An Additive Decomposition in S-Primitive Towers
aa r X i v : . [ c s . S C ] F e b An Additive Decomposition in S-Primitive Towers
Hao Du , Jing Guo , Ziming Li , Elaine Wong Johann Radon Institute (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040, Linz, Austria Key Laboratory of Mathematics and Mechanization, AMSS, Chinese Academy of SciencesSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, [email protected], [email protected], [email protected], [email protected]
ABSTRACT
We consider the additive decomposition problem in primitive tow-ers and present an algorithm to decompose a function in an S-primitive tower as a sum of a derivative in the tower and a re-mainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitelymany logarithmic functions and logarithmic integrals. A functionin an S-primitive tower is integrable in the tower if and only if theremainder is equal to zero. The additive decomposition is achievedby viewing our towers not as a traditional chain of extension fields,but rather as a direct sum of certain subrings. Furthermore, we candetermine whether or not a function in an S-primitive tower hasan elementary integral without solving any differential equations.We also show that a kind of S-primitive towers, known as logarith-mic towers, can be embedded into a particular extension where wecan obtain a finer remainder.
KEYWORDS
Additive decomposition, Primitive tower, Logarithmic tower, Sym-bolic integration, Elementary integrability
ACM Reference Format:
Hao Du , Jing Guo , Ziming Li , Elaine Wong . 2020. An AdditiveDecomposition in S-Primitive Towers. In ISSAC ’20: International Sympo-sium on Symbolic and Algebraic Computation, June 20–23, 2020, Kalamata,Greece.
ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/1122445.1122456
We consider the integrability problem in some class F of functionsin x , where F is assumed to be closed under addition and the usualderivation ′ = ddx . For f ∈ F , we ask if the indefinite integral of f belongs to F . Let F ′ : = { д ′ | д ∈ F } . The problem can thereforebe stated as follows:Given f ∈ F , decide if f ∈ F ′ . (1) Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full cita-tion on the first page. Copyrights for components of this work owned by others thanACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re-publish, to post on servers or to redistribute to lists, requires prior specific permissionand/or a fee. Request permissions from [email protected].
ISSAC ’20, June 20–23, 2020, Kalamata, Greece © 2020 Association for Computing Machinery.ACM ISBN 978-1-4503-9999-9/18/06...$15.00https://doi.org/10.1145/1122445.1122456
We can see that a positive answer to (1) tells us that we cancompute д ∈ F such that f = д ′ . If (1) produces a negative answer,then we say f is not integrable in F .In the latter case, we would still like to be able to say somethingabout the given function. Is there any information to help us un-derstand how far off we are from being successful? The answer liesin the additive decomposition problem:Compute д , r ∈ F such that f = д ′ + r , where(i) r is minimal in some sense;(ii) f ∈ F ′ if and only if r = r a remainder of f in F and write f ≡ r mod F ′ . So, it is clear that an algorithm for solving the problem of additivedecomposition also provides a solution to the integrability prob-lem. Elements in F ′ have a special form, indicating that most func-tions have nonzero remainders. Remainders help us find “closedform” expressions for integrals of elements in F , in the sense thatthe integrals belong to some extensions over F . They also play animportant role in reduction-based methods for creative telescop-ing.The first additive decomposition for the class F = C ( x ) is due toOstrogradsky [13] and Hermite [12]. Given a rational function f ∈F , they were able to compute a remainder r ∈ F of f such that r isproper and has a squarefree denominator, and r is minimal in thesense that if f ≡ ˜ r mod F ′ for some ˜ r ∈ F , then the denominatorof r divides that of ˜ r .There has been rapid development of additive decompositionsin both symbolic integration and summation in recent years [1,3, 4, 7–9, 11, 14, 16]. Most of the articles were motivated by com-puting telescopers based on reduction [2]. In the cited literature,some classes of functions that were studied include hyperexpo-nential [3], algebraic [9], Fuchsian D-finite [7], and D-finite [16].Additive decomposition problems in these classes have been fullysolved. We observe that the space of D-finite functions is not closedunder composition or taking reciprocals. For example, log x is D-finite, but log ( log ( x )) and 1 / log ( x ) are not. In this paper, we con-sider a class of functions that is closed under these two operations.Singer et al. in 1985 and then Raab in 2012 gave some decisionprocedures for finding elementary integrals in some Liouvillian ex-tensions [14, 15] and in the extensions which contain some nonlin-ear generators [14]. They recursively solve Risch differential equa-tions until one of them has no solution, or else the integral can befound. In the implementation of Raab’s algorithm, the former caseoutputs an integrable part and collects all nonzero terms that pre-vent the differential equations from having a solution. Recently, SSAC ’20, June 20–23, 2020, Kalamata, Greece Du, Guo, Li, Wong
Chen, Du and Li [6] were able to construct remainders in someprimitive extensions (they termed them “straight towers” and “flattowers”) without solving any differential equations.In this article, we expand their work [6] to “S-primitive towers”,which can be neither straight nor flat. Instances for S-primitivetowers include differential field extensions generated by finitelymany logarithmic functions and logarithmic integrals. Moreover,we show that a logarithmic tower can be embedded in awell-generated logarithmic tower with the aid of logarithmic prod-uct and quotient rules. We can compute “finer” remainders in suchan extension.
Primitive Towers K ( t , . . . , t n ) S-Primitive TowersLogStraight Flat Well-GeneratedLog Towers K ( u , . . . , u w ) EmbeddingTheorem 5.6
Figure 1: The gray ellipses on the left indicate the classesof functions for which we can construct a remainder. Theembedding gives us a field extension ( n ≤ w ) where a “finer”remainder can be obtained. The organization of this article is as follows. In Sections 2 and 3,we give some relevant definitions associated to primitive towers,and then present a different way to view the towers. In Section 4,we give an algorithm for additive decompositions in S-primitivetowers, and present a criterion for elementary integrability for thefunctions in such a field. In Section 5, we discuss how to find afiner additive decomposition in well-generated logarithmic towers.Concluding remarks are given in Section 6.
