is empty, there cannot be an m ∈ Z \ {0} and a g ∈ A \ {0} with (11). Otherwise, if the basis is not empty, the rank is 1. More precisely, we obtain m > 0 with M = m Z. Hence m is the smallest positive choice such that there is a g ∈ A \ {0} with (11). Therefore we can again decide 2 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 a ∈ A \ {0} and f = (f 1 , . . . , f n ) ∈ A n . Then we define if it is clear from the context, we write V (a, f , W ) and suppress the automorphism σ. As Lemma 3 the following result will be crucial for further considerations. Lemma 4. Let (A, σ) be a difference ring with constant field K and let G ≤ A * such that sconst G A \ {0} ≤ A * . Let V be a K-subspace of A. Then for f ∈ A n and a ∈ G we have that V (a, f , V ) is a K-subspace of K n × V with dim V (a, f , V ) ≤ n + 1.
Proof. Suppose that there are m linearly independent solutions with m > n + 1, say (c i,1 , . . . , c i,n , g i ) with 1 ≤ i ≤ m. Then by row operations over the field K we can derive at least two linearly independent vectors, say v 1 = (0, . . . , 0, g) and v 2 = (0, . . . , 0, h). Hence we have that σ(g) = a g and σ(h) = a h where g, h ∈ sconst G A \ {0} ≤ A * . Consequently, σ( g h ) = g h , thus c = g/h ∈ K * and therefore v 1 = c v 2 ; a contradiction that 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 difference ring (A, σ) with constant field K and G ≤ A * such that sconst G A \ {0} ≤ A * ; given a ∈ G and f ∈ A n . Find a K-basis of V (a, f , A).
In particular, if we can solve Problem PFLDE in (A, σ) for G = {1}, we can solve Problem T and PT in (A, σ). Furthermore, we can solve the multiplicative version of telescoping: if α ∈ G, we can determine g ∈ A \ {0}, in case of existence, such that σ(g) = α g holds. This feature is illustrated by the following example.
1+k on the left hand side of (7), we want to rewrite it in terms of the product P (b) = b−1 j=1 jι j . In a preparation step we constructed already the RΠΣ * -extension ( . Thus we are in the position to consider the corresponding Problem PFLDE in (K(k)[x] t , σ) for G. Activating our algorithms in Example 16 below, we get the basis {(0, g), (1, 0)} of V with g = x(ι+x 2 ) k t −1 . Since g is a solution of σ(g) = u g, g(k) = ι + (−1) k ι k k P (k) −1 is a solution of − ι k k+1 = g(k+1) g(k) . Hence by the telescoping trick we get b k=1 − ι k k+1 = g(b+1) g(1) which produces (7).

The main results
Suppose that we are given a difference ring (G, σ) which is computable and we are given a group G ≤ G * with sconst G G \ {0} ≤ G * . In this article we will restrict to certain classes of RΠΣ * -extensions (E, σ) of (G, σ) equipped with a groupG with G ≤G ≤ E * and sconstGE \ {0} ≤ E * such that we can derive the following algorithmic machinery: (1) Problem O inG can be reduced to Problem O in G; (2) Problems PMT and PFLDE in (E, σ) forG can be reduced to Problem PMT in (G, σ) for G and to the Problems PFLDE in (G, σ k ) for G for all k ≥ 1. In a nutshell, if we choose as base case a difference ring (G, σ) and a group G ≤ G * in which we can solve Problem O in G and Problems PMT and FFLDE in (G, σ) for G, we obtain recursive algorithms that solve the corresponding problems in the larger difference ring (E, σ) and larger groupG.
As it turns out, we will succeed in this task for a subclass of RΠΣ * -extensions (G, σ) ≤ (E, σ) and a properly chosen groupG ≤ E * that can treat all objects (among the general class of RΠΣ * -extensions) that the author has encountered in practical problem solving so far. More precisely, we will restrict to simple RΠΣ * -extensions.
Let (G t 1 . . . t e , σ) be a RΠΣ * -extension of (G, σ) and let G ≤ G * . Then we define It is easy to see thatG = G E G forms a group. More precisely, we obtain the following chain of subgroups: G ≤ G E G ≤ E * . We call G E G also the product-group over G for the RΠΣ *extension (E, σ) of (G, σ). We are now ready to define (G-)simple RΠΣ * -extensions.
So far, in all our examples the difference rings have been built by simple RΠΣ * -extensions. Before we finally turn to the class of simple RΠΣ * -extensions, we present one example which cannot be treated properly with our toolbox under consideration.
In this ring we are given the idempotent elements e 1 = (x − 1)/2 and e 2 = (x + 1)/2 with e 2 1 = 1 and e 2 2 = 1. Finally take α = e 1 + e 2 t. Then observe that α · (e 1 + e 2 /t) = 1, i.e., α ∈ Q(k) t [x] * . Note that ord(α) = 0. Otherwise it would follow that e k 2 = 0; a contradiction that e 2 is idempotent. Consequently T cannot be an R-extension, and we construct the unimonomial extension ( Summarizing, we aim at solving Problems PMT and PFLDE in a G-simple RΠΣ *extension (G, σ) ≤ (E, σ) forG = G E G , and we want to solve Problem O inG. In order to accomplish this task, we will restrict ourselves further to the following two situations.

A solution for single-rooted RΠΣ * -extensions
In most applications R-extensions are not nested, e.g., only objects like (−1) k arise. Moreover, such objects do not occur in transcendental products, but only in sums, like cyclotomic sums [3] or generalized harmonic sums [4]. A formal definition of this special, but very practical oriented class of RΠΣ * -extensions is as follows.
Definition 5. An RΠΣ * -extension (E, σ) of (G, σ) is called single-rooted if the generators of the extension can be reordered to 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 3 RΠΣ * -extension, we will show the following Let (E, σ) be a simple and single-rooted RΠΣ * -extension of (G, σ) with (16) as specified in Definition 5, and letG = G For its proof see page 24. In particular, we obtain the following reduction algorithm summarized in Theorem 3; for a proof of part 1 see page 32 and of part 2 see page 41.

A solution for RΠΣ * -extensions of a strong constant-stable difference field
In the following we restrict to RΠΣ * -extensions where the ground domain G = F is a field. In this setting, we will show that the semi-constants form a multiplicative group.
For a proof of this theorem we refer to page 26. In order to solve Problems PMT and PFLDE we require in addition that (F, σ) is strong constant-stable.
It is called strong constant-stable if it is constant-stable and any root of unity of A is in K.
This means that 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 for further (algorithmic) properties see Corollary 5 below. Given such a tower of RΠΣ * -extensions, we can solve Problems PMT and PFLDE as follows; for the proofs, resp. the underlying algorithms, of part 1 see page 29, of part 2 see page 35 and of part 3 see page 44.
We remark that the underlying machinery of this theorem has been utilized in Examples 8 and 9 to obtain the identities (6) and (7), respectively. Further details will be given below.

