CCOMPUTING PERIODS OF RATIONAL INTEGRALS
PIERRE LAIREZ
Abstract.
A period of a rational integral is the result of integrating, withrespect to one or several variables, a rational function over a closed path. Thiswork focuses particularly on periods depending on a parameter: in this case theperiod under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dworkreduction and apply it to the computation of Picard-Fuchs equations. Theresulting algorithm is elementary and has been successfully applied to problemsthat were previously out of reach.
Introduction
This work studies periods of rational integrals, that is, the result of the integration,with respect to one or several variables, of a rational function over a closed path.I focus especially on the case where the period depends on a parameter. The factthat periods depending on a parameter of rational or algebraic integrals satisfylinear differential equations with polynomial coefficients has emerged from the workof Euler [24, §7] and his computation of a differential equation for the perimeterof an ellipse as a function of eccentricity. Since then, these differential equations,known as Picard-Fuchs equations , have proven to be useful in numerous domainssuch as combinatorics [11], number theory [6] or physics [39]. They play also akey role in mirror symmetry [38]. Research in computer algebra has devoted greatefforts to provide algorithms for computing integrals and, in particular, Picard-Fuchsequations. Nevertheless the practical efficiency of current methods is not satisfactoryin many cases. One reason might be the high level of generality of most algorithms,which apply to the integration of general holonomic functions. Rational functions arecertainly very specific among holonomic functions, but the numerous applicationsof Picard-Fuchs equations as well as the fundamental nature of rational functionsmake it worth developing specific methods for them.
The problem.
Let R be a rational function in the variables x , . . . , x n , denoted x ,and a parameter t , with coefficients in C . Let γ be a n -cycle in C n , e.g. an embeddingof the sphere S n in C n , on which R is continuous when t ranges over some connectedopen set U of C . We can form the following integral, depending on t ∈ U ,(1) P ( t ) def = I γ R ( t, x )d x , where d x stands for d x · · · d x n . Date : January 31, 2015.2010
Mathematics Subject Classification.
Primary 68W30; secondary 14K20, 14F40, 33F10.
Key words and phrases.
Integration, periods, Picard-Fuchs equation, Griffiths-Dwork reduction,algorithms. ( t − t ) y + (1 − t ) y + ty = 0 a r X i v : . [ c s . S C ] A ug PIERRE LAIREZ
Example . For t ∈ C , with | t | < − √ ∞ X n =0 n X k =0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19) t n = 1(2 πi ) I γ d x d y d z − (1 − xy ) z − txzy (1 − x )(1 − y )(1 − z ) , where the cycle of integration γ is (cid:8) ( x, y, z ) ∈ C (cid:12)(cid:12) | x | = | y | = | z | = 1 / (cid:9) . This isthe generating function of Apéry numbers [6].These integrals, for different cycles γ , are called the periods of the integral H R .It is well-known that P ( t ) satisfies a linear differential equation with polynomialcoefficients. It is a consequence of the finiteness of the algebraic de Rham cohomologyof A n \ V ( f ) with C ( t ) as base field [28; 37]. Let L R,γ denote the differential operatorin t and ∂ t which corresponds to the minimal-order equation of P ( t ). That is tosay L R,γ is the non zero operator P rk =0 a k ( t ) ∂ kt with coprime polynomial coefficientsand minimal r , such that L R,γ ( P ) def = r X k =0 a k ( t ) P ( k ) ( t ) = 0 . Every linear differential equation for P ( t ) translates into an operator which is a leftmultiple of L R,γ .It often happens that the description of the cycle γ is analytic or topological,sometimes not even explicit, and, to say the least, unsuitable to a formal algorithmictreatment. In fact there is no harm in simply discarding γ : there exists a differentialequation satisfied by all the periods of H R . In other words, there exists an operatorin t and ∂ t which is a left multiple of all L R,γ . Let L R denote the least common leftmultiple of the L R,γ . The classical result which allows the algorithmic computationof L R is that it is the minimal operator L such that(2) L ( R ) = n X i =1 ∂ i ( B i )for some rational functions B i in C ( t, x ) whose denominators divide a power of thedenominator of R , and where ∂ i denotes ∂/∂x i . This article presents an algorithmthat compute the operator L R , or at least a left multiple of it. Example . In the case of Example 1, the operators L R and L R,γ both equal L R = t ( t − t + 1) ∂ t + 3 t (2 t − t + 1) ∂ t + (7 t − t + 1) ∂ t + ( t − . Note that integrals of algebraic functions are easily translated into integrals of ra-tional functions with one variable more: if W ( t, x ) is a function such that P ( t, x , W ) =0 for some polynomial P in C [ t, x , y ], elementary residue calculus shows that W ( t, x ) = 12 πi I τ y∂ y PP d y over some adequate contour τ and where ∂ y denotes the derivation ∂/∂y , so that I γ W ( t, x )d x = 12 πi I γ × τ y∂ y PP d x d y. OMPUTING PERIODS OF RATIONAL INTEGRALS 3
Contributions.
Following the principle of the reduction of the pole order, I definea family of finer and finer reductions [ ] r , for r (cid:62)
1, that given a rational function R in several variables produces another rational function [ R ] r that differs from R onlyby a sum of partial derivatives of other rational functions (Section 4). The firstreduction [ ] is the Griffiths-Dwork reduction (Section 3).When applied to the case of periods depending on a parameter, these reductionscan solve Equation (2), and hence compute Picard-Fuchs equations of rationalintegrals (Section 6). A major difficulty is to fix an r such that the r th reductionmap [ ] r will be fine enough to ensure the termination of the algorithm. It is solvedby applying a theorem of Dimca (Section 5).The new algorithm has been implemented and shows excellent performance (Sec-tion 7). For example, I applied it to compute 137 periods coming from mathematicalphysics that were previously out of reach [4] (Section 8). Reduction of pole order.
The principle of the method originates from Hermitereduction [29]. It is a procedure for computing a normal form of a univariatefunction modulo derivatives. Hermite introduced his method as a way to compute thealgebraic part of the primitive of a univariate rational function without computing theroots of its denominator, as opposed to the classical partial fraction decompositionmethod. Let [ R ] denote the reduction of a fraction R . It is defined as follows.Let a/f q be a rational function in C ( x ), with f a square-free polynomial and q apositive integer. Every fraction can be written in this way since a and f are notassumed to be relatively prime. If q >
1, then we first write a as uf + vf , usingthe assumption that f is square-free, and we observe that af q = u + q − v f q − − (cid:18) q − vf q − (cid:19) . This leads to the following recursive definition of [ a/f q ]: (cid:20) af q (cid:21) = " u + q − v f q − . When q = 1, the reduction [ a/f ] is defined to be r/f , where r is the remainder inthe Euclidean division of a by f . Hermite reduction enjoys the following properties:it is linear; the fractions [ R ] and R differ only by a derivative of a rational function;and [ R ] is zero if and only if R is the derivative of a rational function.The principle of Hermite reduction gives an efficient way to compute the Picard-Fuchs equation of univariate integrals [8]. Let R be a rational function in C ( t, x ).Hermite reduction can be performed without modification over the field with oneparameter C ( t ). To compute L R , it is sufficient to compute the reductions [ ∂ kt R ],for k (cid:62)
0, until finding a linear dependency relation over C ( t ) r X k =0 a k ( t )[ ∂ kt R ] = 0 . Then the properties of the Hermite reduction assure that L R is P rk =0 a k ( t ) ∂ kt . Thecomputations of all the reductions [ ∂ kt R ] is improved significantly when noting theinductive formula (cid:2) ∂ k +1 t R (cid:3) = (cid:2) ∂ t [ ∂ kt R ] (cid:3) .With several variables, the construction of a normal form modulo derivatives isconsiderably harder than with a single variable. Nonetheless, as soon as we obtain PIERRE LAIREZ such a normal form, it is possible to compute Picard-Fuchs equations as above, byfinding linear relations between the [ ∂ kt R ].Part 1 deals with the construction of the maps [ ] r whereas Part 2 deals with thecomputation of Picard-Fuchs equations. Related works.