Let K be a field of characteristic zero and K ( t ) be the field of ratio-nal functions in t over K . An element of K ( t ) is said to be t -proper if the degree of its denominator in t is higher than that of its nu-merator. In particular, zero is t -proper. For each f ∈ K ( t ) , thereexists a unique t -proper element д ∈ K ( t ) and a unique polyno-mial p ∈ K [ t ] such that f = д + p . (2)Let ′ be a derivation on K . The pair ( K , ′ ) is called a differentialfield. An element c of K is called a constant if c ′ =
0. The set of con-stants in K , denoted by C K , is a subfield of K . Set K ′ : = { f ′ | f ∈ K } , which is a linear subspace over C K . We call K ′ the integrablesubspace of K .Let ( E , δ ) be a differential field containing K . We say that E is a differential field extension of K if δ | K = ′ . The derivation δ is alsodenoted by ′ when there is no confusion. Let ( F , δ ) be anotherdifferential field. An algebraic homomorphism ϕ from K to F issaid to be differential if ϕ ( f ′ ) = ϕ ( f ) δ for all f ∈ K .Let ( K , ′ ) be a differential field and f ∈ K . We call f a log-arithmic derivative in K if f = д ′ / д for some д ∈ K . Let K ( t ) be a differential extension of K where t is transcendental over K and t ′ ∈ K [ t ] . A polynomial p in K [ t ] is said to be t -normal if gcd ( p , p ′ ) =
1. For f ∈ K ( t ) , we say that f is t -simple if it is t -proper and has a t -normal denominator.We next define primitive and logarithmic generators, which arebased on Definitions 5.1.1 and 5.1.2 in [5] , respectively. Definition 2.1.
Let ( K , ′ ) be a differential field, and E be a dif-ferential field extension of K . An element t of E is said to be primitive over K if t ′ ∈ K . A primitive element t is called a primitive generator over K if it is transcendental over K and C K ( t ) = C K . Furthermore,a primitive generator t is called a logarithmic generator over K if t ′ is a C -linear combination of logarithmic derivatives in K . An immediate consequence of Theorem 5.1.1 in [5] is: Proposition 2.2.
Let t be primitive over K . Then t is a primitivegenerator over K if and only if t ′ < K ′ . Assume that t is a primi-tive generator over K . Then p ∈ K [ t ] is t -normal if and only if p issquarefree. For the rest of the section, assume that ( K , ′ ) is a differentialfield, and that t is a primitive generator over K . By Theorem 5.3.1in [5] and Lemma 2.1 in [6] , for each f ∈ K ( t ) , there exists aunique t -simple element h such that f ≡ h mod (cid:0) K ( t ) ′ + K [ t ] (cid:1) . (3)We call h the Hermitian part of f with respect to t , and denote itby hp t ( f ) . It is easy to check that hp t is a C K -linear map on K ( t ) .Because of the uniqueness of Hermitian parts and Lemma 2.1 in[6] , we have the following lemma. Lemma 2.3.
Let f , д ∈ K ( t ) . Then (i) f ∈ K ( t ) ′ + K [ t ] = ⇒ hp t ( f ) = , (ii) f is t -simple = ⇒ f = hp t ( f ) , and (iii) f ≡ д mod ( K ( t ) ′ + K [ t ]) = ⇒ hp t ( f ) = hp t ( д ) . The next two lemmas give some nice properties of proper ele-ments and logarithmic derivatives.
Lemma 2.4. If f ∈ K ( t ) is t -proper, then f − hp t ( f )∈ K ( t ) ′ . Proof.
Since t is a primitive generator over K , the derivative ofa t -proper element of K ( t ) is also t -proper. By (3), f = hp t ( f ) + д ′ + p for some д ∈ K ( t ) and p ∈ K [ t ] . Let r be the t -proper part of д .Thus, f − hp t ( f ) − r ′ = p + ( д − r ) ′ whose left-hand side is t -properand whose right-hand side is a polynomial in t . Thus, both sidesmust be zero. Consequently, f − hp t ( f ) = r ′ ∈ K ( t ) ′ . (cid:3) Lemma 2.5.
Let f ∈ K ( t ) be a logarithmic derivative. (i) If f is t -proper, then f is t -simple. (ii) There exists a t -simple logarithmic derivative д ∈ K ( t ) and alogarithmic derivative h ∈ K such that f = д + h . Proof. (i) The only thing we need to show is that the denom-inator of f is t -normal. By the logarithmic derivative identity [5,Theorem 3.1.1 (v)] , the denominator of f is squarefree, which isalso t -normal by Proposition 2.2.(ii) By irreducible factorization and the logarithmic derivativeidentity, f = (cid:16)Í i m i p ′ i / p i (cid:17) + α ′ / α , where α ∈ K , m i ∈ Z , and p i ∈ K [ t ] is monic irreducible and pairwise coprime. Then each p ′ i / p i is t -proper, because t is primitive over K . Setting д = Í i m i p ′ i / p i and h = α ′ / α yields (ii). (cid:3) n Additive Decomposition in S-Primitive Towers ISSAC ’20, June 20–23, 2020, Kalamata, Greece The following lemma will be useful when we construct our re-mainders. This is the same as Lemma 2.3 in [6].
Lemma 2.6.