A complete machinery: algorithms for the ground difference rings
Both, Theorem 3 and Theorem 5 provide algorithms to reduce Problem PMT and PFLDE (and thus Problems MT, PT and T) from an RΠΣ * -extension (E, σ) of (G, σ) to the ground difference ring (G, σ). Theorem 3 requires less conditions on (G, σ), but considers only single-rooted RΠΣ * -extensions, whereas Theorem 5 requires more properties on (G, σ) but allows nested R-extensions which are of the type as given in Corollary 5. Note that the algorithms for the latter case are more demanding, in particular, one has to solve Problem PFLDE in (G, σ k ) with k > 0 instead of k = 1 only.
We emphasize that both theorems are applicable for a rather general class of difference fields (G, σ). Namely, (G, σ) itself can be a ΠΣ * -field extension of (H, σ) where certain properties in the difference field (H, σ) hold. Here the following remarks are in place.
In particular, if we are given a root of unity from G, it cannot depend on transcendental elements and is therefore from H.
It has been shown in [28] that one can solve Problem PMT in (G, σ) for G * and Problem PFLDE in (G, σ k ) for G * for k > 0 if certain properties hold for the difference field (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 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 * ) E G . So far, the required properties have been verified and the necessary algorithms have been 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 K can be a rational function field over an algebraic number field; see [48].
here K is of the type as given in item 1. Note that in this field one can model unspecified 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 take the ΠΣ * -field (H, σ) = (K(k), σ) over K ∈ {Q, Q(ι)} with σ(k) = k + 1.

Application: representation of d'Alembertian solutions in RΠΣ * -extensions
We illustrate how an important class of d'Alembertian solutions [7], a subclass of Liouvillian solutions [20,38], of a given linear difference operator, can be represented completely automatically in RΠΣ * -extensions. In order to obtain d'Alembertian solutions one starts as follows: first the linear difference operator is factored as much as possible into linear right hand factors. This can be accomplished, e.g., with the algorithms [36,21,22] or, within the setting of ΠΣ * -fields with the algorithms given in [6] which are 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 can be read off. They can be given by a finite number of hypergeometric expressions and indefinite nested sums defined over such expressions. More precisely, each solution is of the form where λ i ∈ N and the the hypergeometric expression h i (k) can be written in the form k j=λi α i (j) with α i (z) being a rational function from K(z). Subsequently, we restrict ourself to a field K which is a rational function field K = Q(n 1 , . . . , 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 a single-rooted simple RΠΣ * -extension as follows.
(1) In [48] an algorithm has been presented that calculates a single-rooted simple RΠextension (G, σ) of (K(k), σ) in which all hypergeometric expressions occurring in the d'Alembertian solutions are explicitly represented.
(2) Then the challenging task is to construct a Σ * -extension of (G, σ) and to represent there the arising sums of the d'Alembertian solutions. Given (G, σ) from step 1, this can be accomplished by applying iteratively Theorem 1.1. Suppose we represented already an inner summand in a Σ * -extension (A, σ) of (G, σ) with β ∈ A. Since (A, σ) is a simple RΠΣ * -extension of (K(k), σ) and (K(k), σ) is a ΠΣ * -field over K, we can solve Problem T with f = β by using the underlying algorithm of Theorem 3 in combination with 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 by the boundary condition (lower summation bound) of the given sum; for further details we refer to Example 8.4. Otherwise, we construct the Σ * -extension (A[t], σ) of (A, σ) with σ(t) = t + β by Theorem 1.1. In total we are given again a single-rooted simple RΠΣ *extension of (K(k), σ). Proceeding iteratively, all nested sums over the hypergeometric terms can be expressed in such an RΠΣ * -extension over (K(k), σ).
Exactly this difference ring machinery is implemented in Sigma and has been used to tackle challenging applications, like [44,39,50,33,11,2,1] mentioned already in the introduction. In particular, this toolbox has been combined with the algorithms worked out in [45,48,51,53,8,58] in order to find representations of d'Alembertian solutions with certain optimality properties, like minimal nesting depth. For a recent summary of all these features (unfortunately, in the setting of difference fields) we refer to [57,59].

Single nested RΠΣ * -extensions
This section delivers relevant properties of single RΠΣ * -extensions. First, Theorem 1 will be proved. In addition, properties of the set of semi-constants of RΠΣ * -extensions are derived to gain further insight in the nature of RΠΣ * -extensions and to prove Theorems 2 and 4 in Section 4.
We start with some general properties in ring theory. A ring A is called reduced if there are no non-zero nilpotent elements, i.e., for any f ∈ A \ {0} and any n > 0 we have that f n = 0. Moreover, a ring is called connected if 0 and 1 are the only idempotent elements, i.e., for any f ∈ A \ {0, 1} we have that f 2 = f . Then we rely on the following properties in rings. A polynomial n i=0 a i x i ∈ A[t] with coefficients from a ring A is invertible if and only if a 0 ∈ A * and a i with i ≥ 1 are nilpotent elements. Thus in a reduced ring, i.e., a ring which has no nilpotent non-zero elements, we have that A[t] * = A * . Moreover, there is a complete characterization of invertible elements in the ring of Laurent polynomials A[t, 1 t ] presented in [23, Theorem 1] (see also [32]). Based on this work we extract Lemma 5. 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, 1 t ] * = {u t r | u ∈ A * and r ∈ Z}. Since the rings are usually not connected, Lemma 5 can be applied only partially.
Example 11. The generators in the ring given in Example 10 can be reordered to has the idempotent elements e 1 , e 2 , it is not connected. Therefore we get relations such as (e 1 +e 2 t)(e 1 + e2 t ) which are predicted in [23,32]. Subsequently, we enumerate further definitions and properties in difference rings and fields that will be used throughout the article. Let (A, σ) be a difference ring. The rising factorial (or σ-factorial) of f ∈ A * to k ∈ Z is defined by 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 proofs are omitted to the reader. Lemma 6. Let (A, σ) be a difference ring, f, h ∈ A * and n, m ∈ Z. Then: ( Let A t be a ring of (Laurent) polynomials.
In addition, for a, b ∈ Z we introduce the set of truncated (Laurent) polynomials by We conclude this section with the following two lemmas.
Lemma 7. Let (A t , σ) be a unimonomial ring extension of (A, σ) of (Laurent) polynomial type. Then for any k ∈ Z and f ∈ A t we have that deg(σ k (f )) = deg(f ).

Σ * -extensions
The essence of all the properties of Σ * -extensions is contained in the following lemma.
Then the following statements are equivalent. ( and take g ∈ constA[t]\constA. Then σ(g) = u g with u = 1 ∈ G. Thus the lemma is proven. ✷ As a consequence we can now establish the first part of our characterisation theorem.
Proof of Theorem 1.1 (see page 7). For G = {1} we have that sconst G A = constA = K. By assumption K is a field and thus sconst G A \ {0} ≤ A * . Therefore we can apply Lemma 10 and with its equivalence (2) ⇔ (3) the theorem follows. ✷ In order to rediscover the difference field version from [24,25], we specialize Lemma 10 to difference fields by exploiting Lemma 8.3.
Finally, by the equivalence (3) ⇔ (1) of Lemma 10 we obtain the following result concerning the semi-constants.
If we specialize to G = A[t] * and assume that A is reduced, we get Theorem 8. For the proof we use in addition the following lemma.

