A Family of Denominator Bounds for First Order Linear Recurrence Systems
aa r X i v : . [ c s . S C ] J u l A Family of Denominator Bounds for First Order LinearRecurrence Systems
Mark van Hoeij a , Moulay Barkatou b , Johannes Middeke c a Department of MathematicsFlorida State UniversityTallahassee, FL 32306, USA b Université de Limoges, XLIM123, Av. A. Thomas87060 Limoges cedex, France c Research Institute for Symbolic Computation (RISC)Johannes Kepler UniversityAltenbergerstraße 69, 4040 Linz, Austria
Abstract
For linear recurrence systems, the problem of finding rational solutions is reduced tothe problem of computing polynomial solutions by computing a content bound or adenominator bound. There are several bounds in the literature. The sharpest bound[8] leads to polynomial solutions of lower degrees, but as shown in [7], this advantageneed not compensate for the time spent on computing that bound.To strike the best balance between sharpness of the bound versus CPU time spentobtaining it, we will give a family of bounds. The J ’th member of this family is similarto [2] when J =
1, similar to [8] when J is large, and novel for intermediate values of J , which give the best balance between sharpness and CPU time.The setting for our content bounds are systems τ ( Y ) = MY where τ is an automor-phism of a unique factorization domain, and M is an invertible matrix with entries in itsfield of fractions. This setting includes the shift case, the q -shift case, the multi-basiccase and others. We give two versions, a global version, and a version that bounds eachentry separately.
1. Introduction
Let A be a unique factorization domain and let τ : A → A be an automorphism. Wedenote the quotient field of A by K and extend τ to K . This paper considers systems ofthe form τ ( Y ) = MY where M ∈ GL n ( K ) . ( sys )The goal is to reduce the problem of computing rational solutions Y ∈ K n of ( sys ) tocomputing polynomial solutions Z ∈ A n of a related system. Email addresses: [email protected] (Mark van Hoeij), [email protected] (Moulay Barkatou), [email protected] (Johannes Middeke)
Preprint submitted to Elsevier July 7, 2020 efinition . We say that B ∈ K is a global content bound for ( sys ) ifall of its rational solutions Y ∈ K n are in B · A n . The denominator d : = den( B ) ∈ A \ { } is then a denominator bound , which means all rational solutions are in d · A n .A vector ( B , . . . , B n ) t ∈ K n is a component-wise content bound if all rational solutionsare in B · A n where B = diag( B , . . . , B n ).Note that 0 is a content bound if and only if there are no non-zero rational solutions.Content bounds and denominator bounds are found in [1], [2], [4], [3], [6], [11], or [10].If a content bound B is an invertible scalar or matrix, then we can substitute Y = BZ in τ ( Y ) = MY obtaining the equivalent system τ ( Z ) = τ ( B − ) MB Z for whichall rational solutions are in A n . This way, a denominator or a content bound reducesrational solutions Y ∈ K n to polynomial solutions Z ∈ A n . However, as illustratedin [7], there is tension between two goals: (1) we want a bound that can be computedquickly, and (2) want to minimize the degrees of the entries of Z . The goal in this paperis to strike a good balance between these two goals.We will formulate our bounds in a fairly general setting, see section 2 below, thoughthe practical utility is mainly for cases that have algorithms for polynomial solutions.