Let p ∈ K [ t ] . If p ∈ K ( t ) ′ , then the leading coefficientof p is equal to ct ′ + b ′ for some c ∈ C K and b ∈ K . As a special case,if p ∈ K ∩ K ( t ) ′ , then p ≡ ct ′ mod K ′ . We denote { , , . . . , n } and { , , , . . . , n } by [ n ] and [ n ] , respec-tively. Let ( K , ′ ) be a differential field and for each i ∈ [ n ] , K i = K i − ( t i ) , where t i is transcendental over K i − and t ′ i ∈ K i . Thenwe have a tower of differential extensions: K ⊂ K ⊂ · · · ⊂ K n q q K ( t ) ⊂ · · · ⊂ K n − ( t n ) . (4)We use K ( ¯ t ) to denote the tower (4), where ¯ t : = ( t , . . . , t n ) refersto the generators in the chain of field extensions (to contrast with K n , which is just the largest field in the chain).We can describe K ( ¯ t ) based on the nature of its generators.If K = ( C ( x ) , d / dx ) and each t i in (4) is a primitive generatorover K i − for all i ∈ [ n ] , then we call K n a primitive extension over K and K ( ¯ t ) a primitive tower . By Definition 2.1, C K n = C K ,which is equal to C . Furthermore, a primitive tower is said to be log-arithmic if each t i is a logarithmic generator over K i − . For brevity,the primitive tower K ( ¯ t ) is also denoted by K n when its genera-tors are clear from the context.For each i ∈ [ n ] , an element of K n from (4) is said to be t i -proper if it is free of t i + , . . . , t n and the degree of its numerator in t i islower than that of its denominator. Denote by T i the multiplicativemonoid generated by t i + , . . . , t n for all i with 0 ≤ i < n , and set T n = { } . For each i ∈ [ n ] , let P i be the additive group consistingof all the linear combinations of the elements of T i whose coeffi-cients are t i -proper. Furthermore, let P = K [ t , . . . , t n ] . All ofthe P i ’s are closed under multiplication. A routine induction basedon (2) shows K n = n Ê i = P i . (5)Let π i be the projection from K n onto P i with respect to (5). Forevery element f ∈ K n , we have that f = n Õ i = π i ( f ) , which is called the matryoshka decomposition of f . Figure 2 illus-trates this namesake. We also call π i ( f ) the i -th projection of f forall i ∈ [ n ] . This new view allows us to describe the followingordering (which will be used to define a remainder).Suppose that ≺ is the purely lexicographic order on T , in which t ≺ t ≺ · · · ≺ t n . Then ≺ is also a monomial order on each T i ,because T i ⊆ T . For f ∈ K n and i ∈ [ n ] , the i -th projection of f can be viewed as a polynomial in K i [ t i + , . . . , t n ] , which allows usto define the i -th head monomial of f , denoted by hm i ( f ) , to be thehighest monomial in T i that appears in π i ( f ) if π i ( f ) is non-zero,and zero if π i ( f ) is zero.We define the i -th head coefficient of f , denoted by hc i ( f ) , to bethe coefficient of hm i ( f ) in π i ( f ) if π i ( f ) is non-zero, and zero π ( f ) + P É π ( f ) + P É π ( f ) P + É · · · + É π n ( f ) = P n = P n ... P P P f ∈ K n Figure 2: Matryoshka Decomposition if π i ( f ) is zero. By the matryoshka decomposition, hc i ( f ) is t i -proper for all i ∈ [ n ] . The head monomial of f , denoted by hm ( f ) , is defined to be thehighest monomial among hm ( f ) , hm ( f ) , . . . , hm n ( f ) , in whichzero is regarded as the lowest “monomial”. Let I f = { i ∈ [ n ] | hm i ( f ) = hm ( f )} . The head coefficient of f , denoted by hc ( f ) , isdefined to be Í i ∈ I f hc i ( f ) . Definition 3.1.
For f , д ∈ K n , denote d f and d д to be the degreesof the denominators of f and д with respect to t n , respectively. Wesay that f is lower than д , denoted by f ≺ д , if either d f < d д , or d f = d д and hm ( f ) ≺ hm ( д ) . We say that f is not higher than д ,denoted by f (cid:22) д , if either f ≺ д , or d f = d д and hm ( f ) = hm ( д ) . Since ≺ on T is a Noetherian total order, the partial order on K n given by Definition 3.1 is also Noetherian, that is, every nonemptyset in K n has a minimal element w.r.t. ≺ . We can use this order todefine a desired remainder of the given function. Let f ∈ K n and R f : = { д ∈ K n | д ≡ f mod K ′ n } . (6)Thus, there exists a minimal element r ∈ R f . We note that such aminimal element is not unique. Definition 3.2.
Given f ∈ K n , a minimal element of R f is saidto be a remainder of f . Moreover, let r ∈ K n . Then we say that r is aremainder if r is a remainder of itself. As usual, simple elements (or Hermitian parts) play an impor-tant role when we construct remainders. Before we move on tothe next section, we first generalize the definition of t -simple ele-ments from the previous section with the help of the matryoshkadecomposition. Definition 3.3.
An element f ∈ K n is said to be simple if π i ( f ) is t i -simple for all i ∈ [ n ] , where t = x . SSAC ’20, June 20–23, 2020, Kalamata, Greece Du, Guo, Li, Wong
Remainders in a tower are described in terms of minimality, whichis not constructive. In this section, we will present an algorithmfor constructing a remainder in an S-primitive tower (see Defini-tion 4.3), based on Hermite reduction and integration by parts. Toknow when to terminate the algorithm, we need to be able to iden-tify the first generator present in a given monomial (this is thesame notion as scale in [6]).
Definition 4.1.
For a monomial M = t d · · · t d n n ∈ T , the in-dicator of M , denoted by ind n ( M ) , is defined to be n if M = , ordefined to be min { i ∈ [ n ] | d i , } . For M ∈ T , we set K (≺ M ) n : = { f ∈ K n | hm ( f ) ≺ M } . Note that K (≺ M ) n is closed under addition. The following lemma describessufficient conditions for reducing a given term with respect to ≺ via integration by parts. Lemma 4.2.
Let K n be primitive, M ∈ T with indicator m , and a ∈ K m − . Then aM ∈ K ′ n + K (≺ M ) n if (i) a ∈ K ′ m − , or (ii) a ∈ span C { t ′ , . . . , t ′ m } . Proof.
