Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions
aa r X i v : . [ c s . S C ] M a y Generalized Hermite Reduction, Creative Telescopingand Definite Integration of D-Finite Functions
Alin Bostan
Inria, [email protected]
Frédéric Chyzak
Inria, [email protected]
Pierre Lairez
Inria, [email protected]
Bruno Salvy ∗ Inria, [email protected]
ABSTRACT
Hermite reduction is a classical algorithmic tool in symbolic inte-gration. It is used to decompose a given rational function as a sumof a function with simple poles and the derivative of another ra-tional function. We extend Hermite reduction to arbitrary lineardifferential operators instead of the pure derivative, and developefficient algorithms for this reduction. We then apply the gener-alized Hermite reduction to the computation of linear operatorssatisfied by single definite integrals of D-finite functions of sev-eral continuous or discrete parameters. The resulting algorithm isa generalization of reduction-based methods for creative telescop-ing.
ACM Reference Format:
Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy. 2018. Gen-eralized Hermite Reduction, Creative Telescoping and Definite Integrationof D-Finite Functions. In
ISSAC ’18: 2018 ACM International Symposium onSymbolic and Algebraic Computation, July 16–19, 2018, New York, NY, USA.
ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3208976.3208992
Ostrogradsky [37] and Hermite [27] showed how to decomposethe indefinite integral ∫ R of a rational function R ∈ Q ( x ) as U + ∫ A , where U , A ∈ Q ( x ) , and where A has only simple poles andvanishes at infinity. Their contributions consist in rational algo-rithms to compute A and U , that is algorithms which do not requireto manipulate the roots in Q of the denominator of R , but merelyits (squarefree) factorization. The rational function A is classicallycalled the Hermite reduction of R . In other words, the Hermite re-duction of R is a canonical form of R modulo the derivatives in Q ( x ) :it depends Q -linearly on R , it is equal to R modulo the derivativesand it vanishes if and only if U ′ = R for some U ∈ Q ( x ) .We call generalized Hermite problem the analogous question forinhomogeneous linear differential equations of arbitrary order c r ( x ) y ( r ) ( x ) + · · · + c ( x ) y ( x ) = R ( x ) , (1) ∗ Full affiliation: Univ Lyon, Inria, CNRS, ENS de Lyon, Université Claude BernardLyon 1, LIP UMR 5668, F-69007 Lyon, France. Most references in symbolic integration attribute to Ostrogradsky an algorithm tocompute U and A based on linear algebra. As a matter of fact, Ostrogradsky intro-duced before Hermite a polynomial method, based on extended gcds. In passing, he in-vented an efficient algorithm for squarefree factorization, rediscovered by Yun [47,48].ACM acknowledges that this contribution was authored or co-authored by an em-ployee, contractor or affiliate of a national government. As such, the Governmentretains a nonexclusive, royalty-free right to publish or reproduce this article, or toallow others to do so, for Government purposes only. ISSAC ’18, July 16–19, 2018, New York, NY, USA © 2018 Association for Computing Machinery.ACM ISBN 978-1-4503-5550-6/18/07...$15.00https://doi.org/10.1145/3208976.3208992 where R and the c i are rational functions in K ( x ) , over some field K of characteristic zero. In operator notation, given L = c r ∂ rx + · · · + c ∈ K ( x )h ∂ x i , the problem is to produce a rational function [ R ] in K ( x ) , that depends K -linearly on R , that is equal to R modulothe image L ( K ( x )) and that vanishes if and only if R is in L ( K ( x )) .Equations like Eq. (1) occur in relation to integrating factors , andultimately to creative telescoping . If L ∗ denotes the adjoint of L , de-fined as L ∗ = Í ri = (− ∂ x ) i c i ( x ) , then for any function f , integrationby parts produces Lagrange’s identity [29, §5.3] uL ( f ) − L ∗ ( u ) f = ∂ x ( P L ( f , u )) , (2)where P L depends linearly on f , . . . , f ( r − ) , u , . . . , u ( r − ) . It fol-lows that if f is a solution of L , then any R ∈ L ∗ ( K ( x )) is an inte-grating factor of f , meaning that R f is a derivative of a K ( x ) -linearcombination of f and its derivatives. The converse holds if L is anoperator of minimal order canceling f , see Proposition 4.2. Contributions
We introduce a generalized Hermite reduction to compute such a [ R ] .Classical Hermite reduction addresses the case L = ∂ x . The algo-rithm operates locally at each singularity and it avoids algebraicextensions, similarly to classical Hermite reduction.Next, we improve Chyzak’s algorithm [19] for creative telescop-ing with the use of generalized Hermite reduction. Recall that cre-ative telescoping is an algorithmic way to compute integrals byrepeated differentiation under the integral sign and integration byparts [7]. Chyzak’s algorithm repeatedly checks for the existenceof a rational solution to equations like (1). A lot of time is spentchecking that none exists. The use of generalized Hermite reduc-tion makes the computation incremental and less redundant.As a simple instance of the creative telescoping problem, let f ( t , x ) be a function annihilated by a linear differential operator L ∈ Q ( t , x )h ∂ x i in the differentiation with respect to x only, and such that ∂ t ( f ) = A ( f ) for another operator A also in Q ( t , x )h ∂ x i . We look for theminimal relation of the form λ f + · · · + λ s ∂ st ( f ) = ∂ x ( G ) , (3)with λ , . . . , λ s ∈ Q ( t ) and G ( t , x ) in the function space spannedby f and its derivatives, with the motive that integrating both sideswith respect to x may lead to something useful: on the right-handside, the integral of the derivative simplifies, often to 0, and on theleft-hand side, the integration commutes with the λ i ∂ it , yielding adifferential equation for ∫ f ( t , x ) d x . In Equation (3), the left-handside is called the telescoper and the function G ( t , x ) the certificate .The new algorithm constructs a sequence of rational functions R , R , . . . in Q ( t , x ) such that ∂ it ( f ) = R i f + ∂ x ( . . . ) . Equation (3)holds if and only if λ R + · · · + λ s R s is an integrating factor of f , SSAC ’18, July 16–19, 2018, New York, NY, USA Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy which in turn is equivalent to the relation λ [ R ] + · · · + λ s [ R s ] = , (4)where [ ] is the generalized Hermite reduction with respect to L ∗ .Starting with s =
0, we search for solutions of the equation aboveand increment s until one is found. Chyzak’s algorithm would solveEquation (3) at each iteration mostly from scratch, whereas thenew algorithm retains the reduced forms [ R i ] from one iteration tothe next, computes [ R s ] from [ R s − ] and solves the straightforwardEquation (4). This approach to creative telescoping generalizes toseveral parameters t , . . . , t e in the integrand and to different kindsof operators acting on them, in the setting of Ore algebras.The order of the telescopers and even the termination of the cre-ative telescoping process are related to the confinement propertiesof the generalized Hermite reduction. Assuming that the poles ofthe rational functions R , R , . . . all lie in the same finite set, wededuce from a result of Adolphson’s an upper bound on the dimen-sion of the subspace spanned by the reductions [ R i ] , which in turnbounds the order of the minimal telescoper. Previous work
Extensions of Hermite reduction.