2. Preliminaries
For a ring R we will use R ∗ to denote the group of units in R . The set of m -by- n matrices with entries in R will be written as R m × n . We use GL n ( R ) for the set of n -by- n invertible matrices over R , while A t denotes the transpose of A . For a , . . . , a n ∈ R letdiag( a , . . . , a n ) ∈ R n × n denote the corresponding diagonal matrix.Let A be a unique factorization domain with quotient field K . Let ( A , τ ) be a di ff erence ring ; that is, τ : A → A is an automorphism. Extending τ to K makes ( K , τ )a di ff erence field. Example . Let F be a field of characteristic 0. The main example is A = F [ x ] with τ defined by τ ( f ( x )) : = f ( x + shift case . Here K = F ( x ). Example . Similarly, if F is a field and q ∈ F ∗ , we can let A = F [ x ] and τ ( f ( x )) : = f ( qx ). This is called the q-shift case . Example . Let ( G , τ ) be a di ff erence ring and let x , . . . , x s be indeterminates over G .Choose units α , . . . , α s ∈ G ∗ and β , . . . , β s ∈ G . Let A = G [ x , . . . , x s ] and extend τ to A by τ ( x j ) = α j x j + β j . If τ | G = id is the identity map, then we refer to this as the multi-basic case . Definition . Given two content bounds B , B ′ for the same system τ ( Y ) = MY , we say that B is sharper than B ′ if it constrains Y to a smaller set (i.e. B · A n ( B ′ · A n ). Example . For the shift case A = Q [ x ] and τ : x x + M = ( x + (2 x + x + ( x + − ( x + x ( x + ( x + − ( x + x + x +
3) ( x + (2 x + x ( x + x + ∈ GL ( Q ( x )) . The rational solutions of τ ( Y ) = MY are V = n ( x + c + c x ) x ( x + x + c − c x ) x + (cid:12)(cid:12)(cid:12)(cid:12) c , c ∈ Q o ⊆ Q ( x ) . V ⊂ B · A ( B ′ · A where B = x + x ( x +
2) and B ′ = x ( x + . Here B is a sharper content bound than B ′ . The component-wise bound C : = x + x ( x + x + x + ! ∈ Q ( x ) is sharper still since V ⊂ diag( C ) A ( B · A .Denominator bounds are more common than content bounds in the literature (see,for example, [1], [2], [4], [3], [6], [11], or [10]). If d is a denominator bound, then 1 / d is a content bound. However, Example 6 shows that a sharp global content bound B need not have that form.
3. The Exponent Function
Let p ∈ A be a prime ( = an irreducible polynomial if A = F [ x ]) and a ∈ A . The valuation of a at p is v p ( a ) = sup { j | p j divides a } . Note that v p ( a ) = ∞ if and only if a =
0. We extend v p : K → Z S {∞} by defining v p ( a / b ) = v p ( a ) − v p ( b ) for fractions a / b ∈ K . Then v p ( a + b ) > min { v p ( a ) , v p ( b ) } and v p ( ab ) = v p ( a ) + v p ( b ) . (1)for all a , b ∈ K . For a matrix A let v p ( A ) denote the minimum valuation of its entries.Then v p ( AB ) > v p ( A ) + v p ( B ) (2)for matrices A , B with matching sizes. Definition . Two elements a , a ∈ K are called associates ,denoted a ∼ a , if a = ua for some unit u ∈ A ∗ . Just like polynomial contents inGauss’ lemma, the content ct( A ) ∈ K of a matrix A ∈ K n × m is defined up to ∼ by thefollowing equivalent properties:(a) A can be written as ct( A ) times a matrix in A n × m whose entries have gcd 1.(b) v p (ct( A )) = v p ( A ) for all primes p .(c) ct( A ) = g / d where d is the least common multiple of the denominators in A , and g is the gcd of the entries of dA .An element B ∈ K is a content-bound for τ ( Y ) = MY if and only if v p ( B ) v p ( Y )for all solutions Y ∈ K n and all primes p in A . So finding B means finding a lowerbound for each v p ( Y ).Let D = n a ∈ A (cid:12)(cid:12)(cid:12)(cid:12) a , τ k ( a ) ∼ a for some k , o . A is a UFD means that every non-zero a ∈ A can be written as a productof finitely many primes, unique up to ∼ . This implies that a ∈ D if and only if all itsprime factors are in D .We will only compute a lower bound for v p ( Y ) at primes p < D . That results in acontent bound up to some factor a ∈ D . This is su ffi cient for the main cases includingthe shift case (then D = F ∗ ), and the q -shift case when q is not a root of unity (then D = { cx m | c ∈ F ∗ , m > } ). Definition . Fix a prime p ∈ A . If c ∈ K then we define its exponent function as: if c = e = ∞ , otherwise e is the function e : Z → Z with e ( k ) = v τ k ( p ) ( c ) for all k ∈ Z .We only use this for primes p < D . If c , e has finite support and can berepresented with a finite list containing: a lower bound ℓ and upper bound m for thesupport of e , and the numbers e ( k ) for k from ℓ to m .For a system τ ( Y ) = MY we recursively define a matrix M j such that τ j ( Y ) = M j Y ,as follows: M = I and M j + = τ j ( M ) M j = τ ( M j ) M . For j < M j = τ j ( M − ) M j + . Examples include: M = M , M = τ ( M ) M , M − = τ − ( M − ) , M − = τ − ( M − ) τ − ( M − ) . After selecting a prime p < D , we denote the exponent function of c j : = ct( M j ) as e j : Z → Z . Example . Let M be as in Example 6, then c = ct( M ) = x − ( x + − ( x + ( x + − .The matrix M is always I so c =
1. From M − = τ − ( M − ) = (2 x − x ( x + x + ( x − x + x + ( x − x ( x + x + (2 x − x + x + we obtain c − = ( x − − ( x + − ( x + p = x we have e ( k ) = − k = − k =
12 if k = − k =
30 otherwise e = e − ( k ) = − k = −
10 if k = − k =
11 if k =
20 otherwise.