It is obvious for M =
1. Assume that M , M = t d m m · · · t d n n for d m , . . . , d n ∈ N and d m >
0. Since K n is a primitive extension over K , we have t ′ j ∈ K j − for each j with m ≤ j ≤ n . Then M ′ = n Õ j = m h j N j , (7)where h j belongs to K j − , and N j is either equal to zero if d j = t d j − j t d j + j + · · · t d n n if d j >
0. There exists д ∈ K m − such that a = д ′ ,because a ∈ K ′ m − . With integration by parts and (7), we see that д ′ M = ( дM ) ′ + Í nj = m (− дh j ) N j . Let M j = t d j j t d j + j + · · · t d n n for all j with m ≤ j ≤ n . Then N j ≺ M j ≺ M implies − дh j N j ≺ M j ≺ M because дh j is free of t j , t j + , . . . , t n , and N j ≺ M j . It follows that Í nj = m (− дh j ) N j ≺ M and aM ∈ K ′ n + K (≺ M ) n .(ii) Let M = t dm N , where d ∈ Z + and N ∈ T m . Since a ∈ span C { t ′ , . . . , t ′ m } , a = д + h , where д ∈ K ′ m − and h = ct ′ m forsome c ∈ C . Then дM ∈ K ′ n + K (≺ M ) n by (i) and hM = ct ′ m t dm N = (cid:16) cd + t d + m (cid:17) ′ N . The lemma holds since hM ∈ K ′ n + K (≺ N ) n and N ≺ M . (cid:3) In order to avoid increasing the order during the process andobtain sufficient and necessary conditions, we need to impose anextra condition on the generators:hm ( t ′ i ) = i ∈ [ n ] . By Lemma 2.4 and the rational additive decomposition, for all i ∈[ n ] , there exists a simple h i in K i − and a д i ∈ K i − such that t ′ i = д ′ i + h i . Let u i = t i − д i . Then u i is a primitive generatorover K i − . Moreover, K ( ¯ t ) = K ( ¯ u ) . Therefore, without loss ofgenerality, we can further assume that each t ′ i is simple in K i − forall i ∈ [ n ] . Definition 4.3.
A tower K ( ¯ t ) is said to be S-primitive if it is aprimitive tower and t ′ i is simple for all i ∈ [ n ] . Our next goal is to construct remainders in S-primitive towersbased on a special property of simple elements.
Lemma 4.4.
Let K n be an S-primitive tower. If f ∈ K ′ n is simple,then f ∈ span C { t ′ , . . . , t ′ n } . Proof.
Since f ∈ K ′ n and π n ( f ) is t n -simple, π n ( f ) = hp t n ( f ) = f ∈ K n − .We proceed by induction on n . If n =
1, then f ∈ K ∩ K ′ is x -simple by Definition 3.3. By Lemma 2.6, there exists a c ∈ C suchthat f ≡ ct ′ mod K ′ . Since both f and t ′ are x -simple, we havethat f = ct ′ by Lemma 2.3 (ii) and (iii).Assume that n > n −
1. For f in K n − ∩ K ′ n , there is a c ∈ C such that f ≡ ct ′ n mod K ′ n − by Lemma 2.6.Then f − ct ′ n ∈ K ′ n − . Since both f and t ′ n are simple, f − ct ′ n isalso simple. By the induction hypothesis, we have that f − ct ′ n ∈ span C { t ′ , . . . , t ′ n − } , which implies that f ∈ span C { t ′ , . . . , t ′ n } . (cid:3) The previous lemma gives us a direct way to determine whetheror not a tower is S-primitive.
Corollary 4.5.
The tower K n is S-primitive if and only if for all i ∈ [ n ] , t ′ i ∈ K i − is simple and t ′ , . . . , t ′ n are C -linearly independent. Proof. If K n is an S-primitive tower, then t ′ i is simple for all i ∈ [ n ] . Furthermore, t ′ i < K ′ i − for all i ∈ [ n ] by Proposition 2.2. So t ′ , . . . , t ′ n are C -linearly independent.We prove the converse by induction. If n =
1, then a non-zeroand simple t ′ clearly implies that K is S-primitive. Suppose n > n −
1. Assume that for all i ∈ [ n ] , t ′ i ∈ K i − is simple and that t ′ , . . . , t ′ n are C -linearly independent.By the induction hypothesis, K n − is S-primitive. By Lemma 4.4, t ′ n < span C { t ′ , . . . , t ′ n − } implies that t ′ n < K ′ n − . Thus, t n is aprimitive generator over K n − by Proposition 2.2. Accordingly, K n is S-primitive. (cid:3) The following lemma gives a sufficient and necessary conditionin S-primitive towers for lowering an element with respect to ≺ modulo the integrable space. Lemma 4.6.
Suppose that K n is an S-primitive tower. Let M ∈ T with ind n ( M ) = m and a ∈ K m − be simple. Then aM ∈ K ′ n + K (≺ M ) n if and only if a ∈ span C { t ′ , . . . , t ′ m } . Proof.
The sufficiency follows from Lemma 4.2 (ii). Conversely,assume that aM ∈ K ′ n + K (≺ M ) n . If M =
1, then m = n and a ∈ K ′ n . ByLemma 4.4, a ∈ span C { t ′ , . . . , t ′ n } . If M ≻ M = t d m m · · · t d n n and d m >
0, we can proceed by induction on n .For the base case, aM ∈ K ′ + K (≺ M ) implies that there exists a t -proper element b ∈ K and p ∈ K [ t ] with deg t ( p ) < d suchthat aM + b + p ∈ K ′ . By Lemma 2.4 and Lemma 2.3 (i), aM + p ∈ K ′ .Then Lemma 2.6 implies that a − ct ′ ∈ K ′ for some c ∈ C . Hence, a = ct ′ , because a and t ′ are both x -simple. n Additive Decomposition in S-Primitive Towers ISSAC ’20, June 20–23, 2020, Kalamata, Greece Assume that n > n −
1. Let N = M / t d n n , which is a power product of t m , . . . , t n − . Since aM ∈ K ′ n + K (≺ M ) n , there is a t n -proper element b and p ∈ K n − [ t n ] withhm ( p ) ≺ M such that aNt d n n + b + p ∈ K ′ n . By Lemma 2.4, wecan assume that b is t n -simple. So, b = p = qt d n n + r such that q ∈ K n − with hm ( q ) ≺ N and r ∈ K n − [ t n ] with deg t n ( r ) < d n . Then we have ( aN + q ) t d n n + r ∈ K ′ n . ByLemma 2.6, there exists c ∈ C such that aN + q − ct ′ n ∈ K ′ n − .Hence, aN ≡ ct ′ n mod (cid:0) K ′ n − + K (≺ N ) n − (cid:1) . (8)If N =
1, then m = n and a ∈ K ′ n . By Lemma 4.4, we have that a ∈ span C { t ′ , . . . , t ′ n } . The lemma holds. If N ≻
1, then ind n − ( N ) = m < n . By (8), aN ∈ K ′ n − + K (≺ N ) n − , because hm ( ct ′ n ) =
1. It followsfrom the the induction hypothesis that a ∈ span C { t ′ , . . . , t ′ m } . (cid:3) We can now specify a remainder in S-primitive towers and provethat the algorithm to construct it will terminate.