Ostrogradsky [37] and Hermite [27]introduced a reduction for rational functions. A century later, itwas extended to larger and larger classes of functions: algebraic [43],hypergeometric [3], hyperexponential [22,26,10,11], Fuchsian [16].Van der Hoeven’s preprint [44] considers a reduction w.r.t. thederivation operator on differential modules of finite type, so as toaddress the general differentially finite case. Our generalized Her-mite reduction is inspired by these works. It has the same archi-tecture as several previous ones [10, 16, 11, 44]: local reductions atfinite places, followed by a reduction at infinity and the compu-tation of an exceptional set to obtain a canonical form. Our firstcontribution in the present paper is to open a new direction ofgeneralization, namely by considering reductions with respect toother operators in K ( x )h ∂ x i than the derivation operator ∂ x , act-ing on the space K ( x ) of rational functions. An extra benefit of ourmethod is to avoid algebraic extensions of K . Index theorems.
The finite-dimensionality of a function spacemodulo the image of a differential operator is crucial to the termi-nation of our reduction and creative-telescoping algorithms. Thisfiniteness, and even explicit bounds, are given by index theorems for differential equations [35]. Rational versions appeared in workby Monsky [36] related to the finiteness of de Rham cohomology,and by Adolphson [6] in a p -adic context, see also [41,45], and §3.6. Creative telescoping by reduction.
The use of Hermite-like reduc-tions for computing definite integrals roots in works by Fuchs [25]and Picard [38, 39]. In the realm of creative telescoping, this lineof research forms what is called the fourth generation of creativetelescoping algorithms. It was first introduced for bivariate ratio-nal functions [9], and later extended to the multivariate rationalcase [12, 32]. For bivariate functions/sequences, the approach wasalso extended to larger classes: algebraic [15, 14], hyperexponen-tial [10], hypergeometric [13,28], mixed [11], Fuchsian [16], differ-entially finite [44]. Our second contribution is the first reduction-based variant, for single integrals, of Chyzak’s algorithm [19] for D-finite functions depending on several continuous or discrete pa-rameters.
The equation M ( y ) = ax + bx + c , with M defined by M ( y ) = ( x − ) y ′′ + ( x − p ( x − )) y ′ + ( p ( x − ) − px − n ) y , has a rational solution y ∈ Q ( n , p , x ) if and only if ax + bx + c isa multiple of p x − px − n − p . This follows in two steps.First, a local analysis reveals that if y has a pole at some α ∈ C ,then so does M ( y ) : for any α ∈ C \ {± } and for any s > M (cid:0) ( x − α ) − s (cid:1) = ( α − ) s ( s + )( x − α ) − s − ( + O ( x − α )) and M (cid:0) ( x ± ) − s (cid:1) = ± s ( s + )( x ± ) − s − ( + O ( x ± )) . Therefore, if M ( y ) is a polynomial then y is also a polynomial.Next, for any s ≥ M ( x s ) = p x s + + O ( x s + ) , as x → ∞ . It fol-lows that if y ∈ Q ( n , p )[ x ] then M ( y ) ∈ Q ( n , p )[ x ] and deg x M ( y ) = deg x y +
2. In particular, every polynomial of degree ≤ M ( Q ( n , p , x )) is a multiple of M ( ) = p x − px − n − p over Q ( n , p ) .In §3, we define the Hermite reductions w.r.t. M of 1, x and x : [ ] = , [ x ] = x , and [ x ] = xp + n + p p , showing that [ p x − px − n − p ] =
0. Similarly, the reduction ofany polynomial w.r.t. M is a Q ( n , p ) -linear combination of 1 and x . We consider the classical integral identity [40, §2.18.1, Eq. (10)] ∫ − e − px T n ( x )√ − x d x = (− ) n πI n ( p ) , where T n denotes the n th Chebyshev polynomial of the first kindand I n the n th modified Bessel function of the first kind. The in-tegrand F n ( p , x ) satisfies a system of linear differential and differ-ence equations, easily found from defining equations for T n ( x ) and e − px : ∂ F n ∂ p = − x F n , nF n + = ∂∂ x (cid:16) ( x − ) F n (cid:17) + ( px + ( n − ) x − p ) F n , ( − x ) ∂ F n ∂ x = ( px + x − p ) ∂ F n ∂ x + ( p x + px − n − p + ) F n . We aim at finding a similar set of linear differential-differenceoperators in the variables n and p for the integral ∫ − F n ( p , x ) d x .Note that F n and all its derivatives w.r.t. x and p and shifts w.r.t. n are Q ( n , p , x ) -linear combinations of F n and ∂ F n / ∂ x .The adjoint of the last equation is M ( y ) =