Π-extensions
Analogously to Lemma 9 we obtain by coefficient comparison Suppose that t is a Π-monomial, but ord(α) = n > 0. Then σ(t n ) = α n t n = t n , which is a contradiction to the first part of the statement. ✷ Requiring in addition that the semi-constants form a group, this result can be sharpened.
The other direction is immediate by Theorem 1.2. ✷ Together with Lemma 8 we rediscover Karr's characterization of Π-field extensions.
Proof. The direction from right to left follows by Theorem 1.2 and the fact that any 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 that deg(q) ≥ deg(p) (otherwise take 1/g instead of g). By Lemma 8, We consider two cases. First suppose that p ∈ F * and q = t m with m > 0. Then What remains to consider is the case that p / ∈ F or q = t m for some m > 0. Define The following holds.
(1) 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 the assumption of our second case. (2) u := σ(a)/a ∈ F * by (20).
(3) a = ut m for all u ∈ F * and m > 0: a could be only of this form, if q = t m for some m > 0, but since gcd(p, q) = 1, t ∤ p.
By properties (1) and (3), it follows that a = n i=k a i t i with a k = 0 = a n where n > k ≥ 0. With property (2) and Lemma 13 it follows that σ(a k ) = u α k a k and σ(a n ) = u α n a n which implies σ( a k an ) = α n−k a k an . Since a k an ∈ F * and n − k > 0, the theorem is proven. ✷ Finally, we characterize the set of semi-constants for Π-extensions.
Theorem 11. Let (A, σ) be a difference ring and G ≤ A * such that sconst G A\{0} ≤ A * . Let (A[t, 1 t ], σ) be Π-extension of (A, σ) with σ(t) = α t for some α ∈ G. Then we have that 1 t ] with σ(g) = u g for some u ∈ G. By Lemma 13 we get σ(g i )α i = u g i and thus σ(g i ) = u α i g i Now suppose that there are r, s ∈ Z with s > r and g r = 0 = g s . As u α s ∈ G, it follows that g s ∈ sconst G A \ {0} ≤ A * . Thus we conclude that σ( gr gs ) = α s−r gr gs with s − r > 0; a contradiction to Theorem 1.
1 t ] * . ✷ So far we obtained a description of the semi-constants for a subgroup G of A * . Now we will lift this result to the group Then by Theorem 11 the theorem is proven. Since G ≤G the inclusion sconstGA[t, Hence there are an m ∈ Z and an h ∈ G such that σ(g) = h t m g. By coefficient comparison it follows that σ(g i )α i = hg i−m . If m ≥ 1, take 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 maximal such 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, again a contradiction. Thus m = 0 and consequently, g ∈ sconst G A[t, 1 t ]. ✷ We conclude this subsection by specializing to the groupG = A t * under the assumption that A is reduced and connected. We remark that this result is not applicable if general R-extensions pop up; see Example 11. However, with this result we obtain further insights summarized in Corollary 2.2, Proposition 2 and Corollary 4 below.  = {h t m | h ∈ sconstA and m ∈ Z}. ✷

R-extensions
We start to derive a characterization of R-extensions.
As in Theorem 12 we will lift this result from the group G ≤ A * tõ We remark that there is the following subtlety. We have to assume that A[x] is reduced in order to prove the result below. In order to take care of this extra property, further investigations will be necessary in Subsection 4.1.

Nested R-extensions
A special case is obtained immediately by applying iteratively Theorem 14.

Proposition 1. Let (A, σ) be a difference ring and G ≤
. . x e be an R-extension of (A, σ) with σ(xi) xi ∈ G and n i = ord(x i ).
In order to treat nested R-extensions, we proceed as follows. Let (A x 1 . . . x e , σ) be an R-extension of (A, σ) with λ i = ord(x i ) and σ(x i ) = α i x i . Moreover, take the polynomial ring R = A[y 1 , . . . , y e ] and define α ′ i = α| x1→y1,...,xi−1→yi−1 . Then we obtain the automorphism σ ′ : R → R by σ ′ | A = σ and σ(y i ) = α i y i , i.e., (R, σ ′ ) is a difference ring extension of (A, σ). Thus by iterative application of the construction used for Lemma 2 it follows that A x 1 . . . x e is isomorphic to R/I where I is the ideal 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 x 1 . . . x e , σ) is isomorphic to In order to show that sconst G E \ {0} ≤ E * holds as claimed in Corollary 1 below, we use Gröbner bases theory. Lemma 14. Let λ i ∈ N\{0}. The zero-dimensional ideal I given in (21) in the polynomial ring R = F[y 1 , . . . , y e ] is radical.
Proof. The ideal I is zero-dimensional. Since F has characteristic 0, it is perfect. We therefore apply the criterion (algorithm) given in [10,Thm. 8.22]. Define f i = y λi i − 1. Proof. The difference ring (E, σ) with E = F x 1 . . . x r is isomorphic to (R/I, σ ′′ ) as defined above with (21) where A = F. Suppose that E is not reduced. Then there are a f ∈ E \ {0} and n > 0 such that f n = 0. Hence there are a 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 14. Hence E is reduced. Thus we can apply Theorem 15 iteratively and it follows that sconst G E \ {0} ≤ E * . ✷

Nested ΠΣ * -extensions
In Corollary 2 we will characterize the set of semi-constants for ΠΣ * -extensions. Part 1 will deal with the general case. Part 2 assumes in addition that the ground ring is reduced and connected. In order to obtain these results, we rely on the following lemmas.
Lemma 15. Let (A t , σ) be a ΠΣ * -extension of (A, σ). If A is reduced, A t is reduced. If A is reduced and connected, A t is reduced and connected.
Proof. Let t be a Π-monomial. Moreover let A be reduced. Now take f = 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 n m . Hence f n m = 0 and thus f m is a nilpotent element in A, a contradiction. Now let A be reduced and connected and take Let m be maximal such that f m = 0. If m > 0, then the coefficient of t 2m in f 2 is f 2 m and thus with f 2 = f we have that f 2 m = 0; a contradiction that A is reduced. Otherwise, if m = 0, we takem minimal such that fm = 0. Note thatm < 0 since f / ∈ A. As above, it follows that f 2 m = 0, a contradiction that A is reduced. Summarizing, if A is reduced (resp. reduced and connected), A[t, 1 t ] is reduced (resp. reduced and connected). If t is a Σ * -monomial, we conclude the same In particular, if A is reduced, a ΠΣ * -extension has the following particular nature: the shift behaviour of Π-monomials does not depend on Σ * -monomials. Proof. Let E = t 1 . . . t e . By iterative application of Lemma 15 it follows that E is reduced. Let t i be a Π-monomial where α = σ(t i )/t i ∈ A t 1 . . . t i−1 depends on a Σ * -monomial t j with j < i. Then we can reorder the generators such that we get here we forget σ and argue purely in the given ring.
In particular, α ∈ H t j = H[t j ] \ H. Since α is invertible, α ∈ H by Lemma 5; a contradiction. 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 the right and obtain again a ΠΣ * -extension. ✷ With this preparation we obtain the following result.
(2) If A is reduced and connected and sconstA \ {0} = A * , then (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 one extraG-simple ΠΣ * -monomial t e+1 on top. DefineG =G This completes the induction step. Similarly, the first equality of part (2) follows by Theorems 8 and 13. The second equality follows by iterative application of Lemmas 5 and 15. Since any field is connected and reduced and sconstA \ {0} = A * by Lemma 12, part (3) follows by part (2). ✷ Restricting to Σ * -extensions, the above result simplifies as follows.