Several existing algorithms are applicable to the computationof L R . The reader may refer to [14] for an extensive survey of “creative telescop-ing” approaches. A first family, originating in the work of Fasenmyer [25] andVerbaeten [45], gave rise to an algorithm by Wilf and Zeilberger [46], refined byApagodu and Zeilberger [3], applicable to proper hyperexponential terms, whichincludes rational functions. The idea is to transform Equation (2) into a linearsystem over C ( t ) by bounding a priori the order of a left multiple of L R and thedegree of the polynomials appearing in the fractions B i . While being an interestingmethod, especially because it gives a priori bounds, the order of the linear systemto be solved is large even for moderate sizes of the input.Zeilberger’s “fast algorithm” [47] for hypergeometric summation is the originof a different family of algorithms, whose key idea is to reduce the resolutionof Equation (2) to the computation of rational solutions of systems of ordinarylinear differential equations. Interestingly, Picard used this idea much earlier in amethod for computing double rational integrals [41]. Chyzak’s algorithm [13] andKoutschan’s semi-algorithm [33]—termination is not proven—belong to this line andapply to D -finite ideals in Ore algebras. Rational functions are a very specific case.A last family of algorithms coming from D -module theory has given algorithmsfor numerous operations on D -modules and, in particular, an algorithm by Oakuand Takayama [40] to compute the de Rham cohomology of the complement ofan affine hypersurface, which would allow, in theory, to compute Picard-Fuchsequations. It is worth noting that an algorithm to compute the integration of aholonomic D -module does not give as such an algorithm applicable to our problem:computing the annihilator of a rational function in the Weyl algebra is far frombeing an easy task [40].The domain of application of each of these three families is much larger than justrational integrals: any comparison with the present algorithm must be done withthis point in mind.The guessing method, or equation reconstruction , a totally different method,applies to the computation of L R,γ . It often happens that beside the integral formulafor P ( t ) one has a way to compute a power series expansion. After computingsufficiently many terms, it is possible to recover L R,γ via
Hermite-Padé approximants.It may be difficult to prove that the operator computed is indeed correct, but nottoo hard to get convinced. The simplicity of this method counterbalances a certainlack of delicacy and justifies its ample use. When the power series expansion of P ( t )is, for some reason, easy to compute, it can find Picard-Fuchs equations which arefar out of reach of any existing algorithms [ e.g. P ( t ) is expensive to compute. For example, I am aware ofno general method allowing to compute directly the first p terms of a diagonal ofa rational function in n variables in less than p n arithmetic operations. However,space complexity can be improved [36].Picard and Simart have studied the case of simple and double integrals ofalgebraic functions and gave methods to compute normal forms modulo derivativesextensively [42]. Chen, Kauers, and Singer [12] gave an algorithm in this direction, OMPUTING PERIODS OF RATIONAL INTEGRALS 5 for double rational integrals. This algorithm is an echo, independently discovered,of one of the methods of Picard [41]. Interestingly, it has two steps: a first one basedon a reduction à la
Hermite and another one based on creative telescoping.Well later after Picard, Griffiths resumed the search for a normal form in thesetting of de Rham cohomology of smooth projective hypersurfaces, defining what isnow known as the Griffiths-Dwork reduction [22, §3; 23, §8; 27, §4]. This reductionis in many respects similar to the Hermite reduction. It can be applied to thecomputation of Picard-Fuchs equations in the same way as Hermite reductionapplies to univariate integrals. The smoothness hypothesis can be worked aroundwith a generic deformation. This leads to an interesting complexity result about thecomputation of Picard-Fuchs equations [9] but to disappointing practical efficiencyin singular cases. The direction of Griffiths and Dwork was extended, in particular,by Dimca [19; 18] and Saito [21], and some results are known in the case of a singularhypersurface.
Acknowledgment.
I am grateful to Alin Bostan and Bruno Salvy for their precioushelp and support, to Mark van Hoeij and Jean-Marie Maillard for their expertisewith differential operators and to the referee for his thorough work.
Part Reduction of periods
Let K be a field of characteristic zero, and let A be the polynomial ring K [ x , . . . , x n ],for some integer n . Let f be an homogeneous element of A and let A f be the local-ized ring A [1 /f ]. The degree of f is denoted N . We focus here on integrals H R d x which are homogeneous of degree zero, this means that R ( λx , . . . , λx n )d( λx ) · · · d( λx n ) = R ( x , . . . , x n )d x · · · d x n , or equivalently that R is a homogeneous rational function of degree − n −
1. Everyintegral can be homogenized with a new variable, see §6.2.This part addresses the problem of finding an algorithm à la
Hermite thatcomputes an idempotent linear map R [ R ], from A f to itself such that [ R ]equals zero if and only if R is in the linear subspace P ni =0 ∂ i A f . This problem issolved by the Hermite reduction when n is 1 and by the Griffiths-Dwork reductionwhen f satisfies an additional regularity hypothesis (see Theorems 3 and 10).To this purpose, a family of maps, denoted [ ] r , is constructed such that [ ] isthe Griffiths-Dwork reduction and such that [ ] r +1 factors through [ ] r . I give anefficient algorithm to compute these maps. Conjecturally, [ ] n +1 satisfies the desiredproperties. Fortunately, other results allow to avoid relying on this conjecture whendealing with periods depending on a parameter.1. Overview
Griffiths-Dwork reduction.
To achieve a normal form modulo derivatives,the guiding principle is the reduction of pole order . Let us first consider the decisionproblem: given a rational function a/f q , decide whether it lies in P ni =0 ∂ i A f . Amajor actor of the study is Jac f , the Jacobian ideal of f . It is the ideal of A generated by the partial derivatives ∂ f, . . . , ∂ n f . The basic observation is that thedifferentiation formula(3) n X i =0 ∂ i (cid:18) b i f q − (cid:19) = P ni =0 ∂ i b i f q − − ( q − P ni =0 b i ∂ i ff q PIERRE LAIREZ implies, by reading it right-to-left, that if a ∈ Jac f and q > a/f q equals a /f q − modulo derivatives, for some polynomial a . Namely, if a = P i b i ∂ i f then af q ≡ q − P ni =0 ∂ i b i f q − mod n X i =0 ∂ i A f . Griffiths [27] proved the converse property in the case when Jac f is zero-dimensional or, equivalently, when the projective variety defined by f is smooth.Under this hypothesis, if q > a/f q ≡ a /f q − , modulo derivatives, for somepolynomial a , then a ∈ Jac f . This gives an algorithm to solve the decision problem,by induction on the pole order q .1.2. Singular cases.
In presence of singularities, Griffiths’ theorem always fails.For example, with f equal to xy − z ,(4) x f = ∂ x (cid:18) x f (cid:19) − ∂ y (cid:18) x yf (cid:19) , but x is not in Jac f , which is here the ideal ( xy, y , z ). This identity is aconsequence of the following particular case of Equation (3):(5) n X i =0 b i ∂ i f = 0 ⇒ n X i =0 ∂ i (cid:18) b i f q (cid:19) = P ni =0 ∂ i b i f q . Tuples of polynomials ( b , . . . , b n ) such that P ni =0 b i ∂ i f are called syzygies (of thesequence ∂ f, . . . , ∂ n f ). Therefore, in order to complete the reduction of pole orderstrategy, we should not only consider elements of the Jacobian ideal, but alsoelements of the form P i ∂ i b i , where ( b b , . . . , b n ) is a syzygy. Such elements arecalled differentials of syzygies .Considering differential of syzygies is not always enough. For example, with f equal to x x − x x x + x x : x f = 1062347276480 89 x + 96 x x + 712 x f + X i =0 ∂ i (cid:18) b i f (cid:19) , for some lengthy polynomials b i , whereas x is not a sum of a differential of a syzygyand of an element of Jac f . Note the exponent 3 appearing in ∂ i ( b i /f ), it is theleast possible.1.3. Higher order relations.
Let M q be the set of rational functions of theform a/f q . Let W q be the subset of M q × M q − defined by W q = (cid:26)(cid:18) ( q − P ni =0 b i ∂ i ff q , P ni =0 ∂ i b i f q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) b i ∈ A (cid:27) . An element (
R, R ) of W q relates a rational function R with a pole order at most q with another rational function R , with pole order at most q −
1, which is equivalentto R modulo derivatives. The following statement is a rewording of Griffiths’ result: Theorem 3 (Griffiths) . Assume that V ( f ) is smooth. For all R in M q , homogeneousof degree − n − , the following assertions are equivalent:(1) R is in P i ∂ i A f ;(2) there exists R in M q − ∩ P i ∂ i A f such that ( R, R ) is in W q . OMPUTING PERIODS OF RATIONAL INTEGRALS 7
The starting point of the method in the general case is to observe that W q contains ordered pairs in the form (0 , R ). Namely, if b , . . . , b n is a syzygy,then (0 , P i ∂ i b i /f q − ) is in W q . They seem to be useless relation in view of Theo-rem 3. However, for all such pairs (0 , R ), the rational function R is in P i ∂ i A f ,since it is equivalent to 0 modulo derivatives.But it is possible, as remarked above, that R is not part of a pair ( R , R )in W q − . This motivates the definition of W q as W q def = W q + (cid:8) ( R, (cid:12)(cid:12) (0 , R ) ∈ W q +1 (cid:9) . Of course, this can be iterated: W r +1 q def = W rq + (cid:8) ( R, (cid:12)(cid:12) (0 , R ) ∈ W rq +1 (cid:9) . The basic property that is preserved through this induction is that for all (
R, R )in W rq , the first element R has a pole of order at most q and is equivalent, moduloderivatives, to the second element R , which has a pole of order at most q − V ( f ) is smooth, then W rq = W q , for all q , but when V ( f ) is singular, thespaces W rq , with r >
1, bring new relations. This construction is somehow exhaustive.The first result is the following, with no assumption on V ( f ): Theorem 4.