4. The J ’th global content bound Algorithm
10 (“The global algorithm”: J ’th global content bound) .Input M ∈ GL n ( K ) and an integer J > Output B ∈ K such that ∃ a ∈ D for which aY ∈ B · A n for any rational solution Y . Inother words, a content bound up to some factor a ∈ D . In the shift-case a = q -shift case if q not a root of unity then a = x m for some m not computedhere. 4 rocedure (a) Compute M j and c j ≔ ct( M j ) for j ∈ {− J . . . J } .(b) Let P be the set of prime factors in the denominators of c and c − .(c) Select one p ∈ P from each τ -equivalence class , where p is τ -equivalentto p if τ k ( p ) ∼ p for some k ∈ Z (recall ∼ from Definition 7).Let O be the resulting set of primes.(d) Let B : = p ∈ O − D (e.1) For each j ∈ {− J . . . J } compute the exponent-function e j : Z → Z of c j at p . Recall that e j has finite support and e j ( k ) = v τ k ( p ) ( c j ).(e.2) Let f be the output of the local algorithm in section 5 with input e − J , . . . , e J .(e.3) If f = ∞ then stop and return B =
0. Otherwise, f : Z → Z has finitesupport and we set B : = B · Q k ∈ Z τ k ( p ) f ( k ) .(f) Return B .The paper [2] gives a denominator bound that is based solely on the denominatorsof M and M − . That is similar to the above algorithm with J =
1, and although it canbe sharper with J =
1, see Example 14, its main novelty is when J >
1. Then the localalgorithm uses more data, allowing it to construct a sharper bound (see the examplein section 7). The goal of the local algorithm in section 5 is to obtain the sharpestcontent bound (up to a factor a ∈ D ) that can be derived from the exponent-functions e − J , . . . , e J . In the shift case, that factor a ∈ D is simply 1.In the q -shift case, if q is a root of unity then τ has finite order so D = A − { } which makes the output trivial. But the root of unity case is usually excluded. If q isnot a root of unity then aY ∈ B · F [ x ] n for some a = x m not computed here. Thenthe output B restricts rational solutions Y not to B · F [ x ] n but to B · F [ x , / x ] n . In the q -case, algorithms to bound the degree of polynomial solutions can also bound m (justreplace x , q with 1 / x , / q ). So in the q -case, finding all solutions in F [ x , / x ] n is notmeaningfully harder than finding all solutions in F [ x ] n .In general, B restricts rational solutions to B · A n where A : = D − A ⊆ K is thelocalization of A at D . This reduces solutions over K to solutions over A .
5. Local Bounds
Fix one prime p < D . A function f is called a local content bound (for M at p ) if v τ k ( p ) ( Y ) > f ( k ) for all solutions Y ∈ K n and all k ∈ Z . (3)The local algorithm below will compute such f as follow: Lemma 12 below will pro-vide an initial f , which is then repeatedly improved with Lemma 11.For j ∈ {− J , . . . , J } , let e j be the exponent function of the content c j of M j . If Y is a rational solution of τ ( Y ) = MY then τ j ( Y ) = M j Y and from Equation (2) we get5 τ − j ( q ) ( Y ) = v q ( τ j ( Y )) > v q ( M j ) + v q ( Y ) = v q ( c j ) + v q ( Y ) for any prime q . For q = τ k + j ( p )we get v τ k ( p ) ( Y ) > v q ( c j ) + v q ( Y ) = e j ( k + j ) + v q ( Y ) > e j ( k + j ) + f ( k + j ) (4)for any local content bound f . We have shown: Lemma 11.