Proposition 4.7.
Let K n be an S-primitive tower, and r ∈ K n with m = ind n ( hm ( r )) . Then r is a remainder if either r = , or π n ( r ) is t n -simple and hc ( r − π n ( r )) is simple and is not a nonzeroelement of span C { t ′ , . . . , t ′ m } . Proof.
Let f ∈ R r as defined in (6). As π n ( r ) is t n -simple, wehave hp t n ( f ) = π n ( r ) by Lemma 2.3 (ii) and (iii). Then the denom-inator of π n ( r ) , which is exactly the denominator of r as a polyno-mial in K n − [ t n ] , divides the denominator of f by Theorem 5.3.1in [5] .We further need to show that hm ( r ) (cid:22) hm ( f ) . Suppose the con-trary. Then r ,
0. Let M = hm ( r ) and a = hc ( r − π n ( r )) .If M =
1, then m = n , a = r − π n ( r ) , and f =
0, which impliesthat r ∈ K ′ n . Then π n ( r ) = a ∈ K n − ∩ K ′ n .By Lemma 4.4, we have that a belongs to span C { t ′ , . . . , t ′ n } . Thus, a = r =
0, a contradiction.Assume that M ≻
1. Since M ≻ hm ( f ) , we have that hm ( r − f ) = M and hc ( r − f ) = hc ( r ) . Then hc ( r − f ) = a because M ≻ ( π n ( r )) =
1. From r − f ∈ K ′ n , we see that a M ∈ K ′ n + K (≺ M ) n . By Lemma 4.6, a belongs to span C { t ′ , . . . , t ′ m } , which implies that a =
0. Then r = π n ( r ) and M =
1, a contradiction. (cid:3)
Theorem 4.8.
Let K n be an S-primitive tower and let f ∈ K n .Then one can construct a remainder of f with the properties describedin Prop. 4.7 in a finite number of steps. Proof.
By Lemma 2.4, π n ( f ) ≡ hp t n ( f ) mod K ′ n . Then f ≡ hp t n ( f ) + ( f − π n ( f )) mod K ′ n . (9)The n -th projection of the right-hand side of the congruence isequal to hp t n ( f ) , which is t n -simple.Let M = hm ( f − π n ( f )) . We proceed by a Noetherian inductionon M with respect to ≺ . If M =
0, then f = π n ( f ) . By (9) andProposition 4.7, hp t n ( f ) is a remainder of f .Assume that M ,
0, and for any д ∈ K n with hm ( д ) ≺ M , thereis a remainder ˜ r of д as described in Proposition 4.7.Let a = hc ( f − π n ( f )) and m = ind n ( M ) . Since a ∈ K m − , its j -thprojection is equal to zero for each j ∈ { m , . . . , n } . By Lemma 2.4, π i ( a ) ≡ h i mod K ′ i for some t i -simple elements h i ∈ K i for all i ∈ [ m − ] with t = x . By Lemma 4.2 (i), f − π n ( f ) ≡ bM mod ( K ′ n + K (≺ M ) n ) , (10)where b = Í m − i = h i . Note that b is simple by Definition 3.3.If b ∈ span C { t ′ , . . . , t ′ m } , then bM is in K ′ n + K (≺ M ) n by Lemma 4.2(ii). So f − π n ( f ) ≡ д mod K ′ n for some д in K (≺ M ) n by (10). Ac-cordingly, д has a remainder ˜ r as described in Proposition 4.7 bythe induction hypothesis. It follows that hp t n ( f ) + ˜ r is a remainderof f .Assume that b < span C { t ′ , . . . , t ′ m } . It follows from (9) and (10)that f ≡ hp t n ( f ) + bM + д mod K ′ n for some д in K (≺ M ) n . More-over, we may further assume that π n ( д ) is t n -simple by Lemma 2.4.The right-hand side of the above congruence is a remainder asdescribed in Proposition 4.7, because b is the head coefficient of bM + ( д − π n ( д )) . (cid:3) We now present an algorithm to decompose an element in anS-primitive tower over K = ( C ( x ) , d / dx ) into a sum of a derivativeand a remainder. The algorithm is a slight refinement of the proofof the above theorem. We refer the reader to the online supplemen-tary material for the implementation. AddDecompInField (cid:0) f , K ( ¯ t ) (cid:1) Input:
An S-primitive tower K ( ¯ t ) , described as a list { x , { t , . . . , t n } , { t ′ , . . . , t ′ n }} , s.t. t ′ i ∈ K i − is simple for all i ∈ [ n ] , and f ∈ K n . Output:
Two elements д , r ∈ K n such that f = д ′ + r and r satisfies the conditions in Proposition 4.7.(1) If f =
0, then return ( , ) .(2) Initialize: M ← hm ( f ) , a ← hc ( f ) , m ← ind n ( M ) , d ← deg t m ( M ) , B ← H ←
0, ˜ c ← a = Í mi = a i be the matryoshka decomposition.(4) Reduction: For all i from 0 to m , compute b i , h i ∈ K i s.t. a i = b ′ i + h i , where h i is t i -simple. Decide whether ∃ c , . . . , c m ∈ C s.t. h i = Í mj = c j t ′ j .Yes: B ← B + b i + Í m − j = c j t j and ˜ c ← ˜ c + c m ;No: B ← B + b i and H ← H + h i .(5) Lower term: ℓ ← f − aM − BM ′ − ˜ cd + · t d + m · (cid:0) M / t dm (cid:1) ′ Recursion: { ˜ д , ˜ r } ← AddDecompInField (cid:0) ℓ, K ( ¯ t ) (cid:1) (6) Return д = BM + ˜ cd + · t m · M + ˜ д and r = H · M + ˜ r . Example 4.9.