RΠΣ * -extensions and their simple and single-rooted restrictions
For simple and single-rooted RΠΣ * -extensions we obtain immediately the

Proof of Theorem 2 (see page 13). Combine Corollary 2.1 and Proposition 1. ✷
In order to get Theorem 4 for general simple RΠΣ * -extensions, further properties are needed. The first lemma states that a tower of simple RΠΣ * -extensions is again a simple RΠΣ * -extension; the proof is immediate. In particular, any simple RΠΣ * -extension can be reordered such that the R-monomials are at the first place and the Σ * -monomials are on top.
Lemma 18. Let (A, σ) be a difference ring with G ≤ A * a group. Any G-simple RΠΣ *extension (E, σ) of (A, σ) can be reordered to the form E = A t 1 t 2 . . . t e with r, p ∈ N (0 ≤ r ≤ p ≤ e) such that the following properties hold: In particular, for any f ∈ G E A which depends on a Π-monomial t i we have that ord(f ) = 0.
Proof. We show the lemma by induction on the number of RΠΣ * -monomials. Suppose that the lemma holds for e extensions. Now let E = A t 1 . . . t e and consider the RΠΣ *monomial t e+1 on top of E. By the induction assumption we can reorder E such that it has the desired form (all R-monomials are on the left, all Π-monomials are in the middle and all Σ * -monomials are on the right). If t e+1 is a Σ * -monomial, the required shape is fulfilled. If t e+1 is an R-monomial, observe that α := σ(t e+1 )/t e+1 ∈ G E A . Since ord(α) = ord(t e+1 ) > 1 by Theorem 1.3, α is free of Π-monomials by the induction assumption and (by definition) free of Σ * -monomials. Thus we can shuffle t e+1 to the left (such that all 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 E A is free of Σ * -monomials by definition and we can shuffle t e+1 to the left such that all Σ * -monomials are to the right. This completes the first part of the lemma. Now let E = A x 1 . . . x e+1 be in the desired ordered form. If x e+1 is a Σ * -monomial, we have that G Thus the second part holds by the 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-monomial and take f ∈ G E xe+1 A . If f ∈ E and f depends on Π-monomials, we have again that ord(f ) = 0 by the induction assumption. To this end, suppose that f depends on x e+1 and we have that ord(f ) = n > 0. Then f = u x m e+1 where m = 0 and u ∈ E * . Since f n = 1, u n x m n e+1 = 1 where u n = 0. Hence x e+1 is not transcendental over E, a contradiction to the definition of a Π-monomial. Consequently, ord(f ) = n = 0. This completes the proof. ✷ For completeness we observe that this reordering is also possible if one relaxes the condition that the RΠΣ * -extension is simple but requires that the ground ring is a field. Proof. First we try to shuffle all R-extensions to the front. Suppose that this fails at the first time. Then there are an R-extension (H, σ) of (F, σ), a ΠΣ * -extension (G, σ) of (H, σ) with G = H y 1 . . . y l and an R-extension (G x , σ) of (G, σ) with α = σ(x)/x in which y l occurs. Note that H is reduced by Corollary 1 and G is reduced by iterative application of Lemma 15. Write α = i f i y i l . Let m = 0 such that f m = 0 and such that |m| ≥ 1 is maximal (we remark that m < 0 can only happen if y l is a Π-monomial). By the choice of m we have that the coefficient of y m n l in α n is f n m . Hence with α n = 1 it follows that f n m = 0, a contradiction to the assumption that G is reduced. Therefore we can shuffle all R-monomials to the left and all ΠΣ * -monomials to the right. Since the nested R-extension is reduced by Corollary 1, we can apply Lemma 16 to reorder the ΠΣ * -monomials further as claimed in the statement. ✷ By definition any (nested) Σ * -extension is a also a simple Σ * -extension. If the ground ring is reduced and connected, we obtain the following stronger result. Proposition 2. Let (G, σ) be a difference ring where G is reduced and connect and where sconstG \ {0} = G * . Then a ΠΣ * -extension (E, σ) of (G, σ) is simple.
Proof. Let E = G t 1 . . . t e . By Lemma 16 we may suppose that the generators are ordered such that t 1 . . . , t p are Π-monomials and t p+1 . . . , t e are Σ * -monomials. By Corollary 2.2 we have that σ(ti) 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 ΠΣ * -ring extension and simple ΠΣ * -ring extension are equivalent. The situation becomes rather different if the ring is, e.g., not connected; see Example 10. However, for single-rooted RΠΣ * -extensions over a difference field, the situation is again tame.
Proof. By definition, the RΠΣ * -extension can be reordered to the form (5). Since G is a field, we have that sconstG \ {0} = G * . Hence we can apply Proposition 2 and it follows that the Π-extension (G t 1 . . . t r , σ) of (G, σ) is simple. Since σ(xi) xi ∈ G * for 1 ≤ i ≤ u, the R-monomials x i are G * -simple. Since also the Σ * -monomials s i are G * -simple, we conclude that (G, σ) ≤ (E, σ) is simple. ✷ To this end, we obtain the following main result.
Proof. By Corollary 1 we have that sconst G H \ {0} ≤ H * . Hence we can apply Corollary 2.1 and the result follows. ✷
(3) Take p = per(α) > 0 and v = ord(α) > 0. Then we have that Consequently, we can choose n = ord(α) per(α) to obtain α (n) = 1. In particular, for any Hence per(α)|i. Therefore the smallest 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 the elements of (A * ) E A of a simple R-extension (E, σ) ≥ (G, σ) by recursion. First, we assume that the orders of the R-monomials in (E, σ) ≥ (G, σ) are already computed and show how the orders of the elements of (A * ) E A can be determined. . . x ze e )). By similar arguments we show that (x z1 1 ) n = · · · = (x ze e ) n and consequently ord(x z1 1 . . . x ze e ) = lcm(ord(x z1 1 ), . . . , ord(x ze e ). Since also ord(x zi i ) = ord(xi) gcd(ord(xi),zi) 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) > 0. ✷ In the next lemma we set up the stage to calculate the period and factorial order. Proof.
Combining the two lemmas from above we arrive at the following result. 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.
. . x e be a simple R-extension of a difference field (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 for some root of unity u i ∈ K * with ord(u i ) | ord(x i ) and m i,j ∈ N.
with z 1 , . . . , z e ∈ N and u ∈ K * . Then (4) If (K, σ) is computable and Problem O solvable in K * , the values of ord(α), per(α) and ford(α) are computable for all α ∈ (K * ) Proof. (1) By definition we have that (27) with m i ∈ N and u i ∈ F * . By Lemma 21 it follows that ord(u i ) > 0 and ord(u i )| ord(x i ). In particular, u i ∈ K * since all roots of unity from F are constants by assumption.
(2) It is immediate that (H, σ) with H = K x 1 . . . x e forms a difference ring. Since constE = constF = K, (H, σ) is a simple R-extension of (K, σ). . . x ze e with u ∈ K * and z i ∈ N. Then by Lemma 21 and the computation of ord(u) the order of α can be computed. Moreover, since per(u) = 1 and ord(u) = ford(u) are given, we can invoke Lemma 22 to calculate the period and factorial order of α. (5) Let α be given as in (24) with with u ∈ F * and m i ∈ N. By Lemma 21 ord(α) > 0 iff ord(u) > 0. By assumption, ord(u) > 0 implies u ∈ K * . Thus, if u / ∈ K, ord(α) = 0. Otherwise, if u ∈ K * , we can apply part 4. ✷ To this end, we are now in the position to prove Theorem 5.1.
Proof of Theorem 5.1 (see page 13). Let (E, σ) be a simple RΠΣ * -extension of (F, σ) where (F, σ) is computable and where any root of unity of F is from K = constF. Reorder it to the shape as given in Lemma 18. In particular, the R-extension (F t 1 . . . t r , σ) of (F, σ) has the shape as given in Corollary 5.1. Let f ∈ (F * )