Fix some J > . If f is a local content bound then v τ k ( p ) ( Y ) > e j ( k + j ) + f ( k + j ) , so the functionf new ( k ) : = max { e j ( k + j ) + f ( k + j ) | − J j J } ( k ∈ Z ) is a local content bound as well. Note that f new ( k ) > f ( k ) since f new ( k ) is the maximum of set that contains e ( k ) + f ( k ) = f ( k ). The following picture illustrates for J = J neighbors of f ( k ) to see if the current lower bound f ( k ) for v τ k ( p ) ( Y ) can be improved: f ( k − f ( k − f ( k − f ( k ) f ( k + f ( k + f ( k + · · · · · · e ( k + e ( k + e − ( k − e − ( k − support of f is the set supp( f ) = { k ∈ Z | f ( k ) , } . Lemma 12.
Take ℓ , m , ℓ − , m − ∈ Z such that supp( e ) ⊆ [ ℓ , m ] and supp( e − ) ⊆ [ ℓ − , m − ] . For every non-zero solution Y ∈ K n of τ ( Y ) = MY, if v τ k ( p ) ( Y ) , thenk ∈ [ ℓ, m ] where ℓ = min { ℓ , ℓ − + } and m = max { m − , m − } . This implies that the function f : Z → Z ∪ {−∞} defined byf ( k ) = −∞ if k ∈ [ ℓ, m ]0 otherwise.is a local content bound.Proof. If there are no non-zero solutions then there is nothing to prove. So let Y bea generic non-zero solution and let f ( k ) = v τ k ( p ) ( Y ). Recall from Equation (4) that v τ k ( p ) ( Y ) > e j ( k + j ) + v q ( Y ) where q was τ k + j ( p ), in other words, f ( k ) > e j ( k + j ) + f ( k + j ) . e j ( k + j ) = j = k + > m we find f ( k ) > f ( k +
1) for all k > m − k we have f ( k ) > f ( k + > f ( k + > · · · > f has finite support.Since e j ( k + j ) = j = − k − > m − we find f ( k ) > f ( k −
1) and thus f ( k − f ( k ) for all k − > m − . Then f ( k ) f ( k + · · · k > m − .Thus f ( k ) = k > m . The proof for ℓ is similar: f ( k ) > f ( k +
1) for all k + < ℓ . Then 0 > · · · > f ( k − > f ( k ) for all k < ℓ . f ( k ) > f ( k −
1) for all k − < ℓ − . Then f ( k ) > k < ℓ − + Algorithm
13 (“The local algorithm”: J th local content bound) .Input The exponent-functions e − J , . . . , e J from step (e.1) in the global algorithm. Output
A local content bound f : Z → Z with respect to p , or ∞ if it is discovered thatthere can be no non-zero rational solutions. Procedure (a) Let ℓ, m and f be as in Lemma 12.(b) Repeat:(b.1) Let f new : Z → Z ∪ {−∞} be the function given in Lemma 11.(b.2) If f ( k ) > k < [ ℓ, m ] then stop and return ∞ .(b.3) If f = f new then stop and return f .Otherwise set f : = f new and Repeat. Example . Let J = p = x . Example 9, which continued Example 6, computed k . . . − − . . . e − ( k ) . . . − − . . . e ( k ) . . . − − − . . . We do not list e since that is always 0. Then. ℓ − = − , ℓ = , and ℓ = min { ℓ , ℓ − + } = m − = , m = , and m = max { m − , m − } = . In the algorithm f : Z → Z S {−∞} successively becomes k . . . − − . . . f ( k ) . . . −∞ −∞ −∞ . . . f ( k ) . . . − −∞ − . . . f ( k ) . . . − − . . . At that point f stabilizes ( f new = f ) and the local algorithm returns f . The globalalgorithm converts f to this content bound B = x + x ( x + d = x ( x + x +
2) from algorithmUniversalDenominator in Maple, which implements [2].7 heorem 15.
Algorithm 13 is correct and terminates.Proof.