There exists an integer r (cid:62) , depending only on f , such that forall q and all R in M q , homogeneous of degree − n − , the following assertions areequivalent:(1) R is in P i ∂ i A f ;(2) there exists R in M q − ∩ P i ∂ i A f such that ( R, R ) is in W rq . The algorithm presented in this article is based on this theorem. The definitionof the W rq gives readily an algorithm to compute these spaces: it is only a matterof linear algebra. The second result is a method to achieve efficiency. The twomain ingredients are the use of Gröbner bases, and the computation of a basis of non-trivial syzygies to catch most elements of W rq at reasonnable cost.1.4. Trivial syzygies.
The space W q is made from W q and elements in theform ( P i ∂ i b i /f q , b , . . . , b n is a syzygy, that is P i b i ∂ i f vanishes.Among syzygies, the trivial syzygies do not bring new relations to the relationsalready in W r . A syzygy b , . . . , b n is called trivial if there exist polynomials c i,j ,with c i,j = − c j,i , such that b i = n X j =0 c i,j ∂ j f. The antisymmetry property implies that this defines a syzygy, and we check that n X i =0 ∂ i b i = n X j =0 n X i =0 ∂ i c ij ! ∂ j f + n X i,j =0 c i,j ∂ i ∂ j f | {z } =0 , so that P ni =0 ∂ i b i is in the Jacobian ideal. Moreover X j ∂ j X i ∂ i c ij ! = 0 . PIERRE LAIREZ
It follows that the ordered pair ( P i ∂ i b i /f q ,
0) is already in W q . Thus, in order tocompute W q , one may discard trivial syzygies. Quantitatively, the trivial syzygiesare numerous among the syzygies—see, for example, Table 2—so that discardingthem is a tremendous improvement. A basis of non-trivial syzygies can be computedefficiently by means of Gröbner bases.1.5. Reduction procedure.
Let R = a/f q be a fraction in M q . The reducedform [ R ] r is defined by induction on q in the following way. We decompose R as R + S where R is minimal in some sense and where S is the first element of apair ( S, T ) in W rq . Then [ R ] r is defined to be R + [ T ] r . By construction [ R ] r ≡ R modulo derivatives. The constraint on the homogeneity degree of R will ensurethat T is zero at some point of the induction.2. Exponential isomorphism
The exponential isomorphism , Theorem 6, allows to manipulate polynomialsrather than rational functions. It is folklore, for an account see [18]. We work in ahomogeneous setting and we deal only with homogeneous fractions R of degree − n − R d x · · · d x n is homogeneous of degree 0.) A fraction a/f q is thereforerepresented solely by its numerator a : if a/f q is homogeneous of degree − n −
1, thenumerator a is homogeneous of degree q deg f − n −
1, so that q may be recoveredfrom a . To the usual partial derivative ∂ i on the rational side corresponds the twisted derivative on the polynomial side ∂ i a def = ∂ i a − ( ∂ i f ) a = e f ∂ i ( ae − f ) . The exponential isomorphism relates, on the one hand, homogeneous fractions a/f q ofdegree − n − f ) Z − n −
1, modulo twisted derivatives.2.1.
Differential forms.
This section is a short reminder about differential forms,or simply forms . Let Ω denote the polynomial differential 1-forms: it is thefree A -module of rank n + 1, and the basis is denoted by the symbols d x , . . . , d x n .The differential map d from A to Ω is defined byd a = n X i =0 ∂ i a d x i . The A -algebra of differential forms, denoted Ω, is the exterior algebra over Ω .Its multiplication is denoted ∧ , it is generated by the d x i and is subject to therelations d x i ∧ d x j = − d x j ∧ d x i . The A -module of p -forms, denoted Ω p , is thesubmodule of Ω generated by the d x i ∧ · · · ∧ d x i p . With the multi-index notation,this is denoted d x I , with I = ( i , . . . , i p ). Ω p is a free module of rank (cid:0) np (cid:1) . Themodule of 0-forms Ω is identified with A . As a module, Ω decomposes as ⊕ np =0 Ω p .Specifically, the module Ω n +1 has rank 1 and is freely generated by d x ∧ · · · ∧ d x n ,denoted ω . The module Ω n has rank n + 1 and is freely generated by the elements ξ i defined by ξ i def = ( − i d x ∧ · · · ∧ d x i − ∧ d x i +1 ∧ · · · ∧ d x n . See, for example [35, chap. 10] and [10, §10], for more general and complete definitions.
OMPUTING PERIODS OF RATIONAL INTEGRALS 9
Exterior derivative.
The differential map d, from A to Ω , extends to anendomorphism of Ω, called exterior derivative , such that for α ∈ Ω p and β ∈ Ω,d( α ∧ β ) = d α ∧ β + ( − p α ∧ d β. In particular d(Ω p ) is included in Ω p +1 and d = 0. For a n -form β , writtenas P i b i ξ i , we check that d β equals ( P i ∂ i b i ) ω . The exterior derivative gives riseto a complex 0 −→ A d −→ Ω −→ · · · d −→ Ω n d −→ Ω n +1 −→ Homogeneity.
The degree of a monomial x I d x J is defined to be | I | + | J | . Aform is called homogeneous of degree k if it is a linear combination of monomials ofdegree k . If α and β are two homogeneous forms of degree k α and k β respectively,then d α is a homogeneous form of degree k α and α ∧ β is a homogeneous form ofdegree k α + k β .2.1.3. Koszul complex.
The exterior product with d f gives a map from Ω p to Ω p +1 ,and since d f ∧ d f vanishes we can consider the chain complex K (d f ) : 0 −→ A d f −→ Ω f −→ · · · d f −→ Ω n d f −→ Ω n +1 −→ , known as the Koszul complex of A with respect to d f , and its cohomology H K (d f )defined by H p K (d f ) = Ω p ∩ ker d f d f ∧ Ω p − . For a n -form β , written as P i b i ξ i , the exterior product d f ∧ β is ( P i b i ∂ i f ) ω .Thus H n +1 K (d f ) is isomorphic to A/ Jac f , with a shift of n + 1 in the naturalgrading, where Jac f is the Jacobian ideal ( ∂ f, . . . , ∂ n f ).Let Syz be the kernel of the product by d f on Ω n . It is the syzygy module ofthe sequence ∂ f, . . . , ∂ n f . Let Syz be d f ∧ Ω n − , the module of trivial syzygies,generated by the elements ∂ i f ξ j − ∂ j f ξ i . In particular H n K (d f ) is Syz / Syz .2.2. Chain complex T p . For an integer q , let T pq be the subspace of Ω p generatedby the homogeneous elements of degree qN . Let T p be the direct sum ⊕ q T pq andlet F q T p be ⊕ q (cid:54) q T pq . Note that d f ∧ maps T nq to T n +1 q +1 and that d maps T nq to T n +1 q .Let S (resp. S ) be the intersection of T n and Syz (resp. Syz ). The component ofdegree qN of an element α of T is denoted α q .The space T n +1 q is the equivalent of M q , as defined in the introductory remarks:the elements of T n +1 q represent numerators of rational functions whose denominatoris f q . We define the linear map h from T n +1 to A f by h : aω ∈ T n +1 q ( q − af q ∈ A f . Of course h is not injective since h ( f α ) = qh ( α ), for α ∈ T n +1 q . Finally let D f , the twisted differential , from T p to T p +1 be the map defined by D f α = d α − d f ∧ α .Note that D f maps F q T n to F q +1 T n +1 . The anticommutation d(d f ∧ β ) = − d f ∧ d β ensures that D f ◦ D f = 0, so that T p forms a chain complex. T nq +1 T n +1 q +1 T nq T n +1 q T n − q − T nq − T n +1 q −
1d ddd f dd f dd f dd f d f Figure 1.