0, with the operator M of §2.1. The reduction w.r.t. M described above makes the followingcomputation possible. First, F n is not a derivative (of a Q ( n , p , x ) -linear combination of F n and ∂ F n / ∂ x ). Indeed, F n is a derivativeif and only if 1 ∈ M ( Q ( n , p )) . Second, no Q ( n , p ) -linear relationbetween F n and ∂ F n / ∂ p is a derivative, because ∂ F n / ∂ p = − xF n and [ ] and [− x ] are linearly independent over Q ( n , p ) . Third, the Q ( n , p ) -linear relation p [ x ] + p [− x ] − ( n + p )[ ] = p ∂ F n ∂ p + p ∂ F n ∂ p − ( n + p ) F n = ∂ G ∂ x (5) eneralized Hermite Reduction, Creative Telescoping and Definite Integration ISSAC ’18, July 16–19, 2018, New York, NY, USA for some Q ( n , p , x ) -linear combination G of F n and ∂ F n / ∂ x . Next,the equation for nF n + and the equation [ px + ( n − ) x − p ] = nx + n / p show that, for some ˜ G as above, F n + + ∂ F n ∂ p − np F n = ∂ ˜ G ∂ x . (6)Equations (5) and (6) can then be integrated from − G and ˜ G ) and the left-hand sides provide the desiredoperators for the integral. These equations classically define, up toa constant factor, the function (− ) n I n ( p ) . Throughout this section, M ∈ K [ x ]h ∂ x i denotes a linear differ-ential operator with polynomial coefficients. We are interested infinding K -linear dependency relations in K ( x ) modulo the rationalimage M ( K ( x )) by means of a canonical form with respect to M . Definition 3.1. A canonical form with respect to M is a K -linearmap [ ] : K ( x ) → K ( x ) such that for any R ∈ K ( x ) :(i) [ M ( R )] =
0; (ii) R − [ R ] ∈ M ( K ( x )) .Applying [ ] to R − [ R ] before using (ii) and (i) results in [[ R ]] = [ R ] .As can be seen from Eq. (1), computing such canonical forms istightly related to the computation of rational solutions of linear dif-ferential equations. In classical solving algorithms [2, 33], boundson the order of poles of meromorphic solutions are given by indi-cial equations. Next, in order to factor the computation for differentinhomogeneous parts, instead of using a “universal denominator”,one could at each singularity identify the polar behaviour of po-tential meromorphic solutions, so as to reduce rational solving topolynomial solving. This idea is what inspired the reduction algo-rithm for computing canonical forms in the present section .We begin in §3.1 with a local analysis of M ( K ( x )) . Then we de-scribe in §3.2 a projection map H : K ( x ) → K ( x ) that we call weak Hermite reduction . It is not quite a canonical form. It missesan exceptional set described in §3.3, from which a canonical form isdeduced. For simplicity, this is first described in the algebraic clo-sure of the base field K , and in §3.4 we show how to perform thecomputations in a rational way, i.e., without algebraic extensions.Finally, in §3.6, we bound the dimension of the quotient E / M ( E ) ,for a ring E of rational functions with prescribed poles. This is rel-evant to getting size and complexity bounds for creative telescop-ing. Let K be an algebraic closure of K . For R ∈ K ( x ) and α ∈ K ,let R ( α ) denote the polar part of R at α . This is the unique poly-nomial in ( x − α ) − with constant term zero such that R − R ( α ) hasno pole at α . Similarly, the polynomial part R (∞) of R is the uniquepolynomial such that R − R (∞) vanishes at infinity. By partial frac-tion decomposition, R = R (∞) + Õ α ∈ K R ( α ) . (7) In the case of systems, analogues of indicial equations are more complicated; severalalternatives for rational solving exist [1,8], that resemble the reduction in [44].
Let also ord α R denote the valuation of R as a Laurent series in x − α .For any α ∈ K , there exists a non-zero polynomial ind α ∈ K [ s ] and an integer σ α such that for any s ∈ Z , M (cid:0) ( x − α ) − s (cid:1) = ind α (− s )( x − α ) − s + σ α ( + o ( )) , as x → α . (8)The polynomial ind α is classically called the indicial polynomialof M at α [46, 29]; we call the integer σ α the shift of M at α . Theindicial polynomial and its integer roots give a detailed understand-ing of the image of M . We similarly define the shift and the indicialpolynomial at ∞ by the equation M ( x s ) = ind ∞ (− s ) x s − σ ∞ ( + o ( )) , as x → ∞ .If M = Í ri = p i ( x ) ∂ ix , then σ α = min ≤ i ≤ r ( ord α p i − i ) and σ ∞ = max ≤ i ≤ r ( i − deg p i ) . For any α ∈ K that is not a root of the leading coefficient p r of M , we have ind α ( s ) = p r ( α ) · s ( s − ) · · · ( s − r + ) and σ α = − r . Let im M = M ( K ( x )) . Let H α : K ( x ) → K ( x ) be the local reductionmap at α defined by H α ( R ) = R if ord α R ≥ α is not a pole of R )and by induction on ord α R , H α ( R ) = ( H α (cid:16) R − cM (( x − α ) − s − σα ) ind α (− σ α − s ) (cid:17) if ind α (− σ α − s ) , c ( x − α ) − s + H α ( R − c ( x − α ) − s ) otherwise,where R = c ( x − α ) − s ( + o ( )) as x → α , with c ∈ K \ { } and s >
0. The induction is well-founded because in either caseof the definition, the argument of H α in the right-hand side has avaluation at α that is larger than ord α R . By construction, we checkthat R − H α ( R ) ∈ im M for any R ∈ K ( x ) .Similarly, let H ∞ : K ( x ) → K ( x ) be the local reduction map at ∞ defined by H ∞ ( R ) = R if ord ∞ R > R (∞) =
0) and byinduction on ord ∞ ( R ) by H ∞ ( R ) = H ∞ (cid:16) R − cM ( x s + σ ∞ ) ind ∞ (− s − σ ∞ ) (cid:17) if ind ∞ (− s − σ ∞ ) , s + σ ∞ ≥ cx s + H ∞ ( R − cx s ) otherwise,where R = cx s ( + o ( )) as x → ∞ . By construction, we checkthat R − H ∞ ( R ) ∈ im M for any R ∈ K ( x ) . The condition s + σ ∞ ≥ M ( x s + σ ∞ ) is a polynomial. Definition 3.2.
The weak Hermite reduction is the linear map H ,seen either as H : K ( x ) → K ( x ) or as H : K ( x ) → K ( x ) , anddefined by H ( R ) = H ∞ (cid:18) R (∞) + Õ α ∈{ poles of R } H α (cid:0) R ( α ) (cid:1)(cid:19) . Proposition 3.3.