Throughout the algorithm f is a local content bound by Lemmas 11 and 12. Ifstep (b.2) returns ∞ then this is correct by Lemma 12. Otherwise the support of f staysinside a finite range [ ℓ, m ]. As long as f ( k ) = −∞ for some k we get f new , f . Soall f ( k ) are in Z before the algorithm can terminate in step (b.3). Since no f ( k ) everdecreases and the support is bounded, it follows that either (a) the algorithm terminatesafter finitely many steps, or (b) some f ( k ) grows without bound. Option (b) leadsto a contradiction, because if f ( k ) grows without bound, then so does f ( k +
1) since f new ( k + > e − ( k ) + f ( k ). Then f ( k + , f ( k + , . . . must also grow without bound,which contradicts the fact that the support of f stays inside [ ℓ, m ].The global algorithm only needs to consider primes in c or c − , otherwise f inLemma 12 would be 0. Correctness of the global algorithm follows from Theorem 15.
6. Component-wise Bounds
We give ˆ Z : = Z S {∞} the structure of a tropical semi-ring ( ˆ Z , ⊕ , ⊗ ) with ⊕ = minand ⊗ = + . We extend this to matrices. If A ∈ ˆ Z m × n and B ∈ ˆ Z n × ℓ then the i j ’th entryof A ⊗ B is ( A ⊗ B ) i j : = n M k = A ik ⊗ B k j : = min { A ik + B k j | k n } . If p is a prime and A ∈ K m × n then V p ( A ) ∈ ˆ Z m × n denotes the matrix whose i j ’thentry is v p ( A i j ). The smallest entry is v p ( A ). Equation (1) implies: V p ( AB ) > V p ( A ) ⊗ V p ( B ) (5)for all A ∈ K m × n and B ∈ K n × ℓ , where the inequality is interpreted for each entryseparately. Example . Let A = Q [ x ], p = x and M = − + x − x + x x x x + x x and Y = x − x . Then V p ( M ) = ∞ ∞ and V p ( Y ) = . Lets check Equation (5) for M and Y : = V p ( x − x x ) = V p ( MY ) > V p ( M ) ⊗ V p ( Y ) = min { + , + , + } min {∞ + , + , + } min { + , + , ∞ + } = . lgorithm
17 ( J ’th component-wise content bound) .Input M ∈ GL n ( K ) and J > Output B ∈ K n such that ∃ a ∈ D with aY ∈ diag( B ) A n for any rational solution Y . Procedure (a) Compute M j for j ∈ {− J . . . J } .(b) Let P be the set of prime factors in the denominators in M and M − .(c) O : = select one p ∈ P from each τ -equivalence class.(d) Let B i : = i ∈ { , . . . , n } .(e) For each p ∈ O − D (e.1) For j ∈ {− J . . . J } , compute the exponent-function E j of M j at p , whichis a function E j : Z → ˆ Z n × n where E j ( k ) : = V τ k ( p ) ( M j ).(e.2) Call the local algorithm below with input E − J . . . E J .(e.3) It returned a function F : Z → ˆ Z n . For i ∈ { . . . n } : If F i (the i ’thcomponent of F ) is ∞ then B i : =
0, otherwise B i : = B i · Q k ∈ Z τ k ( p ) F i ( k ) .(f) Return ( B , . . . , B n ) t .If an entry of M j is zero, then the corresponding entry of E j ( k ) is ∞ for all k ∈ Z .To obtain a finite “support”, we define supp ( E j ) as the set of all k ∈ Z for which E j ( k ) < { , ∞} n × n . This way we can represent E j in finite terms with: integers ℓ j , m j such that supp ( E j ) ⊆ [ ℓ j , m j ], matrices E j ( k ) ∈ ˆ Z n × n for k ∈ [ ℓ j , m j ], and a matrix wedenote as E j ( ∞ ) ∈ { , ∞} n × n such that E j ( k ) = E j ( ∞ ) for all k < [ ℓ j , m j ]. Algorithm
18 ( J th local component-wise content bound) .Input: E − J , . . . , E J . Output: F : Z → ˆ Z n such that F ( k ) V τ k ( p ) ( Y ) for all k ∈ Z and rational solutions Y . Procedure: (a) Let ℓ, m be as in Lemma 12, let c = F : Z → ( ˆ Z ∪ {−∞} ) n F ( k ) : = ( −∞ , . . . , −∞ ) t if ℓ k m (0 , . . . , t otherwise . (b) Repeat:(b.1) F new ( k ) : = max { E j ( k + j ) ⊗ F ( k + j ) | − J j J } (for all k ∈ Z )where the maxima are taken component-wise.(b.2) If F new = F then stop and return F .(b.3) If all negative entries of F and F new are the same, then c : = c + c >
10 then return F new . (For alternatives see subsection 6.1.)(b.4) Let F : = F new and Repeat.9 heorem 19. Algorithm 18 is correct and terminates.Proof.