Rham–Koszul double complex
Remark . The spaces T p + qq arranged within a grid form a double complex, knownas Rham-Koszul double complex [20], with the horizontal differential being d andthe vertical one being d f ∧ , see Figure 1. This arrangement may help visualize someof the proofs in this article.For p (cid:62)
0, let H p Rham ( P n K \ V ( f )) be the p th de Rham cohomology group of thevariety P n K \ V ( f ), and let H p +1 T be the p th cohomology group of the complex T ,that is ( T p ∩ ker D f ) /D f ( T p − ). The following theorem has been proved in numerousoccasions under several appearances, it goes back at least to Dwork. In this exactform, I am aware of proofs by Dimca [19, Theorem 1.8], Malgrange [34] andDeligne [17]. Theorem 6. H p +1 T ’ H p Rham ( P n K \ V ( f )) , for all p (cid:62) . We will only make use of Theorem 6 in the case where p = n . The cohomologygroup H n +1 T is T n +1 /D f ( T n ) and H n Rham ( P n K \ V ( f )) is isomorphic to the subspaceof A f / P i ∂ i A f generated by the homogeneous elements of degree − n −
1, and theisomorphism is the map induced by h : T n +1 → A f : Proposition 7. h ( D f ( T n )) ⊂ P ni =0 ∂ i A f . In other words, the map h induces amap from T n +1 /D f ( T n ) to A f / P i ∂ i A f .Proof. Let β = P ni =0 b i ξ i be an element of T nq , then h ( D f ( n X i =0 b i ξ i )) = n X i =0 h ( ∂ i bω ) − h ( b i ∂ i f ω ) = n X i =0 ( q − ∂ i b i f q − q ! b i ∂ i ff q +1 = ( q − n X i =0 ∂ i (cid:18) b i f q (cid:19) . (cid:3) This way, the goal of computing normal forms modulo derivatives of rationalfunctions can be reformulated as computing normal forms of elements of T n +1 modulo D f ( T n ). See [28] for a general definition and [27] for a definition in the specific case of a complement ofa projective hypersurface
OMPUTING PERIODS OF RATIONAL INTEGRALS 11
Example . With f = x y − z , Equation (4) rewrites x d x d y d z = D f ( x d y d z + x d x d z ) . The rewriting is not always as simple as in this example but Theorem 6 asserts thatit is always possible.2.3.
Filtered maps.
The space T n +1 admits a filtration given by the subspaces F q T n +1 with q ∈ Z . In the next sections, we will define reduction maps which will be fil-tered endomorphisms of T n +1 , that is to say linear maps u : T n +1 → T n +1 suchthat u ( F q T n +1 ) ⊂ F q T n +1 for all q ∈ Z . Two filtered endomorphisms of T n +1 ,say u and v , are equivalent if for all q ∈ Z and all α ∈ F q T n +1 we have u ( α ) ≡ v ( α )modulo F q − T n +1 .For all filtered map u , we can define the associated graded map asGr u : α ∈ T n +1 X q (cid:62) u ( α q ) q ∈ T n +1 . Two filtered maps are equivalent if and only if their associated graded maps areequal. 3.
Griffiths-Dwork reduction
We reword the Griffiths-Dwork reduction, presented in Section 1, in the abovesetting. Let us choose a monomial ordering on A , denoted ≺ . For a linear subspace V of A and an element a of A , let rem( a, V ), be the unique b in A such that a − b isin V and no monomials of b is divided by the leading monomial of some elementof V . If V is an ideal of A , this can be computed using a Gröbner basis of V ,and if it is a finite-dimensional subspace, then Gaussian elimination following themonomial ordering computes rem( a, V ).The elementary step of the Griffiths-Dwork reduction is the following. Let α be anelement of T n +1 q . By definition there is a β in T n such that α = rem( α, d f ∧ T n ) +d f ∧ β . We choose β in such a way that: it depends linearly on α ; β = 0 if α is in D f T n ; and β is in T nq − . The elementary reduction of α in degree q is thendefined to be(6) red GD q ( α ) def = rem( α, d f ∧ T n ) + d β. For α in T n +1 k , for some k different from q , we define red GD q ( α ) = α . The definitionof red GD q depends on the choice of β ; however, the equivalence class of red GD q as afiltered map does not.This elementary reduction is very easy to compute using a Gröbner basis of theJacobian ideal Jac f = ( ∂ f, . . . , ∂ n f ) and its cofactors. Indeed, the multivariatedivision algorithm gives a decomposition of a polynomial a as rem( a, Jac f ) + P ni =0 b i ∂ i f . If α is aω , then rem( α, d f ∧ T n ) is rem( a, Jac f ) ω and β may be chosenequal to P i b i ξ i . In this way, the assumptions on β are naturally satisfied. SeeSection 7 for more details about the implementation.By construction, α − red GD q ( α ) = − D f β , so that red GD q is an idempotent mapwhose kernel is included in D f ( T n ). When translated into a relation betweenfractions, this reflects integration by parts: I b i ∂ i (1 /f q − )d x = − I ∂ i b i /f q − d x . This reduction step can be iterated and for α ∈ F q T n +1 , the Griffiths-Dworkreduction of α , denoted [ α ] GD , is defined as[ α ] GD def = red
GD1 ◦ · · · ◦ red GD q ( α ) . We check the following recursive relation: [ α ] GD = rem( α, d f ∧ T n )+[d β ] GD , where β is the one in the equation 6. Again, the map [ ] GD depend on the choice of β but itsequivalence class, as a filtered map, does not. Proposition 9.
The map [ ] GD is filtered, idempotent and ker[ ] GD ⊂ D f ( T n ) . Inparticular α ≡ [ α ] GD modulo D f ( T n ) for all α ∈ T n +1 . Moreover, for all q (cid:62) and α ∈ F q T n +1 , [ α ] GD ∈ F q − T n +1 if and only if α ∈ D f ( F q − T n ) + F q − T n +1 .Proof. It is straightforward that the map [ ] GD is filtered, idempotent and that ker[ ] GD is included in D f ( T n ). Concerning the second point, let α ∈ F q T n +1 . By contruc-tion [ α ] GD = α + D f β for some β ∈ F q − T n . So if [ α ] GD is in F q − T n +1 then α isin D f ( F q − T n ) + F q − T n +1 .Conversely, let α = D f β + ε , with β ∈ F q − T n and ε ∈ F q − T n +1 . Then α q =d f ∧ β q − , and so rem( α q , d f ∧ T n ) = 0. By Equation (6), red GD q ( α ) ∈ F q − T n +1 ,and so [ α ] GD as well. (cid:3) The Griffiths-Dwork reduction [ ] GD is a multivariate and homogeneous analogueof Hermite reduction. In general, it does not have all the nice properties of Hermitereduction: it may happen that for some α in D f ( T n ) the reduction [ α ] GD is notzero and it may fail at reducing the degree. Nevertheless, Dwork [22, §3; 23, §8]and Griffiths [27, §4] have proven the following: Theorem 10 (Dwork, Griffiths) . If V ( f ) is smooth in P n K then(i) ker[ ] GD = D f ( T n ) ,(ii) for all α in T n +1 the reduction [ α ] GD is in F n T n +1 Remark . This theorem still holds if we replace [ ] GD by any equivalent fil-tered and idempotent map u whose kernel is included in D f ( T n ). Indeed, in thiscase ker Gr u = ker Gr[ ] GD = Gr( D f T n ) Since ker Gr u is Gr(ker u ), this impliesthat ker u = D f ( T n ). Moreover, the point (ii) implies that [ F q T n +1 ] GD ⊂ F q − T n +1 for all q > n . Since [ ] GD and u are equivalent, the same holds for u . And since u isassumed to be idempotent, this implies that u ( T n +1 ) ⊂ F n T n +1 .The hypothesis “ V ( f ) is smooth” is equivalent to the fact that Jac f is a zero-dimensional ideal, that is A/ Jac f is finite-dimensional over K . It is also equivalentto the equality of S and S , respectively the syzygies and the trivial syzygies in T n .The main step of the proof of Theorem 10 is [27, Theorem 4.3]: Theorem 12 (Dwork, Griffiths) . If V ( f ) is smooth in P n K then D f ( T n ) ∩ F q T n +1 is contained in D f ( F q − T n ) for all q (cid:62) . In the singular case, it is never true that ker[ ] GD = D f ( T n ). Worse still, thecokernel T n +1 / ker[ ] GD is never finite dimensional. Indeed, we have F q T n +1 F q T n +1 ∩ ker[ ] GD + F q − T n +1 ’ ( A/ Jac f ) qN − n − , so the quotient is finite dimensional if and only if Jac f is a zero-dimensional ideal. OMPUTING PERIODS OF RATIONAL INTEGRALS 13 − d f ∧ β q +2 ) β q +2 β q +2 − d f ∧ β q +1 ) β q +1 β q α q β q − α q − − d f d − d f d − d f d − d f Figure 2. A n -form β ∈ T nq − + · · · + T nq +2 such that D f β ∈ F q T n +1 , thus giving anelement α of W q . Computation of higher order relations
Construction.