The algorithmic machinery II: Problem PMT
We aim at proving Theorems 3.1 and 5.2, i.e., providing recursive algorithms that reduce Problem PMT from a given RΠΣ * -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 \ {0} ≤ G * . This property guarantees that for any f ∈ G n a Z-basis of M (f , G) with rank ≤ n exists; see Lemma 3. In particular, we rely on the fact that there are algorithms available that solve Problem PMT in (G, σ) for G.
For the underlying reduction method we use the following two lemmas.

Lemma 23. Let (A[t]
, σ) be a Σ * -extension of (A, σ) and let H ≤ A * be a group such Then Proof. "⊆": Let (m 1 , . . . , m n ) ∈ M (f , A t ). Hence we can take g ∈ A t \ {0} with σ(g) = f m1 1 . . . f mn n g, i.e., g ∈ sconstH A t \ {0} withH = H A t A . Thus by Theorem 12 it follows that g =g t m with m ∈ Z andg ∈ sconstH A \ {0} ≤ A * . Hence . . h mn n α −mg t m1 e1+···+mn en . Sinceg = 0, we conclude that σ(g) = 0. By coefficient comparison it follows then that m 1 e 1 + · · · + m n e n = 0, i.e., (m 1 , . . . , m n ) ∈ M 2 . Thus σ(g) = h m1 1 . . . h mn n α −mg and consequently (m 1 , . . . , m n , m) ∈ M ((h 1 , . . . , h n , 1 α ), A), i.e., (m 1 , . . . , m n ) ∈ M 1 . "⊇": Let (m 1 , . . . , m n ) ∈ M 1 ∩ M 2 . Thus we can takeg ∈ A \ {0} and m ∈ Z such that σ(g) = h m1 1 . . . h mn n α −mg . Moreover, we have that e 1 m 1 + · · · + e n m n = 0. Thus σ(g t m ) = (h 1 t e1 ) m1 . . . (h n t en ) mng and therefore (m 1 , . . . , m n ) ∈ M (f , A t ). ✷ Now we are in the position to get the underling algorithm resp. the Proof of Theorem 17. Let (G, σ) be a difference ring and let G ≤ G * such that sconst G G \ {0} ≤ G * and suppose that Problem PMT is solvable in (G, σ) for G. Now let (E, σ) be a G-simple ΠΣ * -extension of (G, σ) as in the theorem withG = G E G and let f ∈G n . By Corollary 2.1 it follows that sconstGE \ {0} ≤ E * and together with Lemma 3 it follows that M (f , E) = M (f , sconstGE) is a Z-module. The calculation of a basis of M (f , E) will be accomplished by recursion/induction. If E = A, nothing has to be shown. Otherwise, let (A, σ) be a G-simple ΠΣ * -extension of (G, σ) in which we know how one can solve Problem PMT for H = G A G , and let E = A t where t is a H-simple ΠΣ * -monomial. We have to treat two cases. First, suppose that t is a Σ *monomial. Then it follows thatG = G E G = G A G = H ≤ A * and thus f ∈ H n . Hence we can activate Lemma 23 and it follows that M (f , E) = M (f , A). Thus by assumption we can compute a basis. Second, suppose that t is a H-simple Π-monomial. Then we can utilize Lemma 24: We calculate a basis of M 2 by linear algebra. Moreover, we compute a basis of M ((h 1 , . . . , h n , 1 α ), A) by the induction assumption (by recursion). Hence we can derive a basis of M 2 and thus of M 1 ∩ M 2 = M (f , A t ). This completes the proof. ✷ Note that the reduction presented in Lemma 24 is accomplished by increasing the dimension of M 1 by one. In general, the more Π-monomials are involved, the higher the dimension will be in the arising Problems PMT.
Looking closer at the reduction algorithm, we can extract the following shortcut, resp. a refined version of Theorem 1.2.
Corollary 6. Let (A, σ) be a difference ring and G ≤ A * be a group such that sconst G A\ {0} ≤ 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 E A = G H A and the following holds.
Proof. Note that G E A ≤ H * . Hence by iterative application of Lemma 23 the first statement is proven. The second statement follows by statement 1 and Theorem 1.2. ✷ The following remark is in place. If one restricts to the special case that (G, σ) is a ΠΣ * -field with G = G * , the presented reduction techniques boil down to the reduction presented in [24,Theorem 8]. The major contribution here is that Theorem 17 can be applied for any computable difference ring (G, σ) with the properties given in Theorem 17. Subsequently, we utilize this additional flexibility to tackle (nested) R-extensions.