As in section 5, entries can not decrease and the algorithm does not stop ifany entries = −∞ remain. Apart from replacing scalars with matrices and vectors,correctness is proved in the same way as well. As for termination, negative entries canonly increase finitely many times, which makes c in step (b.3) a simple terminationmechanism. For more sophisticated versions, see subsection 6.1 below. ff The question in this subsection is how to ensure termination without an arbitrarycut-o ff counter c in step (b.3). We sketch one approach with an example, and an alter-native that is easier to implement.Let M = x
00 1 ! , take p = x , and let P n = τ − ( p ) · · · τ − n ( p ) = ( x − · · · ( x − n ).Up to constants, the only rational solution of τ ( Y ) = MY is (0 , t . Now ( P n , t isa valid content bound for any n since Y = P n . In every loop,Algorithm 18 constructs an F new that is strictly sharper than F (if F encodes ( P n , t then F new encodes ( P n + J , t ). So if we remove step (b.3) without implementing analternative, then the algorithm will not terminate for M .During the computation F looks as follows. Since M is a 2 by 2 matrix, F has twocomponents F and F , each of which is a function Z → ˆ Z S {−∞} . After the first loop F is identically 0, while F looks like this . . . , , , , . . . , , , . . . which encodes P n where n is the number of 1’s. This n increases by J in each loop.We now sketch the first approach to ensure termination without an arbitrary cut-o ff . Outside a finite range of k ’s, the matrices E j ( k ) are constant (recall E j ( ∞ ) rightbefore Algorithm 18). If a su ffi ciently long repeating pattern of positive entries in F ( k )’s outside of this range forms during the computation, then, since the E j ( k ) areconstant here, it is not hard for the algorithm to prove that this pattern will continueindefinitely. In the example, when at least n = E ( ∞ ) that this pattern can onlygrow in each loop. But that means that Y , the first entry of Y , must be divisible by apolynomial P n whose degree keeps increasing. That implies Y =
0, so we can replace F by the function that is identically + ∞ . With this strategy, only finitely many entries < { , ∞} can occur, because if more than a bounded number appear, the algorithm canconstruct a proof from E − J ( ∞ ) . . . E J ( ∞ ) that the pattern will continue, allowing it toreplace a component of F by + ∞ .We decided not to spell out the details of this approach, because there is a simplerapproach which accomplishes a similar outcome. Let the degree of a rational functionbe the degree of the numerator minus the degree of the denominator. To computerational solutions Y , one needs to compute a degree-bound for the entries of Y . Forinstance, if Y is a polynomial of degree
3, then the information that Y is divisibleby P is equivalent to the “sharper” bound that Y is divisible by P , since both imply Y =
0. So one can design a version of Algorithm 18 where the arbitrary cut-o ff c > ff informed by a degree-bound.Among these alternatives, while the arbitrary cut-o ff approach is the least elegant,we presented it as the default because it takes the least amount of implementation e ff ort,10nd its practical performance, except in very rare cases, will likely be the same as thealternatives sketched in this subsection.