Let W q be the subspace of T n +1 q + T n +1 q − defined by(7) W q def = D f ( T nq − ) = (cid:8) − d f ∧ β + d β (cid:12)(cid:12) β ∈ T nq − (cid:9) . Following the idea developed in Section 1, we define, for r (cid:62) q (cid:62) W r +1 q def = W q + W rq +1 ∩ F q T n +1 . Compared to Section 1, the space M q has been replaced by T n +1 q and the prod-uct M q × M q − by the direct sum T n +1 q ⊕ T n +1 q − . Proposition 13.
For all r (cid:62) and q (cid:62) , W rq = D f (cid:0) r X k =1 T nq + k − (cid:1) \ F q T n +1 . Proof.
By induction on r . For r = 1, the claim reduces to W q = D f ( T nq − ), whichis the definition. Then, let us prove that the right-hand side satisfies the recurrencerelation defining W rq , that is: D f (cid:0) r +1 X k =1 T nq + k − (cid:1) \ F q T n +1 = D f ( T nq − ) + D f (cid:0) r X k =1 T nq + k − (cid:1) \ F q T n +1 , which follows simply from D f ( T nq − ) ⊂ F q T n +1 . (cid:3) Figure 2 depicts what are elements of W rq . Example . With f = xy − z , we find that W = 0 and W = (cid:10) x y, xy , y , xyz, y z, xz , yz , z , (cid:11) ω. Thus W ∩ T = h ω i and W = h ω i . Reductions of order r . The higher order analogue of red GD q , denoted red rq is the linear map T n +1 → T n +1 defined byred rq α def = rem( α, W rq ) , for α in T n +1 q , and red rq α = α for α ∈ T rk with k = q . As for the Griffiths-Dworkreduction, we define for α in F q T n +1 [ α ] r def = red r ◦ · · · ◦ red rq ( α ) . This reduction map enjoys the following properties, to be compared with Proposi-tion 9 relative to [ ] GD . Proposition 15.
Let r (cid:62) . The map [ ] r is filtered and idempotent, its kernel isincluded in D f ( T n ) and [ ] r +1 ◦ [ ] r = [ ] r +1 . Moreover, for all q (cid:62) and α ∈ F q T n +1 , [ α ] r ∈ F q − T n +1 if and only if α ∈ D f ( F q + r − T n +1 ) + F q − T n +1 .Proof. It is straightforward that the map [ ] r is filtered and idempotent. Since W rq ⊂ D f ( T n ), for all q , we have ker[ ] r ⊂ D f ( T n ). And since W rq ⊂ W r +1 q we have [ ] r +1 ◦ [ ] r = [ ] r +1 .Let α ∈ F q T n +1 such that [ α ] r ∈ F q − T n +1 . From the definition, [ α ] r ≡ red rq α (mod F q − T n +1 ) and red rq α ≡ α (mod W rq ). So α ≡ W rq + F q − T n +1 )and α ∈ D f ( F q + r − T n +1 ) + F q − T n +1 .Conversely, let us assume that α = D f β + α , with β in F q + r − T n +1 and α in F q − T n +1 . The form β splits as β + ε , with β ∈ P rk =1 T nq + k − and ε ∈ F q − T n .We check that D f β ∈ F q T n +1 , so D f β ∈ W rq , by Proposition 13. And red rq ( D f β ) ∈ F q − T n +1 , by definition of red rq . Thusred rq ( α ) = red rq ( D f β ) + red rq ( D f ε + α ) ∈ F q − T n +1 , and [ α ] r , which equals [red rq ( α )] r , is in F q − T n +1 as well. (cid:3) Corollary 16. D f ( T n ) = S r (cid:62) ker[ ] r .Proof. Let β ∈ T n such that D f β = 0. Let q (cid:62) D f β ∈ F q T n +1 . Let r (cid:62) β ∈ F q + r − T n +1 . By Proposition 15,[ D f β ] r is in F q − T n , and it is also in D f ( T n ) because D f β ≡ [ D f β ] r modulo D f ( T n ).By induction on q , there exists an s (cid:62) r such that [[ D f β ] r ] s = 0. Since [ ] s ◦ [ ] r = [ ] s ,the result follows. (cid:3) Remark . The reductions [ ] GD and [ ] do not necessarily coincide, but they areequivalent filtered maps.Thus, we have a family of finer and finer reductions which generalize the Griffiths-Dwork reduction and which are exhaustive in the sense that they reduce to zeroevery D f β if r is large enough. However, two problems remains. The first on ispractical: as defined, the computation of [ ] r , for a given r , involves the resolution ofhuge linear systems, both when computing the spaces W rq and when computing red rq .This is in contrast with [ ] GD which only involve the computation of a Gröbner basisand reductions modulo it for computing red GD q . The §4.3 describe a faster way tocompute [ ] r . The second problem is theoretical: how to set the parameter r ? Thisis addressed in Section 5. OMPUTING PERIODS OF RATIONAL INTEGRALS 15
Faster computation.
There are two ingredient for computing [ ] r faster thanwith plain linear algebra. The first is the use of red GD , whose implementation isefficient and which readily perform a great deal of reductions. Secondly, we discardtrivial syzygies, as explained in §1.4.Let A q be a complementary subspace of S q in S q , that is S q equals S q ⊕ A q .Let X q def = d A q − and, for all q (cid:62) r (cid:62) X r +1 q def = d A q − + red GD q (cid:0) X rq +1 ∩ F q T n +1 (cid:1) . Since d A q − = D f ( A q − ), it is clear that X q ⊂ D f ( F q − T n ), and by induction on r ,we obtain that X rq ⊂ D f ( F q + r − T n ). Moreover, we have red GD q α = α for all q andall α ∈ X rq . Finally, let ρ rq : T n +1 → T n +1 the linear map defined by ρ rq ( α ) def = rem(red GD q ( α ) , X rq ) , for α in T n +1 q , and ρ rq ( α ) = α for α ∈ T rk with k = q . For α ∈ F q T n +1 we define[ α ] r def = ρ r ◦ · · · ◦ ρ rq ( α ) . This paragraph aims at proving the following:
Theorem 18.
For all r (cid:62) , the map [ ] r is filtered and idempotent, its kernel isincluded in D f ( T n ) and [ ] r +1 ◦ [ ] r = [ ] r +1 . Moreover, it is equivalent to [ ] r , inparticular, for all q (cid:62) and α ∈ F q T n +1 , [ α ] r ∈ F q − T n +1 if and only if α ∈ D f ( F q + r − T n +1 ) + F q − T n +1 . Corollary 19. D f ( T n ) = S r (cid:62) ker[ ] r .Proof. The proof is the same as Corollary 16. (cid:3)
The map [ ] r is easier to compute than [ ] r because the linear algebra involvedin the computation of X rq arises in much lower dimension than the one for W rq .It comes at the cost of using red GD and of computing the space A q of non trivialsyzygies, which can be done efficiently through Gröbner basescomputations, seeSection 7.The main fact which allows to discard trivial syzygies is the following: Lemma 20. red GD q (d S q ) ⊂ d S q − , for all q (cid:62) .Proof. Recall that S q = d f ∧ T nq − , so let β ∈ T nq − . The differential anti-commuteswith d f ∧ so that d(d f ∧ β ) = − d f ∧ d β . By definition red GD q (d(d f ∧ β )) is thus d γ forsome γ ∈ T nq − such that d f ∧ γ = − d f ∧ d β . Thus γ = − d β + ε , for some ε ∈ S q − .Since d(d β ) = 0, we obtain that red GD q (d(d f ∧ β )) = d ε . (cid:3) Let G q ⊂ T n +1 be the kernel of red GD q . It is a subspace of T n +1 q ⊕ T n +1 q − . Proposition 21. W rq = X rq + G q + d S q − , for all q (cid:62) and r (cid:62) .Proof. We proceed by induction on r . When r = 1, it boils down to provingthat D f ( T nq − ) = d A q − + G q + d S q − , that is D f ( T nq − ) = G q + d S q − , using thefact that d A q − + d S q − = d S q − . Let β ∈ T nq − . By definition of red GD q ,red GD q ( D f β ) = − red GD q (d f ∧ β ) + d β = d( β − β ) , for some β ∈ T nq − such that d f ∧ β = d f ∧ β . Thus β − β lies in S q − and red GD q ( D f β )is in d S q − . Moreover, since red GD q is idempotent, D f β − red GD q ( D f β ) is in G q , and in the end D f β ∈ G q + d S q − . Conversely, S q − ⊂ T nq − , so it remains to provethat G q ⊂ D f ( T nq − ), which is easy from the definitions.Now let r (cid:62)
1. By definition, and by the induction hypothesis W r +1 q = W q + W rq +1 ∩ F q T n +1 = G q + d A q − + d S q − + ( X rq +1 + d S q + G q +1 ) ∩ F q T n +1 . And we have( X rq +1 + d S q + G q +1 ) ∩ F q T n +1 = X rq +1 ∩ F q T n +1 + d S q . Indeed d S q ⊂ F q T n +1 , and if α ∈ X rq +1 and α ∈ G q +1 are such that α + α ∈ F q T n +1 ,then α = 0 because α + α = red GD q +1 ( α + α ) = red GD q +1 ( α ) + red GD q +1 ( α ) = α + 0 . Thus W r +1 q = G q + d A q − + d S q − + d S q + X rq +1 ∩ F q T n +1 . For any linearsubspace A ⊂ T n +1 , the decomposition α ∈ A as red GD q α + ( α − red GD q α ) showsthat G q + red GD q ( A ) = G q + A . Thus W r +1 q = G q + d A q − + d S q − + red GD q (d S q ) + red GD q ( X rq +1 ∩ F q T n +1 ) , and the statement follows, by Lemma 20 and the definition of X r +1 q . (cid:3) We may now prove Theorem 18.