A reduction strategy for R-extensions and thus for RΠΣ * -extensions
First, we treat the special case where the R-extensions are single-rooted.  In order to tackle the more general case that the R-extensions are nested (and that they might occur also in Π-extensions), we require additional properties on the difference rings: they must be strong constant-stable; see Definition 6. In order to derive the underlying algorithms in Theorem 5.2 below, we utilize the following structural property of the semiconstants. They factor into two parts: a factor which depends only on the R-monomials with constant coefficients and a factor which is free of the R-monomials.
(1) Let 1 ≤ i ≤ e. By Proposition 3.1 it follows that ord(u i ) > 0. Together with the assumption that per(u i ) > 0 we have that ford(u i ) > 0 by Lemma 20.3. Moreover, by Proposition 3.2 it follows that per(x i ) > 0. Again with ord(x i ) > 0 and per(x i ) it follows that ford(x i ) > 0 by Lemma 20.3. Therefore r > 0.
(2) By the choice of 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 22.3. Moreover, by Lemma 6 we conclude that (r) g =ũ g withũ := v (r) . Since G is closed under σ, we have thatũ ∈ G. Write g = s∈S g s x s where S ⊆ N e is finite, g s ∈ F * and for (s 1 , . . . , s e ) ∈ S and x = (x 1 , . . . , x e ) we use the multi-index notation x s = x s1 1 . . . x se 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 ) =ũ g i for any i ∈ S. Note that g i ∈ sconst G (A, σ r )\ {0} ≤ 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 sg for some c s ∈ K * andg ∈ sconst G (A, σ r ) \ {0} ≤ A * with σ r (g) =ũg.
Proof. Since u i ∈ K * by Corollary 5.1, per(u i ) = 1. Define G = F * which is closed under σ. In particular, sconst G (F, σ k ) \ {0} = F * for any k > 0. Moreover, by Corollary 1 it follows that sconstGE\{0} ≤ E * . Thus we may apply Lemma 26. The corollary follows by observing that λ ∈ A * 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.
where . . . ,f n , α 1 , . . . , α s ), F) Proof. Let (m 1 , . . . , m n ) ∈ M (f , E), i.e., there is a g ∈ sconst G E \ {0} with σ(g) = f m1 1 . . . f mn n g. Hence by Corollary 7 it follows that g =g h withg ∈ F * and h ∈ K[x 1 ] . . . [x e ] * . In particular, σ(g) =f m1 1 . . .f mn n λg for λ ∈ K * being an rth root of unity. Hence we can take m n+1 , . . . , m n+s ∈ N such that λ = α ). Here we will apply Lemma 27. By Example 12.3 we get ford(x) = 8. With u 1 = 1 we determine r = 8 by (29). We definef 1 = k,f 2 = −1/(k + 1) and h 1 = h 2 = x. All 8th roots of unity of K are generated by α 1 = ι. As worked out in the above lemma, we have to determine a basis of Here we use, e.g., the algorithms worked out in [24] (this is the base case of our machinery) and obtain the basis Thus (m 1 , . . . , m n ) = (m ′ 1 , . . . , m ′ n ) + (λ 1 z 1 , . . . , λ n z n ) where (m ′ 1 , . . . , m ′ n ) ∈M and (λ 1 z 1 , . . . , λ n z n ) = z 1 (λ 1 b 1 ) + · · · + z n (λ n b n ). Consequently, (m 1 , . . . , m n ) is an element of the left hand side of (36). Since the number of vectors of the span on the left hand side is finite, we can derive a Z-basis of (36). ✷ Remark 2. A basis of M (f , H) can be obtained more efficiently as follows. We start with a vector space which is given by the basis B = {λ 1 b 1 , . . . , λ n b n } where b i ∈ K n is the ith unit vector. Now go through all elements fromM . Take the first element m fromM . 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 4), put it in B and transform the set again to a Z-basis. More precisely, the rows of the matrix should give a matrix which is in Hermite normal form. In this way, one can again check easily if an element is in span(B). We proceed until all elements ofM are visited and update step by step B as described above. By construction we have that our span(B) equals the left hand side of (36) and thus equals M (f , H). We remark that B consists always of e linearly independent vectors. However, the Z-span is more and more refined. Since (1, 0, 0) / ∈ span(B), we check if there is a g ∈ K[x] \ {0} with σ(g) = x 1 x 0 ι 0 g: this is not the case. We continue with (2, 0, 0). Here we have that (2, 0, 0) / ∈ span(M ). Now we check if there is a g ∈ K[x] \ {0} with σ(g) = x 2 x 0 ι 0 g. Plugging in g = g 0 + g 1 x + g 2 x 2 + g 3 x 3 into σ(g) = x 2 g gives the constraint (g 0 − g 2 )x 0 + ιx(g 1 + ιg 3 ) + x 2 (−g 0 − g 2 ) + x 3 (−g 1 − ig 3 ) = 0 which leads to the solution g = x + ι x 3 . Proof of Theorem 5.2 (see page 13). By Lemma 18 we can reorder the generators of the RΠΣ * -extension such that (Ē, σ) is an F * -simple R-extension of (F, σ) and (E, σ) is a G-simple ΠΣ * -extension of (Ē, σ) with G = (F * )Ē F . LetĒ = F[x 1 ] . . . [x e ] with u i , α i and f ∈ G n withf i and h i as given in Lemma 27. By assumption we can compute a basis of M 1 as given in Lemma 27. Since Problem O is solvable in K * , we can compute o i = ord(x i ) and λ i = ord(u i ) by Corollary 5.4. Thus we can use Proposition 4 to compute a basis of M 2 as posed in Lemma 27. Hence we can compute a basis of (33). Summarizing, we can solve Problem PMT in (Ē, σ) for G. In particular, we have that sconst GĒ \ {0} ≤Ē * by Corollary 1. Hence by Theorem 17 we can solve Problem PMT for (E, σ) in G Ē E . Since G Ē E = (F * ) E F by Lemma 17, the theorem is proven. ✷ To this end, we work out the following shortcut, resp. refined version of Theorem 1.3.
Corollary 8. Let (F, σ) be a strong constant-stable difference field with constant field K, and let G ≤ F * be a group with sconst ] be a G-simple R-extension of (A, σ) and let (E, σ) be a G H F -simple ΠΣ * -extension of (H, σ).
where α ∈ K * is a root of unity and m i ∈ N. Thus u ∈ K * ∩ G and hence f ∈ (K * ∩ G) H F .
(2) Let f ∈ (G H F ) n be as given above. By part 1, where the α i ∈ K are roots of unity and m i,j ∈ N. Suppose that (after reordering of the ΠΣ * -monomials) where the t i are Π-monomials and the s i are Σ * -monomials. By Corollary 6 we have that for some u = a x µ1 1 . . . x µr r with µ i ∈ N and with a being a root of unity in H and thus by assumption being from K. By Corollary 5.3 we get µ := ord(u) > 0; in addition we have that µ ′ = ford(u) > 0. By Theorem 16 it follows that g = q t ν1 1 . . . t ν k k with q ∈ sconst G H \ {0} 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. Then g µ depends also on t m which contradicts to constH t 1 . . . t k = constH. Consequently g = q ∈ sconst G H \ {0}. By Corollary 7 it follows that g =g h with h ∈ K[x 1 ] . . . [x r ] * andg ∈ F * with σ(h) = λ u h where λ ∈ K * is a root of unity. Recall that µ ′ = ford(u) > 0 and hence µ ′′ = lcm(µ ′ , ord(λ)) > 0. Since σ µ ′′ (h) = h and (F, σ) is constant-stable, it follows that h ∈ K * . Therefore g ∈ K[

The algorithmic machinery III: Problem PFLDE
We aim at proving Theorems 3.2 and 5.3, i.e., providing recursive algorithms that reduce Problem PFLDE from a given RΠΣ * -extension to its ground ring (resp. field). If we are considering single-rooted RΠΣ * -extension (Theorem 3.2), we rely heavily on 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) \ {0} ≤ G * . This property allows us to assume that for any f ∈ G n and any u ∈ G the K-space V = V (u, f , (A, σ)) has a basis with dimension ≤ n + 1; see Lemma 4. In particular, our reduction algorithm is based on the assumption that there are algorithms available that solve Problem PFLDE in (G, σ) for G, i.e., one can compute a basis of V . For general RΠΣ * -extensions over a strong constant-stable difference field (G, σ) (Theorem 5.3) we need stronger properties: all what we stated above must hold not only for σ but for the automorphisms σ l with l ≥ 1.