For any R ∈ K ( x ) :(i) H ( R ) = H ∞ ◦ H α ◦ · · · ◦ H α n ( R ) , where α , . . . , α n ∈ K arethe poles of R ;(ii) R − H ( R ) ∈ im M and H ( M ( R )) ∈ im M ;(iii) H ( H ( R )) = H ( R ) .Moreover:(iv) for any α ∈ K and for any s > , ind α ( s ) , and σ α − s > ⇒ H (cid:0) M (( x − α ) − s ) (cid:1) = (v) for any s ≥ , ind ∞ ( s ) , ⇒ H (cid:0) M ( x s ) (cid:1) = . SSAC ’18, July 16–19, 2018, New York, NY, USA Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy
Proof.
By linearity and Equation (7), Property (i) follows from theformulas H ( R (∞) ) = H ∞ ( R (∞) ) and H ( R ( α ) ) = H ∞ ( H α ( R ( α ) )) de-rived from the definition of H . The first part of Property (ii) followsfrom corresponding properties for H α and H ∞ ; the second part isa consequence of applying the first to M ( R ) .As for the idempotence, we observe, first, that every H ( R ) is alinear combination of some ( x − α ) − s , with ind α (− s − σ α ) = x s , with s + σ ∞ ≥ ∞ (− s − σ ∞ ) =
0; and second,that H is the identity on such monomials.As for (iv), the condition ind α ( s ) , α M (( x − α ) − s ) = − s − σ α , and then by definition of H α , H α (cid:0) M (( x − α ) − s ) (cid:1) = H α (cid:0) M (( x − α ) − s ) − M (( x − α ) − s ) (cid:1) = . The last property is proved similarly. (cid:3) If H were a canonical form, H ( M ( R )) would be 0 for any R ∈ K ( x ) .But this property fails, and more work is required to refine H intoa canonical form. Definition 3.4.
The space Exc M of exceptional functions is the K -linear subspace of K ( x ) defined by Exc M = H ( im M ) . Lemma 3.5.
For any R ∈ K ( x ) , R ∈ im M if and only if H ( R ) ∈ Exc M . Proof.
The direct implication is the definition of Exc M . For theconverse, assume H ( R ) = H ( M ( U )) for some U . As ( R − M ( U )) − H ( R − M ( U )) = M ( V ) for some V by Prop. 3.3 (ii), R = M ( U + V ) . (cid:3) The generalized Hermite reduction is not a canonical form, butit is strong enough to ensure that Exc M is finite-dimensional over K . Proposition 3.6.
Over K , the vector space Exc M is generated by thefinite family(a) H ( M (( x − α ) − s )) with α ∈ Sing ( M ) , s > and ind α (− s ) = ,(b) H ( M (( x − α ) − s )) with α ∈ Sing ( M ) , < s ≤ σ α ,(c) H ( M ( x s )) with s ≥ and ind ∞ (− s ) = ,where Sing ( M ) ⊂ K is the set of singularities of M (the zeroes of itsleading coefficient). Proof.
The elements ( x − α ) − s ( α ∈ K , s >
0) and x s ( s ≥
0) form abasis of K ( x ) . In particular, Exc M , by definition, is generated by the H (( x − α ) − s ) and H ( x s ) . By Proposition 3.3 (iv) and (v), H ( M (( x − α ) − s )) = α (− s ) , s < σ α . Similarly, H ( M ( x s )) = ∞ (− s ) ,
0. Moreover, any α ∈ K such that ind α hasa negative root or σ α > M . Therefore, theonly nonzero generators of Exc M belong to the set given in thestatement. (cid:3) Example 3.7.
Let M = x ∂ x . We compute ind α ( s ) = − α s forany α ∈ K and ind ∞ ( s ) = s . Moreover σ α = − α < { , ∞} , σ = σ ∞ = −
9. It follows thatExc M = Vect (cid:8) H ( M ( x − )) , . . . , H ( M ( x − )) (cid:9) = Vect (cid:8) , x , . . . , x (cid:9) . Lemma 3.8.
Given a finite-dimensional K -linear subspace W ⊂ K ( x ) ,there is a unique idempotent linear map ρ W : K ( x ) → K ( x ) suchthat: (i) W = ker ρ W ; (ii) for any R ∈ K ( x ) , the degree of the numer-ator of ρ W ( R ) is minimal among all S ∈ K ( x ) with R − S ∈ W . The following proof gives an algorithm for computing ρ W . Algorithm 1
Rational weak Hermite reduction.
Input R ∈ K ( x ) ; M a linear differential operator. Output
The rational weak Hermite reduction of R . function WHermiteRed( R , M ) if R = then return else if R is a polynomial then write R as cx s + (lower degree terms) if ind ∞ (− s − σ ∞ ) , and s + σ ∞ ≥ thenreturn WHermiteRed (cid:16) R − cM ( x s + σ ∞ ) ind ∞ (− s − σ ∞ ) , M (cid:17) else return cx s + WHermiteRed ( R − cx s , M ) else P ← an irreducible factor of the denominator of R .write R as AP s Q , with A , Q ∈ K [ x ] and s maximal. if ind P (− s − σ P ) = then U ← A / Q mod P . return U / P s + WHermiteRed ( R − U / P s , M ) else R ← A / Q / ind P (− s − σ P ) mod P return WHermiteRed (cid:0) R − M ( R / P s + σ P ) , M (cid:1) Proof.
When W ⊂ K [ x ] , the value ρ W ( R ) is the result of Gaussianelimination applied in the monomial basis to the polynomial partof R with the elements of W .In the general case, we write W = Q − V , for some subspace V ⊂ K [ x ] and Q ∈ K [ x ] , and define ρ W ( R ) = Q − ρ V ( QR ) . The twoproperties are easily checked. (cid:3) Definition 3.9.
The generalized Hermite reduction with respect to M is the map [ ] : K ( x ) → K ( x ) defined by [ R ] = ρ Exc M ( H ( R )) . Theorem 3.10.
The map [ ] is a canonical form with respect to M . Proof.