Proof of Theorem 18.
It is straightforward that [ ] r is filtered and idempotent,that ker[ ] r ⊂ D f ( T n ) and that [ ] r +1 ◦ [ ] r = [ ] r +1 .To prove that [ ] r and [ ] r are equivalent, it is enough to prove that red rq and ρ rq are equivalent. And indeed, if α ∈ F q T n +1 then ρ rq ( α ) ≡ rem( α, G q + X rq ) mod F q − T n +1 and red rq ( α ) ≡ rem( α, d S q − + G q + X rq ) mod F q − T n +1 , using Proposition 21. Since d S q − ⊂ F q − T n +1 the claim follows. (cid:3) In what follows, [ ] r will stand for [ ] r . Except in terms of computational complexity,they have the same properties.4.4. Quantitative facts.
It is useful to introduce the spaces E rq def = F q T n +1 D f ( F q + r − T n ) ∩ F q T n +1 + F q − T n +1 . It is clear that E q is F q T n +1 /F q − T n +1 , which is isomorphic to T n +1 q . Moreover,as a reformulation of Proposition 9, the space E q is E q = coker(Gr[ ] GD ) q def = F q T n +1 { α ∈ F q T n +1 | [ α ] GD ∈ F q − T n +1 } ’ T n +1 q d f ∧ T nq − . And by Proposition 15, this generalizes to the isomorphism E rq ’ coker(Gr[ ] r ) q .In other words, E rq is F q T n +1 modulo elements which are reducible to F q − T n +1 by [ ] r The space E r +1 q is a quotient of E rq , and the dimension fall represents howmany new relations in degree qN are computed by [ ] r +1 compared to [ ] r . For r = 2,we check that E q ’ T n +1 q d f ∧ T nq − + d S q = T n +1 q d f ∧ T nq − + d A q . OMPUTING PERIODS OF RATIONAL INTEGRALS 17 q q > E q (cid:0) q − (cid:1) ∼ q dim E q q + 48dim E q E q E rq , r (cid:62) Table 1.
Some dimensions related to Example 22
The dimension of E q is (cid:0) Nq − n (cid:1) , which is equivalent to N n q n /n ! when q → ∞ .The dimension of E q is O ( q ν ), where ν is the dimension of the singular locus of V ( f )in P n K . There is no easy estimate of the dimension of E q , but dim A q − is also O ( q ν ).By contrast, dim S q ∼ ( n + 1) N n q n /n !. For the computation of [ ] (or rather [ ] ),it is thus a substantial improvement to consider the non-trivial syzygies A q ratherthan all the syzgies S q . Example . To illustrate precisely what does bring the maps [ ] r in comparaisonwith [ ] GD , let us consider the polynomial ff def = 2 x x x ( x − x )( x − x )( x − x ) − x ( x − x x + x x x )coming from an integral for the Apéry numbers, see Example 1. In this case n = 3and N = 6. The dimension of the singular locus of V ( f ) in P K is 1.The dimensions of the first few E rq are shown in Table 1. This illustrates thesuccessive dimension falls. Noticeably, at r = 3 a new relation appears in F T n +1 .It is (2 x − x − x ( x − x )) ω , which equals D f β for some β in F T n but nosuch β is small enough to be reproduced here.Illustrating the same polynomial f , Table 2 shows the numbers of syzygies andnon-trivial syzygies at a given degree. It also displays the difference dim E q − dim E q ,that is how many new relations are really generated from the syzygies.5. Extensions of Griffiths’ theorems
Given α in T n +1 , how can we compute a r such that if α is in D f ( T n ) then [ α ] r equals zero? Corollaries 16 and 19 are lacking effective bounds and do not answerthis question. Dimca proved two theorems [18, Th. B and Cor. 2; 19, Th. 2.7] whichgeneralize Theorem 10. While they do not give a full answer, they allow to giveenough guarantees on [ ] r to design algorithms that terminates. Theorem 23 (Dimca) . There exists an integer C , depending only on f , suchthat D f ( T n ) ∩ F q T n +1 ⊂ D f ( F q + C − T n ) for all q (cid:62) . This statement is to be compared with Theorem 12. Given f and q , it is easy toprove that there exists a C such that D f ( T n ) ∩ F q T n +1 ⊂ D f ( F q + C − T n ), because theleft-hand side is a finite dimensional space and it is included in ∪ C (cid:62) D f ( F q + C − T n ).It is remarkable that one can choose a C which does not depend on q . Let r f bethe least such C . Corollary 24. ker[ ] r f = D f ( T n ) . q q > S q ∼ q dim A q q + 24dim E q − dim E q q + 42 Table 2.
Gain of dimension by discarding trivial syzygies and number of new relationsgenerated by the syzygies in the Example 22
Proof.
Let β ∈ T n and q (cid:62) D f β ∈ F q T n +1 . ByTheorem 23, there exists β ∈ F q + r f − T n such that D f β = D f β . Thus, by The-orem 18, [ D f β ] r f is in F q − T n +1 , and besides, it is also in D f ( T n ). By inductionon q , [[ D f β ] r f ] r f = 0. Since [ ] r f is idempotent, the claim follows. (cid:3) Unfortunately, this integer r f , while explicit, is not easy to compute: in Dimca’sproof it is expressed in terms of a resolution of the singularities of the projectivevariety V ( f ). By contrast, the point (ii) of Theorem 10 fully generalizes to singularcases: Theorem 25 (Dimca) . D f ( T n ) + F n T n +1 = T n +1 . Corollary 26.
For all α ∈ T n +1 , the reduction [ α ] r f lie in F n T n +1 .Proof. By Theorem 25, there exists β ∈ T n such that α + D f β is in F n T n +1 .Since [ α ] r f = [ α + D f β ] r f − [ D f β ] r f , the claim follows from Corollary 24. (cid:3) For some applications, such that the computation of annihilating operators ofperiods with a parameter, Theorem 25 gives an efficient workaround to the lackof a priori bounds for r f . Consider an algorithm which computes reductions [ α ] r ,for some forms α and some fixed integer r , and does it as long as the reductionsit computes are linearly independent. Then either all the [ α ] r are in the finitedimensional space F n T n +1 , and then the algorithm terminates; or some [ α ] r is notin F n T n +1 , and then r < r f , by Theorem 25. When the second case is encountered,we abort the algorithm, increment r and start over. This may happen only if r < r f ,and when it happens r increases. So it may happen only finitely many times andthe algorithm terminates.Concerning the integer r f Dimca [18] conjectured that
Conjecture 27. r f (cid:54) n + 1 . As far as I know, computations on explicit examples confirm this conjecture.Moreover the bound is tight when n = 2. A proof of this conjecture would havevery interesting algorithmic consequences: the reduction algorithm is extendable tothe computation of the whole cohomology of T , not just the top cohomology. Onlythe bound r f (cid:54) n + 1 is lacking for obtaining an efficient algorithm for computingthe de Rham cohomology of the complement of a projective hypersurface. Part Periods with a parameter
We apply the reduction algorithm to the computation of Picard-Fuchs equations.
OMPUTING PERIODS OF RATIONAL INTEGRALS 19 Algorithms
Setting.