Find an additive decomposition for f = ( x ) Li ( x ) + Li ( x ) − x log ( x )( log ( x )) + log ( log ( x )) . Then f belongs to the S-primitive tower K = C ( x )( log ( x ) |{z} t , Li ( x ) |{z} t , log ( log ( x )) | {z } t ) , https://wongey.github.io/add-decomp-sprimitive/ SSAC ’20, June 20–23, 2020, Kalamata, Greece Du, Guo, Li, Wong and we can write f = /( t t ) + ( t − xt )/ t + t ∈ K . By theabove algorithm, we have that f = (cid:18) xt + t − t − xt + x t (cid:19) ′ + t t |{z} r . (11) The nonzero remainder r implies that f has no integral in K . An element f ∈ K is said to have an elementary integral over K if there exists an elementary extension E of K and an element д of E such that f = д ′ (see [5, Definition 5.1.4] ). We can use the re-mainder from Theorem 4.8 to determine whether or not a functionhas an elementary integral. Theorem 4.10.
Let K n be S-primitive and C be algebraically closed.Let f ∈ K n have a remainder r as described in Proposition 4.7. Then f has an elementary integral over K n if and only if r ∈ span C { t ′ , . . . , t ′ n } + span C { д ′ / д | д ∈ K n } . (12) Proof.
The sufficiency is obvious. Conversely, there exists an h ∈ span C { д ′ / д | д ∈ K n } such that f ≡ h mod K ′ n by Liou-ville’s Theorem [5, Theorem 5.5.2] . Since r is a remainder of f ,we have that h ≡ r mod K ′ n . By Proposition 4.7 and Lemma 2.5,we know that π n ( r ) and π n ( h ) are t n -simple, which, together withLemma 2.3 (ii) and (iii), implies that π n ( r ) = π n ( h ) . Since hm ( h ) =
1, we have that hm ( r ) (cid:22) ( r ) =
0, then r =
0. Otherwise, hm ( r ) =
1. By Proposition 4.7, r is simple. Since h is simple, r − h ∈ K ′ n is also simple. By Lemma 4.4, r − h ∈ span C { t ′ , . . . , t ′ n } , which implies (12). (cid:3) Example 4.11.
Let us reconsider the function f and the tower K in Example 4.9 under the assumption that C is algebraically closed.The remainder is r = t ′ / t . By Theorem 4.10, f has an elementaryintegral over K . It follows from (11) that ∫ f dx = x log ( log ( x )) + Li ( x ) − Li ( x ) − x Li ( x ) + x log ( x ) + log ( Li ( x )) . The Mathematica implementation by Raab based on work in [14]computes the same result. But the “int( )” command in Maple andthe “Integrate[ ]” command in Mathematica both leave the integralunevaluated.
A repeated use of Lemma 2.5 (ii) easily reveals a logarithmic towerto be S-primitive. Hence,
AddDecompInField can be applied toall logarithmic towers. In this section, we show that a logarithmictower can be differentially embedded into a logarithmic tower thatwe will term “well-generated” (see Definition 5.5) with the aid ofthe logarithmic derivative identity and the matryoshka decompo-sition. An element in the latter tower may have a “finer” remainder.The logarithmic derivative identity is actually a differential versionof logarithmic product and quotient rules, while the matryoshkadecomposition guides us how to apply the rules appropriately.
Example 5.1.
Consider the following function in x : f = log (( x + ) log ( x )) x log ( x ) . For this function, there are two possible ways to construct the towerover Q ( x ) containing f : (i) t = log ( x ) , t = log (( x + ) t ) ; f = t xt , (ii) u = log ( x ) , u = log ( x + ) , u = log ( u ) ; f = u + u xu .In the first tower, f is already a remainder by Proposition 4.7. In thesecond tower, AddDecompInField computes a remainder u /( xu ) that is lower than f . This is because we can decompose log (( x + ) log ( x )) as a sum of log ( x + ) and log ( log ( x )) in the second tower,but neither of the two summands is contained in the first. We can use the matryoshka decomposition to describe a prim-itive tower in terms of a matrix, which will be used to rearrangeour generators in an order that would yield a finer remainder byapplying
AddDecompInField . Definition 5.2.
Let K ( ¯ t ) be primitive. The n × n matrix A = (cid:16) π i ( t ′ j ) (cid:17) ≤ i ≤ n − , ≤ j ≤ n is called the matrix associated to K ( ¯ t ) . t ′ t ′ · · · t ′ n ↓ ↓ ↓ © « ª®®®®®®¬ P → ⋆ ⋆ · · · ⋆ P → ⋆ · · · ⋆... . . . ... P n − → ⋆ Figure 3: A labeled associated matrix of a primitive tower.The ⋆ represents a possibly nonzero element. The associated matrix records all information about the deriva-tion on K ( ¯ t ) , because π n ( t ′ ) = · · · = π n ( t ′ n ) = . Since t ′ j ∈ K j − for all j ∈ [ n ] , the associated matrix A is in upper triangular formas in Figure 3. Furthermore, if K ( ¯ t ) is a logarithmic tower, thenthe entries of A are all logarithmic derivatives by Lemma 2.5 (ii).For the following discussion, we will invoke the superscript no-tation to distinguish between different sets of generators (for ex-ample, π ¯ ti for projections in K ( ¯ t ) ). Definition 5.3.