We check the properties of Definition 3.1. Let R ∈ K ( x ) .First, [ M ( R )] = H ( M ( R )) ∈ Exc M (Lemma 3.5) and then ρ Exc M ( H ( M ( R ))) =
0, by Proposition 3.3 (ii) and the constructionof ρ Exc M . Second, R − [ R ] ∈ im M because R − H ( R ) ∈ im M (Propo-sition 3.3) and H ( R ) − ρ Exc M ( H ( R )) ∈ Exc M ⊂ im M . (cid:3) In most cases, computing Hermite reduction as it is defined abovewould require to work with algebraic extensions of the base field.If P ∈ K [ x ] is a monic irreducible polynomial and α a root of P ,the reduction can be performed simultaneously at all roots of P without introducing algebraic extensions.The indicial equation is obtained by considering the leading co-efficient of the P -adic expansion of M ( P − s ) , see [46, §4.1, p. 107].More precisely, there is a unique polynomial ind P ( s ) with coeffi-cients in K [ x ]/( P ) and a unique integer σ P such that for any s > M (cid:0) P − s (cid:1) = ind P (− s ) P − s + σ P + O (cid:0) P − s + σ P + (cid:1) , as P -adic expansions. Since P is irreducible, ind P ( s ) , for a given s ,is either 0 or invertible modulo P . For an irreducible polynomial P ∈ K [ x ] , and for R = U P − s + O ( P − s + ) , we define H P ( R ) = ( U P − s + H P ( R − U P − s ) if ind P (− σ P − s ) = H P (cid:0) R − M ( U ind α (− σ P − s ) − P − s − σ P ) (cid:1) otherwise, eneralized Hermite Reduction, Creative Telescoping and Definite Integration ISSAC ’18, July 16–19, 2018, New York, NY, USA where ind α (− σ P − s ) − is computed mod P . This is the part ofour reduction which most closely resembles the original Hermitereduction, with successive coefficients obtained by modular inver-sions. Definition 3.11.
The rational weak Hermite reduction is the linearmap H rat : K ( x ) → K ( x ) , defined by H rat ( R ) = H ∞ (cid:0) R (∞) + Õ P H P (cid:0) R ( P ) (cid:1)(cid:1) , where the summation runs over the irreducible factors of the de-nominator of R and R ( P ) ∈ K [ x , P − ] denotes the polar part of the P -adic expansion of R .The maps H and H rat satisfy the same properties, mutatis mu-tandis . In particular, the latter can be used to compute a canoni-cal form in the same way as H . Yet, both reductions are not equal(see also §3.5.1). For example, over Q with M = ( x + ) ∂ x + x , R = ( x + ) − and i + = H ( R ) = i (cid:0) ( x + i ) − − ( x − i ) − (cid:1) whereas H rat ( R ) = R . Partial fraction decomposition and actual Hermite reduction canbe performed together. This is described in Algorithm 1. Togetherwith the algorithm for the map ρ Exc M , described in the proof ofLemma 3.8, we obtain an algorithm, denoted CanonicalForm , tocompute the map ρ ◦ H rat that is a canonical form modulo M . A notion of Hermite reduc-tion that is independent from the base field is obtained by replac-ing U · P − s with d s − d x s − UP in the definition of H P . Another benefit ofthis choice is that it is not necessary that P is irreducible to performthe reduction, but simply that ind P ( s ) is either 0 or invertible. Thedenominators that appear in the computation can be factored onthe fly into factors with the required property: when some ind P ( s ) is neither 0 nor invertible modulo P , a gcd computation gives anon-trivial divisor of P . The hypothesis that thedifferential operator M has polynomial coefficients is importantfor the correctness of Algorithm 1. To compute canonical formsmodulo an operator M with rational coefficients, it is sufficient tofind a polynomial Q such that MQ has polynomial coefficients andthen, to compute canonical forms modulo MQ with the algorithmsabove. Indeed, the image of K ( x ) by MQ and M are the same. Thesmallest such Q is the gcd of the denominators of the coefficientsof the adjoint of M . The following observation can be usedto speed up the computation.
Lemma 3.12.
Let L , M ∈ K [ x ]h ∂ x i and A , B in K ( x ) such that MA = BL . If [ ] L is a canonical form w.r.t. L , then [ ] M : R ∈ K ( x ) 7→ B [ R / B ] L is a canonical form w.r.t. M . Proof.
We check the properties of Def. 3.1: [ M ( y )] M = B [ L ( A − y )] L is 0 and R − [ R ] M = B ( R / B − [ R / B ] L ) is in B ( im L ) = im M . (cid:3) Lemma 3.12 may be used with A = B = Î α ( x − α ) m α , where m α is the smallest negative integer root of the indicial polynomial of M at α , and 0 if none exists. This is mostly useful for equations oforder 1, since the corresponding α is not a singularity of the new operator, which becomes smaller. The rational function A plays therole of the shell in previous reduction-based algorithms [10, 11]. Let P ∈ K [ x ] be a squarefree polynomial and let E P = K (cid:2) x , P − (cid:3) .Let ker M ⊂ K ( x ) be the space of rational solutions of M . Let r bethe order of M and d the maximal degree of its coefficients. Proposition 3.13 (Adolphson [6, Sec. 5, Prop. 1]). dim K E P / M ( E P ) = dim K ( E P ∩ ker M ) − σ ∞ − Õ P ( α ) = σ α ≤ ( deg P + ) · r + d . Sketch of the proof.
Let Z = { α ∈ K | P ( α ) = } . Given deg P + s ∞ and s α ( α ∈ Z ), let E P ( s ) denote the subspaceof all R ∈ E P such that the pole order at α is at most s α for α ∈ Z ∪ {∞} , that is all elements R ∈ E P ( s ) of the form R = Í α ∈ Z Í s α s = c α , s ( x − α ) s + Í s ∞ s = c ∞ , s x s . We choose s α and s ∞ large enough so that ker M ⊂ E P ( s ) . Let t α = s α − σ α ( α ∈ Z ∪ {∞} ). We check M ( E P ( s )) ⊆ E P ( t ) and that a basisof E P ( t )/ M ( E P ( s )) induces a basis of E P / M ( E P ) . The bounds − σ α ≤ r , − σ ∞ ≤ d and dim ker M ≤ r give the inequality. (cid:3) The method of creative telescoping is an approach to the compu-tation of definite sums and integrals of objects characterized bylinear functional equations. The notion of linear functional equa-tion is formalized by
Ore algebras . In this part, we consider the Orealgebra A = K ( x )h ∂ x , ∂ , . . . , ∂ e i , where ∂ x is the differentiationwith respect to x and ∂ , . . . , ∂ e are arbitrary Ore operators. In themost typical case, K = Q ( t , . . . , t e ) and each ∂ i is either the dif-ferentiation with respect to t i or the shift t i t i + f in a function space on which A acts, theannihilating ideal of f is the left ideal ann f ⊆ A of all operatorsthat annihilate f . For example, the annihilating ideal in K ( x )h ∂ x i of f = sin ( x ) is generated by ∂ x + ′′ ( x ) = − sin ( x ) .A left ideal I is D-finite if the quotient A /I is a finite-dimensionalvector space over K ( x ) . A function is called D-finite if its annihilat-ing ideal is D-finite. We refer to [18, 20, 21] for an introduction toOre algebras, creative telescoping and their applications.Given a D-finite function f , the problem of creative telescoping is the computation of a generating set of the telescoping ideal of f w.r.t. x , or of its residue class in A / ann f . This is by definition theleft ideal T f ⊂ K h ∂ , . . . , ∂ e i of all operators T such that T + ∂ x G ∈ ann f for some G ∈ A ; equivalently, T f = ( ann f + ∂ x A ) ∩ K h ∂ , . . . , ∂ e i . Example 4.1.