Let K be a field of characteristic zero with a derivation δ . Typically K is Q ( t ) and δ is the usual derivation with respect to t . Let K h δ i be the algebraof differential operators in δ : it is the associative algebra with unity generatedover K by δ and subject to the relations δx = xδ + δ ( x ) for all x in K , where δ ( x )denotes the application of δ to x whereas δx is the operator that multiplies by x and then applies δ . On K ( x , . . . , x n ), let ∂ i denote the derivation with respectto x i . The derivation δ extends to K ( x , . . . , x n ) uniquely by setting δ ( x i ) = 0. Inparticular δ ◦ ∂ i = ∂ i ◦ δ .This section describes an algorithm which takes as input a rational function R in K ( x , . . . , x n ) and outputs an operator L in K h δ i such that there exist otherrational functions C , . . . , C n with L ( R ) = n X i =1 ∂ i C i . Moreover, the irreducible factors of the denominators of the C i divide the denomi-nator of R . Such an operator will be called an annihilating operator of the periodsof R , or a differential equation for H R . The minimal annihilating operator of H R iscalled the Picard-Fuchs equation (of H R ). The output operator L is not necessarilythe Picard-Fuchs equation but it is of course a left multiple of it.Being based on the reduction algorithm of Part 1, the algorithm does not computethe C i . It is worth a word because while only L matters, the size of the C i , say thesize of a binary dense representation, is usually much larger than the size of L [9,Rem. 11]. To be able to compute L without computing the C i is certainly a goodpoint toward practical efficiency. The fractions C i are called a certificate : they allowto check a posteriori that L is indeed an annihilating operator of H R .6.2. Homogenization.
The reduction algorithm works in an homogeneous setting.If we are interested in computing the Picard-Fuchs equation of the integral of aninhomogeneous function, the problem can be homogenized as follows. Let R hom bethe homogenization of R in degree − n − R hom = x − n − R (cid:18) x x , . . . , x n x (cid:19) ∈ K ( x ) , where x denotes x , . . . , x n hereafter. The rational function R hom ( x ) is homo-geneous of degree − n −
1, that is R hom ( λx , . . . , λx n ) = λ − n − R hom ( x , . . . , x n ),or, equivalently, R hom = b/g where b and g are homogeneous polynomials suchthat deg b + n + 1 = deg g .Let us write R hom as a/f q , with a and f two homogeneous polynomials and q an integer. Usually f will be chosen square-free but we don’t have to. Let N bethe degree of f . Since R hom is homogeneous of degree − n −
1, the degree of a is qN − n −
1. This is the main reason for considering homogeneous fractions: thedegree of the denominator determines the degree of the numerator, there is no hidden pole at infinity. The degree − n − Lemma 28. If L ∈ K h δ i is a annihilating operator of H R hom then L is also aannihilating operator of H R . Algorithm 1.
Computation of annihilating operators of the periods of a rational function,smooth case
Input — a/f q a homogeneous rational function in K ( x ) of degree − n −
1, with V ( f )smooth in P n K Output — L ∈ K h δ i the Picard-Fuchs equation of H R procedure PicardFuchs ( a/f q ) ρ ← [ aω ] GD for m from 0 to ∞ doif rank K ( ρ , . . . , ρ m ) = m + 1 then ρ m +1 ← [ δ ( ρ m )] GD else compute a , . . . , a m − ∈ K such that P m − k =0 a k ρ k = ρ m return δ m − P m − k =0 a k δ k Proof.
Assume that L ( R hom ) equals P ni =0 ∂ i ( b i /f m ), for some polynomials b i andsome integer m . Substituting x by 1 gives L ( R ) = ∂ ( b /f m ) | x =1 + n X i =1 ∂ i ( b i /f m | x =1 ) . Since R hom is homogeneous of degree − n −
1, we may assume that each b i /f m ishomogeneous of degree − n . Euler’s relation gives − nb /f m = n X i =0 x i ∂ i ( b /f m ) = n X i =0 ( ∂ i ( x i b /f m ) − b /f m ) . This proves that 0 = ∂ ( b /f m ) | x =1 + P ni =1 ∂ i ( x i b /f m | x =1 ), and the claim follows. (cid:3) The Picard-Fuchs equation of H R hom may not be the Picard-Fuchs equationof H R , but only a left multiple. However, it is the case if x divides f , which ispossible to assume, up to replacing f by x f and a by x q a . From now on I focusexclusively on the homogeneous case.6.3. Computation of Picard-Fuchs equations.
The derivation δ is extended tothe spaces T p of differential forms by δ : α ∈ T p α δ − f δ α ∈ T p , where • δ denotes component-wise differentiation. It commutes with the map h , andthe differential D f , as a consequence of δ commuting with ∂ i .To highlight the difference between the smooth and the singular cases, I recallfirst how the Griffiths-Dwork reduction applies to the computation of Picard-Fuchs equations. Let a/f q be a homogeneous fraction of degree − n −
1. Wedefine ρ = [ aω ] GD and ρ k +1 def = [ δ ( ρ k )] GD . Since δ commutes with D f , it is clearthat ρ k ≡ δ k ( aω ) modulo D f ( T n ). Hence Theorem 10 implies that ρ k = [ δ k ( aω )] GD .Thus, by Theorems 6 and 10 and, for u , . . . , u m in K , m X k =0 u k δ k ( a/f q ) ∈ n X k =0 ∂ k A f if and only if m X k =0 u k ρ k = 0 . See definition in §2.2.
OMPUTING PERIODS OF RATIONAL INTEGRALS 21
This leads to Algorithm 1.
Proposition 29.
Algorithm 1 applied to a fraction R satisfying the regularityassumption terminates and outputs the Picard-Fuchs equation of H R .Proof. Correctness has just been proven. Termination follows from Theorem 10,point (ii), which implies that the ρ i lie in a finite-dimensional space, so they arelinearly dependent. (cid:3) If Conjecture 27 were proven, it would be enough to replace [ ] GD by [ ] n +1 , orits efficient variant [ ] n +1 , in Algorithm 1 to obtain an algorithm which provablyoutputs the Picard-Fuchs equation of a rational integral in the singular case. Whileassuming this conjecture gives good results in practice, the absence of a proof isembarrassing.It is worth mentioning the treatment of singular cases by a generic deformation:to compute a differential for H R , for some R = a/f , we may change R into R λ = af + λ P ni =0 x deg fi , where λ is a free variable. The denominator of R λ always satisfy the smoothnesshypothesis, so Algorithm 1 applies, over K ( λ ), and gives the Picard-Fuchs equationof H R λ , say L in K ( λ ) h δ i . Then ( λ a L ) | λ =0 , where a is the unique integer whichmakes this evaluation neither zero nor singular, is a differential equation for H R . Thismethod achieves a good computational complexity, that is polynomial complexitywith respect to the generic size of the output [9], but its practical efficiency isterrible because most Picard-Fuchs that are interesting to compute are much smallerthan the generic Picard-Fuchs equation.Another approach, using the reductions [ ] r , is to loop over r . We begin byfixing r to an initial value, for example 1, and we introduce another variable M , apositive integer. Then we compute ρ , ρ , etc. as in Algorithm 1 but replacing [ ] GD by [ ] r , up to ρ M . If there is no linear dependency relation between the ρ k then weincrease both r and M and repeat the procedure. At some point, the parameter r will exceed r f and M will exceed the order of the Picard-Fuchs equation of H R .There, a relation will be found between the ρ k and it will give the Picard-Fuchsequation. It is possible that a relation is found before the condition r (cid:62) r f is met:it gives of course a differential equation, but it need not be the minimal one.Theorem 25 and its corollary allow for an interesting variant of this approach.As above, we loop over r . For a given value of r , the forms ρ , ρ , etc. arecomputed as in Algorithm 1 but using [ ] r instead of [ ] GD . Contrary to the previousapproach, the number of ρ i we compute before moving to the next value of r is notbounded a priori . Instead, we compute ρ , ρ , etc. as long as ρ k stays in F n T n +1 .Since F n T n +1 is finite dimensional, we have the following alternative: either thereexists a relation between the ρ k , or there exists a k such that ρ k is not in F n T n +1 .In the first case, the relation gives a differential equation for H R . In the second case,we increase r and start over the computation of the ρ k ’s. Corollary 24 assures thatas soon as r (cid:62) r f , the second condition is never met, so a relation will eventually befound. Algorithm 2 details the procedure. Theorem 30.
Algorithm 2 terminates and outputs an annihilating operator of H R . Algorithm 2.