A reduction strategy for ΠΣ * -extensions
We show the following theorem.
(1) If t is a Σ * -monomial and Problem PFLDE is solvable in (A, σ) for G, then Problem PFLDE is solvable in (A t , σ) for (2) If t is a Π-monomial and Problems PFLDE and PMT are solvable in (A, σ) for G, and let σ(t) = α t + β with α ∈ G and β ∈ A. Moreover let u ∈G, i.e., and let f = (f 1 , . . . , f n ) ∈G n . By Theorem 11 we have that sconstGA t \ {0} ≤ A t * and hence by Lemma 4 a basis of V (u, f , A t with dimension ≤ n + 1 exists. We show how such a basis can be computed with the assumptions given in the theorem.

Degree bounds
The essential step is to search for degree bounds: we will determine a, b ∈ Z such that holds; for the definition of the truncated set of (Laurent) polynomials see (18). For technical reasons we also require that the constraints hold where the m is given by (38) and wherẽ a = min(ldeg(f 1 ) . . . , ldeg(f n )) andb = max(deg(f 1 ) . . . , deg(f n )).
The recovery of these bounds (see Lemmas 28 and 30 below) is based on generalizations of ideas given in [24]; for further details and proofs in the setting of difference fields see also [43,46]. If t is a Σ * -monomial, we have that A t = A[t], α = 1 andG = G; in particular we have m = 0 in (38). In this case, we can utilize the following lemma.
If t is a Π-monomial, we have that A t = A[t, 1 t ] and β = 0. First suppose that u / ∈ A, i.e., m ∈ Z \ {0} as given in (38). If f i = 0 for all i, it is easy to see that g = 0 is the only choice. Hence take a = max(0, m) and b = max(−1, −1 + m), and (39) and (40) hold. Otherwise, if not all f i are 0, we can use the following fact; the proof is left to the reader.
Note that in this scenario we have thatã,b ∈ Z for (41). Hence by setting a =ã and b =b, we can conclude that (39) and (40) hold. What remains to consider is the case u ∈ G with m = 0. Here we utilize for some γ ∈ sconst G A\ {0}, then ν is uniquely determined and we have that min(λ, ν) ≤ λ andμ ≤ max(µ, ν). If there is not such a ν, we have that λ ≤λ andμ ≤ µ.
Summarizing, we obtain bounds a, b ∈ Z such that (39) and (40) hold for a Σ * -monomial. For a Π-monomial we need in addition that Problem PMT is solvable in (A, σ) for G.

Degree reduction
The following reduction has been introduced in [24] in the setting of difference fields. Subsequently, we present the basic ideas in the setting of difference rings; further technical details can be found in [42,Thm. 3.2.2]. We want to determine all c 1 , . . . , c n ∈ K = constA and holds. If b < a, g = 0 and a basis of V (u, f , A t ) = V (u, f , {0}) can be determined by linear algebra. Otherwise, we continue as follows. Due to (40), it follows that λ = max(b, b + m) is the highest possible exponent in (44). Letf i be the coefficient of the term t λ in f i . Then by coefficient comparison w.r.t. t λ in (44) we get the following constraints: if m = 0, if m < 0, α b σ(g b ) = c 1f1 + · · · + c nfn .
Thus a basis of can be determined if one can solve Problem PFLDE for (A, σ) in G. Now we plug in this partial solution (i.e., the possible leading coefficient g b with the corresponding linear combinations of the f i ), and end up at a new first-order parameterized 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 highest coefficient g b−1 . Hence we proceed recursively by updating λ → λ − 1 and b → b − 1 and determine a basis of the reduced problem (with highest degree λ − 1). Finally, given a basis of this solution space and given the basis of (48), one can determine a basis of V (v, f , A t a,b ). Summarizing, solving various instances of Problem PFLDE with the degree reductions b → b − 1 → · · · → a − 1 enables one to determine a basis of V (u, f , A t ). This concludes the proof of Theorem 18.
Example 17 (Cont. Ex. 16). We know that g = g −1 t −1 . Plugging in g into σ(g)+ x k+1 = 0 yields σ(g −1 ) + x 2 k k+1 g −1 = 0. Therefore we look for a basis of V ( −x 2 k k+1 , (0), K(k)[x]). By using the algorithms presented in Subsection 7.2 we get the basis {(1, x(ι+x 2 )/k), (0, 1)}. This finally gives the basis (0, x(ι + x 2 )/k/t), (1,0) Example 18 (Cont. Ex. 15). We want to find a basis of [s] and f = (y k 2 s). Hence we make the Ansatz (c 1 , g 0 + g 1 S) ∈ V with indeterminates c 1 ∈ Q and g 0 , g 1 ∈ A such that holds. Doing coefficient comparison w.r.t. S 1 yields the constraint σ(g 1 ) − g 1 = c 1 0; compare (46). Thus we get all solutions by determining a basis of In this particular instance, the Q-basis {(1, 0), (0, 1)} is immediate utilizing the fact that the constants are precisely Q. Summarizing, the solutions are (c 1 , g 1 ) ∈ Q 2 . Consequently, our Ansatz can be refined with (c 1 , g 0 + c 2 S) ∈ V where c 1 ∈ Q, c 2 (= g 1 ) ∈ Q and g 0 ∈ A such that σ(g 0 + c 2 S) − (g 0 + c 2 S) = c 1 k 2 sy holds. Bringing the c 2 S part to the right hand side yields the new equation 7 with h = σ(s) − s = x y k+1 ∈ A. In other words, we need a basis of V (1, h, A) with h = (y k 2 s, x y k+1 ) ∈ A 2 . Now we apply again the reduction method, but this time in the smaller ring A without the Σ * -monomial S. We skip all the details, but refer to a particular subproblem that we will consider in Example 19. Finally, we get the basis h, A). Thus we can reconstruct the basis {(1, g), (0, 1)} of V with g given in (14).
Note that the reduction of Theorem 18 simplifies to Karr's field version given in [24] if one specializes A to a field and sets G = A * = A \ {0}. However, our version works not only for a field, but for any computable difference ring (A, σ) as specified in Theorem 18. Subsequently, we will exploit this enhancement in order to treat (nested) R-extensions.