In §2.2, we use the Ore algebra K ( x )h ∂ x , ∂ , ∂ i , with ∂ = d / dp and ∂ = S n the shift w.r.t. n . The annihilating ideal I of F n ( p , x ) is generated by three operators, one for each functionalequation. It is D-finite and the quotient A /I has dimension 2, withbasis 1 and ∂ x . The telescoping ideal of F n ( p , x ) (or, equivalently,of 1 ∈ A /I ) is generated by p ∂ p + p ∂ p −( n + p ) and pS n + p ∂ p − n . SSAC ’18, July 16–19, 2018, New York, NY, USA Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy
Let
I ⊆ A be a D-finite ideal and let r be the dimension of A /I over K ( x ) . We denote L ( γ ) the multiplication of an operator L ∈ A and a residue class γ ∈ A /I .Let γ ∈ A /I be a cyclic vector with respect to ∂ x . This meansthat Γ = (cid:8) γ , ∂ x ( γ ) , . . . , ∂ r − x ( γ ) (cid:9) is a basis of A /I ; or, equivalently,that every f ∈ A /I can be written A f ( γ ) for some A f ∈ K ( x )h ∂ x i .Let L ∈ K [ x ]h ∂ x i be a minimal annihilating operator of γ , thatis L ( γ ) = L has order r (because Γ is a basis, there is nonon-zero lower order annihilating operator for γ ). A cyclic vectoralways exists when I is D -finite [17,5]. It plays a role analogous tothat of primitive elements for 0-dimensional polynomial systems.For 1 ≤ i ≤ e , we define a K -linear map λ i : K ( x ) → K ( x ) as follows. First, we can write ∂ i ( γ ) = B i ( γ ) for some operator B i ∈ K ( x )h ∂ x i . Next, let σ i and δ i be the maps such that ∂ i R = σ i ( R ) ∂ i + δ i ( R ) for any R ∈ K ( x ) . Finally, we define for R ∈ K ( x ) λ i ( R ) = B ∗ i ( σ i ( R )) + δ i ( R ) , where B ∗ i ( σ i ( R )) ∈ K ( x ) is the result of applying the adjoint oper-ator B ∗ i to σ i ( R ) , not the operator B ∗ i σ i ( R ) . Proposition 4.2.
With the notation above:(i) f = A ∗ f ( ) γ + ∂ x ( Q ) , for some Q ∈ A /I .Moreover, for any R ∈ K ( x ) :(ii) ∂ i ( Rγ ) = λ i ( R ) γ + ∂ x ( Q ) , for some Q ∈ A /I .(iii) Rγ ∈ ∂ x ( A /I) if and only if R ∈ L ∗ ( K ( x )) . Proof.
Using that f = A f ( γ ) , Lagrange’s identity (2) shows that1 A f ( γ ) − A ∗ f ( ) γ = ∂ x ( Q ) for some Q . This gives (i) . Similarly, us-ing the commutation rule for ∂ i and the definition of B i yields (ii) .Property (iii) is shown by Abramov and van Hoeij [4, Prop. 3]. (cid:3) Example 4.3 (Continuing Example 4.1).
The element 1 ∈ A /I is acyclic vector since { , ∂ x } is a basis of the quotient.Actual computations are performed using a Gröbner basis of I and linear algebra in the finite-dimensional K ( x ) -vector space A /I . We now present our algorithm (Algorithm 2) based on generalizedHermite reduction for the computation of the telescoping ideal T f for an element f of some D-finite quotient A /I . The element f isoften 1, as in Example 4.1.In the same way as Chyzak’s algorithm [19], ours iterates overmonomials in ∂ , . . . , ∂ e by a strategy reminiscent of the FGLM al-gorithm [24]. Each iteration finds either a new generator of K h ∂ , . . . , ∂ e i/T f or a new element in T f . Let [ ] be the generalized Hermite reductionwith respect to L ∗ , the adjoint of the minimal annihilating opera-tor of the cyclic vector γ . Since every visited monomial µ (but thefirst) can be written ∂ i ν for a previously visited monomial ν , wedefine F inductively by the formula F µ = [ λ i ( F ν )] and the basecase F = A ∗ f ( f ) . With Prop. 4.2 and Theorem 3.10, we check that µ ( f ) = F µ γ + ∂ x ( Q µ ) , for some Q µ ∈ A /I and that a µ + · · · + a s µ s ∈ T f ⇔ a F µ + · · · + a s F µ s = . (9) If ∂ i is the differentiation w.r.t. t i , then σ i ( R ) = R and δ i ( R ) = ∂ R / ∂ t i .If ∂ i is the shift t i t i + , then σ i ( R ) = R | ti ← ti + and δ i ( R ) = . Algorithm 2
Reduction-based creative telescoping algorithm
Input I a D-finite ideal of A and f ∈ A /I Output
Generators of the telescoping ideal T f function CreativeTelescoping( I , f ) γ ← a cyclic vector of A /I with respect to ∂ x L ← the minimal operator annihilating γλ , . . . , λ e ← maps as in Prop. 4.2 F ← CanonicalForm ( A ∗ f ( ) , L ∗ )L ← [ ] ⊲ list of monomials in ∂ , . . . , ∂ e G ← {} ⊲ Gröbner basis being computed Q ← {} ⊲ Generators of the quotient R ← {} ⊲ Set of reducible monomials while L , ∅ do Remove the first element µ of L if µ is a not multiple of an element of R thenif µ , then Pick i such that µ / ∂ i ∈ QF µ ← CanonicalForm ( λ i ( F µ / ∂ i ) , L ∗ ) if ∃ a K -linear rel. between F µ and { F ν | ν ∈ Q } then ( a ν ) ν ∈ Q ← coeff. of the relation F µ = Í ν ∈ Q a ν F ν Add µ − Í ν ∈ Q a ν ν to G ; Add µ to R else Add µ to Q for ≤ i ≤ e do Append the monomial ∂ i µ to L return G Theorem 4.4.