Let K ( ¯ t ) be primitive and f ∈ K n \ { } . The significant index of f is si ¯ t ( f ) : = max { i ∈ [ n ] | π i ( f ) , } . The vector sv ( ¯ t ) : = (cid:16) si ¯ t ( t ′ ) , . . . , si ¯ t ( t ′ n ) (cid:17) is called the significant vector of K ( ¯ t ) . Suppose sv ( ¯ t ) is equal to ( k , . . . , k n ) . The sequence sc ( ¯ t ) : = (cid:16) π ¯ tk ( t ′ ) , . . . , π ¯ tk n ( t ′ n ) (cid:17) is called the the significant component sequence of K ( ¯ t ) . n Additive Decomposition in S-Primitive Towers ISSAC ’20, June 20–23, 2020, Kalamata, Greece The significant vector and significant component sequence areunique with respect to the generators by the matryoshka decom-position.
Example 5.4.
Consider the field C ( x ) ( log ( x ) , log ( log ( x )) , log (( x + ) log ( x ))) . We set t = log ( x ) , t = log ( t ) , and t = log (( x + ) t ) . Then C ( x )( t , t , t ) is a logarithmic tower whose significant vector is equalto ( , , ) and whose significant component sequence is ( / x , /( xt ) , /( xt )) . Definition 5.5.
A logarithmic tower K ( ¯ t ) is said to be well-generated if (CLI) sc ( ¯ t ) is C -linearly independent, (MI) sv ( ¯ t ) is (weakly) monotonically increasing, and (ONE) each column of its associated matrix contains exactly one non-zero element. © « • · · · • • · · · • . . . • · · · • ª®®®®®®¬ Figure 4: The associated matrix of a well-generated toweris in the form of a “staircase” where the • ’s are C -linearlyindependent and other entries are zero. We will show that a logarithmic tower K ( ¯ t ) can be embeddedinto a well-generated one. To this end, we impose the usual lexico-graphical order on two significant vectors [10, Chapter 2, Defini-tion 3] . Theorem 5.6.
Let K ( ¯ t ) be a logarithmic tower. Then there existsa well-generated logarithmic tower K ( ¯ u ) , where ¯ u = ( u , . . . , u w ) and n ≤ w ≤ n ( n + )/ , and a differential homomorphism ϕ from K ( ¯ t ) into K ( ¯ u ) with ϕ | K = id K . Proof.
This proof will be separated into two parts. The firstpart will show that each primitive (specifically, logarithmic) toweris isomorphic to one where properties (CLI) and (MI) are satisfied.This will enable us to embed the resulting logarithmic tower into awell-generated one, which makes up the second part of the proof.If K ( ¯ t ) does not satisfy (CLI) and (MI), then we can show thereexists v , . . . , v n ∈ K n such that K ( ¯ v ) is primitive, K ( ¯ v ) = K ( ¯ t ) ,and sv ( ¯ v ) is lower than sv ( ¯ t ) . Since the order of the significant vec-tors is Noetherian, we can eventually reach a primitive tower thatsatsifies both (CLI) and (MI).We start by supposing that sc ( ¯ t ) is C -linearly dependent. Sincesi ¯ t ( t ′ ) =
0, there exists an i ∈ { , . . . , n } and constants c , . . . , c i − such that sc i = Í i − j = c j · sc j , where sc j is the j -th element in sc ( ¯ t ) .We remove the last non-zero projection of t ′ i by setting v k : = t k for all k ∈ [ n ] \ { i } and v i : = t i − Í i − j = c j t j . Thus, K ( ¯ v ) = K ( ¯ t ) .Also, si ¯ v ( v ′ k ) = si ¯ t ( t ′ k ) for all k in [ n ]\ { i } and si ¯ v ( v ′ i ) < si ¯ t ( t ′ i ) . Weconclude that K ( ¯ v ) is a primitive tower with a lower significantvector than K ( ¯ t ) .Next, we assume that sv ( ¯ t ) is not monotonically increasing. Thenthere exist an i ∈ [ n ] such that si ¯ t ( t ′ ) ≤ · · · ≤ si ¯ t ( t ′ i ) and si ¯ t ( t ′ i + ) < si ¯ t ( t ′ i ) . We switch the i -th and ( i + ) -st generators by setting v k : = t k for all k ∈ [ n ] \ { i , i + } and v i : = t i + ; v i + : = t i . Thus, K ( ¯ v ) = K ( ¯ t ) . Also, si ¯ v ( v ′ j ) = si ¯ t ( t ′ j ) for j ∈ [ i − ] andsi ¯ v ( v ′ i ) < si ¯ t ( t ′ i ) . Thus, K ( ¯ v ) is a primitive tower with a lowersignificant vector than K ( ¯ t ) .If the original primitive tower from the argument is logarithmic,then the new generators from the above process are also logarith-mic generators. This implies the new tower must be logarithmicsatisfying (CLI) and (MI), and this is what we assume about K ( ¯ t ) from this point forward.For the second part of the proof, we show that K ( ¯ t ) can be em-bedded into a well-generated tower. We find the C -basis of the as-sociated matrix (cid:16) π i ( t ′ j ) (cid:17) by letting b = π ( t ′ ) and identifying all C -linearly independent elements b , . . . , b w , ordered by searchingthe matrix from left to right and top to bottom. Since K ( ¯ t ) is prim-itive, n ≤ w ≤ n ( n + )/