A reduction strategy for R-extensions and thus for RΠΣ * -extensions
The proof and the underlying algorithm for Theorem 3.2 is based on Proposition 5. Let (A, σ) be a computable difference ring and G ≤ A * with sconst G A \ {0} ≤ 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 a simplified version of the degree reduction presented for Theorem 18. Let u ∈ G and f = (f 1 , . . . , f n ) ∈ A[t] n By definition, it follows that a solution g ∈ A[t] and c 1 , . . . , c n ∈ K = constA of (44) is of the form g = b i=a g i t i with a := 0 and b := d − 1. Thus the bounds are immediate (under the assumption that d has been determined; see Section 5). Since 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 can determine a basis of (48). By recursion we finally obtain a basis of V (u, f , A[t]). ✷ Proof of Theorem 3.2 (see page 13). Since Problem PFLDE is solvable in (G, σ) for G, it follows by iterative applications of Theorem 18 and Corollary 2.1 that Problem PFLDE is solvable in (H, σ) forG with H = G t 1 . . . t r and that sconstGH \ {0} ≤ H * . Thus by iterative applications of Proposition 5 and Theorem 14 we conclude that Problem PFLDE is solvable in (H, σ) forG withH = H x 1 . . . x u and that sconstGH \ {0} ≤H * . Finally, by applying iteratively Theorem 18 and Corollary 2.1 it follows that Problem PFLDE is solvable in (E, σ) forG. Note that in Proposition 5 we have to know the values ord(x i ) = ord(α i ) with α i ∈ G (either as input or by computing them first by solving instances of Problem O in G). ✷ Finally, we present an algorithm for Theorem 5.3 which is based on the following lemma.
Proof. Let K = constA, let E = A x 1 . . . x r , let f = (f 1 , . . . , f n ) ∈ E n and let u = v x m1 1 . . . x mr r ∈ G E A with v ∈ G and m i ∈ N. We will present a reduction method to obtain a basis of V (u, f , E). Let α = x m1 1 . . . x mr r . Then by Lemma 21 it follows that ord(α) > 0 can be computed by the given values of ord(x i ) with 1 ≤ i ≤ r. Hence we can activate Lemma 22.4 and can compute ford(α) > 0. Now take λ = lcm(ford(α), per(x 1 ), . . . , per(x r )).
Note thatṼ is a K subspace of K n × A[x 1 , . . . , x r ]. Thus V (u, f , E) is a subspace ofṼ over K. Hence we concentrate first on the task to find a finite description of this solution set. More precisely, we show that it has a finite basis and show how one can compute it. For this task define M := {(n 1 , . . . , n e ) ∈ N e | 0 ≤ n i < ord(x i )}. Write g = i∈M g i x i andf j = i∈Mf j,i x i in multi-index notation. Since σ λ (x i ) = x i , it follows by coefficient comparison that for i ∈ M we have that σ λ (g i ) − w g i = c 1f1,i + · · · + c nfn,i .
(2) A different approach is to consider an R-extension (F[x], σ) of (F, σ) of order d as a holonomic expression [61,15,30] over a difference field. Then as worked out in [47,17], a solution g = d−1 i=0 g i x i and c i ∈ constF of (9) leads to a coupled system of first-order difference equations in terms of the g i that can be uncoupled explicitly. More precisely, there is an explicitly given formula that constitutes a higher-order parameterized linear difference equation in g d−1 and the parameters c i . Solving this difference equation in terms of g d−1 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. For further details on the holonomic Ansatz in the context of algebraic ring extensions (also on handling such objects in the basis of idempotent elements [60,19]) we refer to [17]. The advantage of the reduction technique proposed in Proposition 6 is that it can be applied in one stroke for nested R-extensions. In particular, Problem PFLDE can be 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 we obtain the following result.
Theorem 19. Let (E, σ) with E = F t 1 . . . t e be a simple RΠΣ * -extension of a constant-stable and computable difference field (F, σ). Suppose that for all R-monomials the period is positive, and the order and period of the R-monomials are given explicitly. Then Problem PFLDE in (E, σ) for (F * ) E F ≤ E * is solvable if one of the following holds: (1) All t i are RΣ * -monomials and PFLDE is solvable in (F, σ k ) for F * for all k > 0.
(2) Problem PMT is solvable in (F, σ) for F * and Problem PFLDE is solvable in (F, σ k ) for F * for all k > 0.
Proof. Let H = (F * ) E F . Recall that by Theorem 4 we have that sconst H E \ {0} ≤ E * , i.e., Problem PFLDE is applicable in (E, σ) for H. By Lemma 18 we can reorder the generators of the RΠΣ * -extension such that (Ē, σ) is an F * -simple R-extension of (F, σ) and (E, σ) is a G-simple ΠΣ * -extension of (Ē, σ) with with G = (F * )Ē F . Note that the multiplicative group F * is closed under σ, sconst (F * ) (F, σ l ) = F * for all l > 0 and sconst (F * )Ē \ {0} ≤Ē * by Corollary 1. Thus we can apply Proposition 6 and PFLDE is solvable in (H, σ) for G with the requirements stated in the two cases (1) and (2), respectively. Applying Theorem 18 and Corollary 2.1 iteratively shows that Problem PFLDE is solvable in (E, σ) for G Ē E = H; again we need the requirements stated in the cases (1) and (2), respectively. ✷ Proof of Theorem 5.3 (see page 13). Let (E, σ) be a simple RΠΣ * -extension of (F, σ) where (F, σ) is computable and strong constant-stable. Then by Corollary 5 (part 3 and 4) the periods and orders of all R-monomials are positive and can be computed. Thus Theorem 19.2 is applicable which completes the proof. ✷ We remark that in Theorem 5.3 one can drop the condition that Problem PMT is solvable in (F, σ) for F * if in the RΠΣ * -extension no Π-monomials occur, i.e., one applies case 1 and not case 2 of Theorem 19.

Conclusion
We provided important building blocks that extend the well established difference field theory to a difference ring theory. In this setting one can handle in addition objects such as (4). We elaborated algorithms for the (multiplicative) telescoping problem (Problems T and MT) and the (multiplicative) parameterized telescoping problem (Problems PT and PMT). In particular, Problem PT enables one to apply Zeilberger's creative telescoping paradigm in the rather general class of simple RΠΣ * -extensions. In order to derive these algorithms we showed that certain semi-constants (resp. semi-invariants) of the difference rings under consideration form a multiplicative group.
A future task will be to push forward the difference ring theory and the underlying algorithms in order to relax the requirements in Theorem 3 and 5 (i.e., that the RΠΣ *extensions are simple and/or that the ground difference ring is strong constant-stable). In any case, the currently developed toolbox widens the class of indefinite nested sums and products in the setting of difference rings. We are looking forward to see new kind of applications that can be attacked with this machinery.