On input I , Algorithm 2 terminates if and only if thetelescoping ideal T f is D-finite. It outputs a Gröbner basis of T f forthe grevlex monomial ordering. Proof.
By construction, when a monomial is added to the set R , itis not a multiple of another monomial in R . By Dickson’s lemma [23],this may happen only finitely many times.The way L is filled ensures that when a monomial µ is visited,every smaller monomial has been visited or is a multiple of a re-ducible monomial. This implies, by induction, that Q is the set ofall non-reducible monomials that are smaller than µ , when µ is vis-ited.If T f is not D-finite, then there are infinitely many non-reduciblemonomials and the algorithm does not terminate. Otherwise, thealgorithm terminates, since neither Q nor R may grow indefinitely.To check that G is a Gröbner basis, we note that: G ⊂ T f , by theequivalence (9); the leading monomials of the elements of G arethe elements of R ; and every leading monomial of an element T f (that is a reducible monomial) is a multiple of an element of R . (cid:3) As stated, Algorithm 2 computes re-lations by increasing total degree in ( ∂ , . . . , ∂ e ) . To choose a differ-ent term order, it is sufficient to change the selection of the mono-mial µ at the beginning of the loop and select the smallest one forthe given order instead. Instead of waiting for the list L to be empty, one can stop as soon as a relation is found, and thenit is the minimal one for the chosen term order. This variant does eneralized Hermite Reduction, Creative Telescoping and Definite Integration ISSAC ’18, July 16–19, 2018, New York, NY, USA not require a D-finite ideal to terminate. Another possibility is tostop as soon as the degree of µ is larger than a predefined bound,returning all the relations that exist below this bound. While an important point of the reduction-based approaches to creative telescoping is to avoid the compu-tation of certificates (in contrast with Chyzak’s and Koutschan’salgorithms that require their computation), it is also possible tomodify the algorithm so that it returns a certificate for each ele-ment of the basis. Indeed, a certificate of the generalized Hermitereduction of §3 can be propagated through the algorithms.
In the general case, the telescoping ideal T f of a D-finite func-tion f need not be D-finite. However, when the auxiliary opera-tors ∂ , . . . , ∂ e are differentiation operators (as opposed to shiftoperators for example), then T f is always D-finite if f is; this is awell-known result in the theory of D-finiteness and holonomy [42].We give here a new proof of this fact, as a corollary of a more gen-eral sufficient condition for general Ore operators. Definition 4.5.
A D-finite function f is singular (w.r.t. ∂ x ) at α ∈ K if every nonzero operator L ∈ K ( x )h ∂ x i such that L ( f ) = α . The singular set (w.r.t. ∂ x ) of f , denoted Sing ( f ) , isthe set of all singular points of f .Let Θ = { , ∂ , ∂ , . . . , ∂ , ∂ ∂ , . . . } be the set of all monomialsin the variables ∂ , . . . , ∂ e . Theorem 4.6.
For any D-finite function f , if Ð µ ∈ Θ Sing ( µ ( f )) isfinite, then T f is D-finite. Proof.
Let γ be a cyclic vector of A / ann f w.r.t. ∂ x with mini-mal annihilating operator L ∈ K ( x )h ∂ x i of order r . For µ ∈ Θ ,let A µ ( f ) ∈ K ( x )h ∂ x i of order < r be such that µ ( f ) = A µ ( f ) ( γ ) , asin §4.1, and let R µ = A ∗ µ ( f ) ( ) ∈ K ( x ) , so that µ ( f ) = R µ γ + ∂ x ( G µ ) ,for some G µ ∈ A / ann f . By Proposition 4.2, the K -linear map ϕ : K h ∂ , . . . , ∂ n i → K ( x ) defined by ϕ ( µ ) = R µ induces aninjective map K h ∂ , . . . , ∂ n i/T f → K ( x )/ im L ∗ . The telescopingideal T f is D-finite if and only if the image of this map is finite-dimensional. In view of Proposition 3.13, it suffices to show thatthe poles of all the R µ lie in a finite subset of K . This is obtainedby proving that Ø µ ∈ Θ poles ( R µ ) ⊆ Sing ( L ) ∪ Ø µ ∈ Θ Sing ( µ ( f )) , (10)where Sing ( L ) is the set of zeroes of the leading coefficient of L .Indeed, let µ ∈ Θ and α ∈ K that is not in the right-hand side. Wenow prove that no coefficient of A µ ( f ) has a pole at α , from whereit follows that neither has R µ = A ∗ µ ( f ) ( ) . By the hypothesis on α ,there exists M ∈ K ( x )h ∂ x i an annihilating operator of µ ( f ) regu-lar at α . It satisfies MA µ ( f ) ( γ ) = M ( µ ( f )) = L it follows that MA µ ( f ) = BL for some operator B . As a conse-quence, 0 , , . . . , r − MA µ ( f ) .Write A µ ( f ) = Í r − i = a i ∂ ix , for some a i ∈ K ( x ) and let j be the max-imal index with ord α a j = min i ord α a i . Then j ∈ { , . . . , r − } and ord α A µ ( f ) (cid:0) ( x − α ) j (cid:1) = min i ord α a i . Then this last quantityis a zero of the indicial polynomial of M , which implies that it isnonnegative and thus that none of the a i has a pole at α . (cid:3) Integral (11) (12) (13) (14) (15) (16) (17) redct
13 s > 1h > 1h 1.5 s 1.5 s 165 s 53 s
HF-CT
19 s 253 s 45 s 232 s 516 s >1h >1h
HF-FCT s* 2.3 s 5.3 s >1h 2.3 s* 5.4 s 2.2 s*
Table 1
Comparative timings on several instances of creative telescoping.Rows are redct (new algorithm); Koutschan’s
HolonomicFunctions , usingfunctions
Annihilator and
CreativeTelescoping (HF-CT); idem, using
FindCreativeTelescoping (HF-FCT), a heuristic that does not necessarilyfind the minimal operators (indicated by *).