7. Example, an eigenring system
To factor an operator L = τ + a τ + a ∈ Q ( x )[ τ ] with the eigenring [12, 4] methodwe need rational solutions for the system τ ( Y ) = MY where M = − b − a b − a − a a b a a b a b a b with b = τ ( a ) . For our example let a = x ( x + x + x + x + x − x + x +
1) and a = − ( x + x + x + x − x − x + x − x + x + . The global content bounds for J B global J = = x − x ( x + ( x + x + pqB global J = = x − x ( x + x + x + pqB global J = = x − x ( x + x + pqB global J = = x − x ( x + pq where p = x + x + q = τ ( p ). The bound from [2] (Maple’s UniversalDenom-inator) is the same as B global J = . Among global content bounds, B global J = is sharp (it equalsthe content of the set of all entries of all rational solutions). But our component-wisecontent bounds are sharper still. For J = J = x − x ( x + p , x ( x + x + q , x − x ( x + x + p , x ( x + ( x + q ! t x + x − p , x + x ( x + q , x − p , x + xq ! t The J = J = B global J = ). If an operator L is the LCLM (Least Common Left Multiple) of smaller operators then L can be factoredwith the eigenring method. This example was constructed as LCLM( τ − x ( x + / ( x + , τ − ( x + / x ). Thisconstruction ensures that M will have at least two (exactly two here) independent rational solutions. J = M − but this is done inall variations. For J = M and M − .After that, we have to compute valuations of their entries, as these valuations formthe entries of the exponent-functions E j . If Q ∈ Q ( x ) is an entry of M j , then a fullfactorization of Q immediately gives its valuation at every prime q ∈ Q [ x ]. However, afull factorization also computes information we do not need, since the only valuationswe use are at primes q of the form τ k ( p ) with p as in the algorithm.We need to compute valuations rapidly in order for the component-wise algorithmto be quick. With modular techniques one can quickly compute an upper bound for thevaluation of a rational function Q at any q , correctness can then be proved with a trialdivision.For special matrices such as the example M above, only a few rational functionsneed to be factored for the J = a and a (then usethat b is a shift of 1 / a ). The same is true for “exterior power systems” which are used[5] to factor general di ff erence operators (the eigenring method only factors specialcases, in particular LCLM’s). The first author’s factoring implementation [9] is set upin a way where rational solutions are already polynomials, but the implementation stillcomputes a component-wise content bound because it significantly reduces the degreesof the polynomials that the algorithm has to find. Acknowledgements
Mark van Hoeij was supported by NSF grant 1618657. Johannes Middeke wassupported by the Austrian Science Fund (FWF) grant SFB50 (F5009-N15).
References [1] Abramov, S. A., 1995. Rational solutions of linear di ff erence and q -di ff erenceequations with polynomial coe ffi cients. Programming and Computer Software21 (6), 273–278, translated from Russian.[2] Abramov, S. A., Barkatou, M. A., 1998. Rational solutions of first order lineardi ff erence systems. In: Proceedings of ISSAC’98.[3] Abramov, S. A., Khmelnov, D. E., 2012. Denominators of rational solutions oflinear di ff erence systems of an arbitrary order. Programming and Computer Soft-ware 38 (2), 84–91.[4] Barkatou, M., 1999. Rational solutions of matrix di ff erence equations: the prob-lem of equivalence and factorization. In: Proceedings of ISSAC’99. Vancouver,BC, pp. 277–282.[5] Bronstein, M., 2006. Factorization and hypergeometric solutions oflinear recurrence systems. (Ideas due to Manual Bronstein, pre-sented at the conference in memory of Manuel Bronstein). URL http://dx.doi.org/10.1016/j.aam.2007.11.004 [7] Ghe ff ar, A., Abramov, S. A., 2011. Valuations of rational solutions of linear dif-ference equations at irreducible polynomials. Adv. Appl. Math. 47 (2), 352–364.URL https://doi.org/10.1016/j.aam.2010.07.002 [8] van Hoeij, M., 1998. Rational solutions of linear di ff erence equations. In: Proc.of ISSAC’98. pp. 120–123.[9] van Hoeij, M., 2020. Factoring recurrence operators (RFactors) implementationavailable at URL [10] Middeke, J., 2017. Denominator bounds and polynomial solutions for systems of q -recurrences over K ( t ) for constant K . In: Proceedings of ISSAC. Kaiserslauern,Germany, pp. 325–332.[11] Schneider, C., Middeke, J., 2017. Waterloo Workshop on Computer Algebra. sub-mitted, Ch. Denominator Bounds for Systems of Recurrence Equations using ΠΣ -Extensions, https://arxiv.org/abs/1705.00280 .[12] Singer, M., 1996. Testing reducibility of linear di ffff