2. Since K ( ¯ t ) satisfies (CLI) and (MI), thereexist ℓ , . . . , ℓ n ∈ [ w ] such that ℓ = , ℓ n = w , ℓ < ℓ < · · · < ℓ n and (cid:0) b ℓ , . . . , b ℓ n (cid:1) = sc ( ¯ t ) . (13)By the definition of the associated matrix and the ordering of { b , . . . , b w } , for all j ∈ [ n ] there exist c j , k ∈ C such that t ′ j = b ℓ j + ℓ j − Õ k = c j , k · b k . (14)Let u , . . . , u w be algebraically independent indeterminates over K , and ¯ u : = ( u , . . . , u w ) . Let v j : = u ℓ j + Í ℓ j − k = c j , k · u k for all j ∈ [ n ] . Then v , . . . , v n are algebraically independent over K ,because u ℓ j does not appear in the expressions defining v , . . . , v j − . It follows that ϕ : K ( ¯ t ) → K ( ¯ u ) defined by f ( t , . . . , t n ) 7→ f ( v , . . . , v n ) is a monomorphism and ϕ | K = id K . For every k ∈[ w ] , we define u ′ k = ϕ ( b k ) . (15)Since u , . . . , u w are algebraically independent over K , the tower K ( ¯ u ) is a differential field by Corollary 1 ′ in [17, page 124] . By (14), ϕ ( t ′ j ) = v ′ j for all j ∈ [ n ] . Thus, ϕ is a differential monomorphism.Lastly, we show that K ( ¯ u ) is a well-generated tower over K .Set ℓ =
0. For each k ∈ [ w ] , there exists a j ∈ [ n ] such that ℓ j − < k ≤ ℓ j . Then s : = si ¯ t ( b k ) ≤ si ¯ t ( t ′ j ) < j and b k is t s -proper. Since ϕ is a monomorphism, it preserves degrees. By (15), u ′ k is u ℓ s -proper,where ℓ s ≤ ℓ j − < k since s < j . Hence, u ′ k ∈ K ( u , . . . , u k − ) .Since ϕ is differential and b k is a logarithmic derivative, u ′ k is alsoa logarithmic derivative by (15). In particular, u ′ k is u ℓ s -simple byLemma 2.5 (i). Moreover, b , . . . , b w are C -linearly independent,and so are ϕ ( b ) , . . . , ϕ ( b w ) because ϕ is a monomorphism. It fol-lows from (15) that u ′ , . . . , u ′ w are C -linearly independent, whichimplies that K ( ¯ u ) is a logarithmic tower by Corollary 4.5. In addi-tion, π i ( u ′ k ) = k ∈ [ w ] and i ∈ [ w ] \ { ℓ s } , because u ′ k is u ℓ s -proper. Consequently, K ( ¯ u ) is well-generated. (cid:3) SSAC ’20, June 20–23, 2020, Kalamata, Greece Du, Guo, Li, Wong
The proof of this theorem shows that a logarithmic tower F can be algorithmically embedded in a well-generated tower E bya differential homomorphism ϕ . Let f be an element of F with aremainder r . Our additive decomposition can be applied to ϕ ( f ) in E to get a remainder whose order is not higher than that of ϕ ( r ) ,and this is what we mean by “finer”.The next example illustrates the results of the embedding algo-rithm and AddDecompInField in both towers.
Example 5.7.
Consider the logarithmic tower F = C ( x ) (cid:16) log ( x ) |{z} t , log ( xt ) | {z } t , log (cid:0) ( x + )( t + ) log ( xt ) (cid:1)| {z } t (cid:17) . By Theorem 5.6, there exists a well-generated tower E = C ( x ) (cid:16) log ( x ) |{z} u , log ( x + ) | {z } u , log ( u ) | {z } u , log ( u + ) | {z } u , log ( u + u ) | {z } u (cid:17) and a differential homomorphism ϕ from F to E given by ϕ ( t ) = u , ϕ ( t ) = u + u and ϕ ( t ) = u + u + u . The associated matrices of F and E are, respectively, © « x x x + t ′ t t ′ t + + t xt t ª®®®®¬ and © « x x + u ′ u u ′ u +
00 0 0 0 00 0 0 0 ( u + u ) ′ u + u ª®®®®®®¬ . Let f = ( t + ) + t t xt ( t + ) t and f = t x be two elements of F . Then ϕ ( f ) and ϕ ( f ) are ( u + ) + u ( u + u ) xu ( u + )( u + u ) and u + u + u x , respectively. Using AddDecompInField , we compute the respectiveremainders of f and f to obtain r = f and r = t −( x + ) + x ( t + ) + −( t + ) xt . In the same vein, we get the remainders of ϕ ( f ) and ϕ ( f ) , ˜ r = and ˜ r = u −( x + ) + −( u + ) x ( u + u ) , respectively. Note that ϕ ( r ) , but ˜ r = , which implies that ˜ r ≺ ϕ ( r ) . While ˜ r and ϕ ( r ) have the same order, we observe that ˜ r has fewer nonzero projections than ϕ ( r ) . In this article, we have introduced the matryoshka decompositionto develop an additive decomposition in an S-primitive tower. Thedecomposition algorithm is based on Hermite reduction and inte-gration by parts. It provides an alternative method for determin-ing in-field (resp. elementary) integrability in (resp. over) an S-primitive tower without solving any differential equations. More-over, we embed a logarithmic tower into a well-generated one. Theembedding enables us to compute finer remainders.We observe that the notion of remainders is defined accordingto a partial order among multivariate rational functions. It would be possible to refine this notion so that remainders possess certainuniqueness. Moreover, we plan to investigate whether our additivedecomposition is applicable to compute telescopers for elements inan S-primitive tower, as carried out in [6]. We also hope to developan additive decomposition in exponential extensions.
ACKNOWLEDGMENTS
We are grateful to Shaoshi Chen, Christoph Koutschan and ClemensRaab for their valuable comments and suggestions. H. Du and E.Wong were supported by the Austrian Science Fund (FWF): F5011-N15. J. Guo and Z. Li were supported by two NFSC Grants 11688101and 11771433.
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A APPENDIX
For the convenience of the reviewers, this section lists definitions,a lemma, some theorems and a corollary that we use from otherbooks and papers but did not explicitly state in this paper. It willnot appear in a formal publication.