All examples were run on thesame machine, with the latest versions of Maple and Mathematica.
Recall that for ≤ i ≤ e , the Ore operator ∂ i satisfies a com-mutation relation ∂ i a = σ i ( a ) ∂ i + δ i ( a ) for any a ∈ K , where σ i isan endomorphism of K and δ i is a σ i -derivation of it. When ∂ i isa differentiation operator, σ i = id K . Corollary 4.7. If ∂ , . . . , ∂ e are differentiation operators, then T f is D-finite for any D-finite function f . Proof.
It is sufficient to check that
Sing ( µ ( f )) ⊂ Sing ( f ) for anymonomial µ ∈ Θ and then conclude by Theorem 4.6.Let M ∈ K [ x ]h ∂ x i be an annihilating operator of д = µ ( f ) regu-lar at α ∈ K \ Sing ( f ) . The commutation rules for the differential op-erators imply that ∂ i M = M ∂ i + R , for some R ∈ K [ x ]h ∂ x i . In partic-ular, we obtain the inhomogeneous differential equation M ( ∂ i ( д )) = R ( д ) for ∂ i ( д ) . Since α is neither a singularity of M nor of R ( д ) , itfollows that it is not a singularity of ∂ i ( д ) . (cid:3) For the case of general Ore operators, we obtain with a similarproof the following result.
Corollary 4.8.
For any D-finite function f , if there is a finite set S ⊂ K such that: (i) σ i ( S ) ⊆ S for any ≤ i ≤ e and (ii) Sing ( f ) ⊆ S ,then T f is D-finite. We present the results of a preliminary Maple implementation called redct . Comparison is done with Koutschan’s HolonomicFunctions package [31], the best available code for creative telescoping. Tim-ings are given in Table 1 . Koutschan’s examples.
Koutschan’s example session [30] contains 40integrals on which we tested our code. In most cases, our code com-pares well with
HolonomicFunctions . There are 37 easy cases, allof whose telescopers are found in 3.5 sec. by redct , while 16 sec.are needed by
HolonomicFunctions (but that includes certificates).The three other examples are (the nature of the parameters is indi-cated in the brackets, C ( α ) n denotes Gegenbauer polynomials, and Available with example sessions at https://specfun.inria.fr/chyzak/redct/. When our code does not terminate, time is spent computing the exceptional set.This seems to be due to apparent singularities of the operators, that become truesingularities of their adjoint. Ways of circumventing this issue are under study.
SSAC ’18, July 16–19, 2018, New York, NY, USA Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy J , I , etc. Bessel functions): ∫ J m + n ( tx ) T m − n ( x )√ − x d x [diff. t , shift n and m ] , (11) ∫ C ( λ ) n ( x ) C ( λ ) m ( x ) C ( λ ) ℓ ( x )( − x ) λ − d x [shift n , m , ℓ ] , (12) ∫ ∞ x J ( ax ) I ( ax ) Y ( x ) K ( x ) d x [diff. a ] . (13) Longer examples.
We mention a few examples, some involvingGegenbauer polynomials [40, 2.21.18.2, 2.21.18.4], that take moretime. The advantage of a reduction-based approach becomes visi-ble. ∫ n + x + n + (cid:16) ( x + ) ( x − )( x − ) ( x − ) (cid:17) n √ x − e x + x ( x − )( x − ) d x [shift n ] , (14) ∫ C ( µ ) m ( x ) C ( ν ) n ( x )( − x ) ν − / d x [shift n , m , µ , ν ] , (15) ∫ x ℓ C ( µ ) m ( x ) C ( ν ) n ( x )( − x ) ν − / d x [shift ℓ , m , n , µ , ν ] , (16) ∫ ( x + a ) γ + λ − ( a − x ) β − C ( γ ) m ( x / a ) C ( λ ) n ( x / a ) d x , [diff. a , shift n , m , β , γ , λ ] . (17) A closer look at our algorithm reveals several aspects of the com-plexity of creative telescoping. To simplify the discussion, we re-strict to the bivariate case and measure the arithmetic complex-ity , obtained by counting arithmetic operations in Q . We look forbounds in terms of the input size (order and degree of the operatorsat hand).In this setting, the complexity of computing T f is not boundedpolynomially (whatever the algorithm). Consider for instance, theintegral representation of Hermite polynomials H n ( t ) = n i √ π ∫ i ∞− i ∞ ( t + x ) n e x d x . If one computes a telescoper over Q ( n , t ) , then our algorithm pro-duces the classical differential equation y ′′ + ny = ty ′ . However,if n is a given positive integer then the minimal telescoper is thefirst-order factor H n ( t ) ∂ t − H ′ n ( t ) , with coefficients of degree n . Itssize is exponential in the bit size of the input. Thus, no algorithmcomputing the minimal telescoper can run in polynomial complex-ity .However, in the frequent cases like this one where the set S ofsingularities discussed in Corollary 4.8 is bounded polynomially interms of the size of the input, then the dimension of the quotientand therefore the order of the telescopers is bounded polynomially asa consequence of Adolphson’s result (Proposition 3.13). The non-polynomial cost of minimality thus resides only in the degree ofthe coefficients. Note that in the differential case, polynomial timecomputation of non-minimal telescopers is also achieved by well-known methods in holonomy theory, e.g., [34, proof of Lemma 3].In our algorithm, the non-polynomial complexity arises first inthe computation of the exceptional set Exc M and next in the reduc-tions by the elements of this set. Removing this part of the compu-tation and using the weak Hermite reduction yields a weak form of the algorithm that does not find minimal telescopers but runs inpolynomial complexity, if the set S has polynomial size. Acknowledgement.
This work was supported in part by FastRe-lax ANR-14-CE25-0018-01.
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