Computation of annihilating operators of the periods of a rational function
Input — a/f q a homogeneous rational function in K ( x ) of degree − n − Output — L ∈ K h δ i a differential equation for H R procedure PicardFuchs ( a/f q ) for r from 1 to ∞ do ρ ← [ a ω ] r . Compute the subspaces X qr as they are needed. for m from 0 to ∞ while deg ρ m (cid:54) n deg f doif rank K ( ρ , . . . , ρ m ) = m + 1 then ρ m +1 ← [ δ ( ρ m )] r else compute a , . . . , a m − ∈ K such that P m − k =0 a k ρ k = ρ m return δ m − P m − k =0 a k δ Implementation
Algorithm 2 has been implemented in the computer algebra system Magma [7],with Q ( t ) as base field K , with the usual derivation. To be able to treat largeexamples—like the ones in Section 8—the coefficient swell makes it necessary toimplement a randomized evaluation-interpolation scheme which splits a computationover Q ( t ) into several analogous computations over different finite fields. However itcomes at a price: since we lack tight a priori bounds on the size of the output—order,degree, size of the coefficients—the reconstruction step is not certified to be correct,even though the probability of failure can be made arbitrarily small. There arealso several ways to cross-check the result independently. The variant is describedin §7.2. In the introduction, I mentionned the guessing method which allows, insome cases, to compute an annihilating operator of a given period but gives noguarantee about its correctness. The nature of the risk of failure is very differentthough. In the evaluation-interpolation method, the algorithm is randomized andthe probability of failure can be made arbitrarily small. It is even less probablethat the algorithm returns twice the same wrong result. It is not possible to foolthe algorithm on purpose with a specific input. In the guessing method, we do notknow how to evaluate the risk of failure and the algorithm is deterministic so anerror will be repeated again and again. It is in principle possible to fool the methodwith input designed for this purpose.When a risk of failure is not acceptable, it is possible to compute certificateswhich can be used a posteriori to prove that what has been computed is correct,see §7.3.7.1. Implementation of [ ] r using Gröbner bases. Let M be the module Ω n +1 ⊕ Ω n , that is the free module generated by ω and the ξ i , recall the definitions in §2.1A convenient way to implement the reduction [ ] r is to compute a reduced Gröbnerbasis say G , of the submodule P of M generated by the ∂ i f ω − ξ i , that is d f ∧ ξ i − ξ i .We choose on M a monomial ordering, denoted (cid:31) , such that for all multi-indices I and J , and all integer j (8) | I | + 1 (cid:62) | J | + N = ⇒ x I ω (cid:31) x J ξ j . The implementation is available at http://github.com/lairez/periods . See [16, chap. 5] for details about Gröbner bases for modules, the division algorithm, etc.
OMPUTING PERIODS OF RATIONAL INTEGRALS 23
For example, any position-over-term (POT) ordering with ω (cid:31) ξ (cid:31) ξ (cid:31) · · · isfine. But a term-over-position (TOP), with ω (cid:31) ξ (cid:31) ξ (cid:31) · · · , extending a gradedordering on A works as well. This gives some flexibility in the implementation.Let rem G denote the remainder on division by G . The condition (8) on the order isenough to ensure that (cid:31) behaves like an order eliminating ω .The reason is the following. If we give to ω the degree 1 and to each ξ i thedegree N , then P is a homogeneous submodule of M . Thus any reduced Gröbnerbasis G of P , whatever the monomial order, contains only homogeneous elements andthe remainder on division by G of a homogeneous element of degree d is homogeneousof degree d . In particular we have the Lemma 31.
Let α be an element of Ω n +1 . Then the coefficient of ω in rem G α iszero if and only if α ∈ d f ∧ Ω n . In this case α = d f ∧ rem G α .Proof. By definition of G there exist polynomials c i such that α = rem G ( α ) + n X i =0 c i (d f ∧ ξ i − ξ i ) . If the coefficient of ω in rem G ( α ) is zero then rem G ( α ) is in Ω n . Identifying thecomponents gives α = d f ∧ n X i =0 c i ξ i = n X i =0 c i ∂ i f ! ω and rem G ( α ) = X i c i ξ i . Conversely, assume that α = d f ∧ β , for some β in Ω n . We may assume that α is homogeneous of degree d and that β is homogeneous of degree d − N . Inparticular α − β is in P and rem G ( α − β ) = 0, since G is a Gröbner basis of P . Bylinearity rem G ( α ) equals rem G ( β ).For the grading introduced above, the element β is homogeneous of degree d − n ,thus so is rem G ( β ). Furthermore, the leading monomial of rem G ( β ), with respectto (cid:31) , is at most the leading monomial of β , which has the form x I ξ i with | I | = d − N − n . The claim follows since no monomial of the form x J ω has degree d − n (with the alternative grading) and is less than x I ξ i , thanks to hypothesis (8). (cid:3) In the same way we prove that
Lemma 32.
The intersection G ∩ Ω n is a Gröbner basis of Syz . Together with a Gröbner basis of Syz , this Gröbner basis can be used to computea basis of S q / S q in the following way. Using the Gröbner bases, we compute the set S def = { lm( α ) | α ∈ S q } \ (cid:8) lm( α ) (cid:12)(cid:12) α ∈ S q (cid:9) . Then, for each element α of S we pick an element of S q whose leading monomialis α . Those elements form a basis of S q / S q .Gröbner bases in the module M can be emulated by Gröbner bases in thepolynomial ring A with two extra variables, say u and v . Let A be A [ u, v ], let ω denote u n +1 and ξ i denote u n − i v i +1 . Let M be the A -submodule of A generatedby ω and ξ i . Let P be the ideal of A generated by ∂ i f ω − ξ i and all themonomials u p v q , with p + q = n + 2. Let ϕ be the A -linear map from M to M sending ω to ω and ξ i to ξ i . Finally, let G be a Gröbner basis with respect toany graded monomial ordering (cid:31) , say the graded reverse lexicographic ordering,with u (cid:31) v (cid:31) x (cid:31) · · · (cid:31) x n . Algorithm 3.
Computation of [ ] r Input — α an element of T n +1 and q an integer Output — red GD q ( α ) as defined in §3 procedure RedStep ( α , q ) α ← α − α q ρ + β ← rem G ( α q ), with ρ ∈ Ω n +1 and β ∈ Ω n return α + ρ + d β Input — r (cid:62) q (cid:62) Output — a basis of X rq , as defined in §4.3 procedure BasisX ( r , q ) if r = 1 thenreturn (cid:8) d β (cid:12)(cid:12) β ∈ (a basis of S q − / S q − ) (cid:9) else X ← BasisX ( r − , q + 1) return Echelon ( BasisX (1 , q ) ∪ { RedStep ( α, q ) ∈ X | deg α = qN } ) Input — α an element of T n +1 , r a positive integer Output — [ α ] r as defined in §4.3 procedure Reduction ( α , r ) q ← deg α/N and α ← α − α q ρ ← rem( RedStep ( α q , q ) , BasisX ( r, q )) return ρ q + Reduction ( α + ρ q − , r )If (cid:31) , the monomial ordering for M , is the TOP ordering proposed above, thenwe have ϕ (rem G α ) = rem G ϕ ( α ), and the proof is left to the reader.The computation of X rq and [ ] r is detailed in Algorithm 3. The function Echelon takes as input a finite subset S of T n +1 and outputs a basis in echelon form of Vect( S ),with respect to the monomial order (cid:31) : that is, a basis B of Vect( S ) such that forall element b of B , the leading monomial of b does not appear with a non-zerocoefficient in the other elements of B .7.2. Evaluation and interpolation scheme.
Let h ( t ) = p/q be an elementof Q ( t ) such that q is a monic polynomial. Let d be the maximum of deg p and deg q ,and M be the maximum of the absolute values of numerators and denominatorsof the coefficients of p and q . Given distinct primes p , . . . , p n , distinct rationalnumbers u , . . . , u m and the evaluations a i,j ≡ h ( u j ) (mod p i ), the fraction h can bereconstructed given that no p i divides the denominator of some coefficient of q , no u j annihilates q , Q mi =1 p i > M and m > d . To do so, we first compute a i in F p i ( t ) suchthat a i ≡ h (mod p i ), using Cauchy interpolation [26, §5.8]. Then, by the Chineseremainder theorem, we compute A such that A ≡ h (mod Q i p i ). And then, usingrational reconstruction [26, §5.10] to each coefficient of A , we recover h . Without apriori bounds on h , it is still possible to try to reconstruct it with the method above.Assume that we obtain a result h , and let M and d be the analogues of M and d for h . Under randomness assumptions, the bigger Q mi =1 p i − M and m − d are,the higher is the probability that h = h .Any algorithm which inputs and outputs elements of Q ( t ) and which performsonly field operations—addition, multiplication, negation, constant one, zero test, OMPUTING PERIODS OF RATIONAL INTEGRALS 25 inversion—in Q ( t ) can be turned into a randomized evaluation-interpolation algo-rithm, simply by evaluating the input at t = u and reducing it in F p , for several p and u , and proceeding to the computation over F p . Indeed, the execution of thealgorithm requires a finite number of operations, either field operations, whichcommute with ν , or zero test. For generic values of p and u , these tests yield thesame result on evaluated or unevaluated data. For specific values of p and u , anon-zero quantity can be evaluated to zero, so the computation over F p may fail orreturn a result which is not the evaluation of the result of the computation over Q ( t ).It is important to be able to test that in order to exclude bad evaluations becausethe reconstruction process does not handle possibly wrong evaluations.The number of evaluation points ( p, u ) is chosen, a priori or on-the-fly, so thatthe reconstruction of the outputs is possible with high probability of success. If apriori bounds on the output are known it may be possible to certify the result. Ifno bounds are known, then the evaluation-interpolation algorithm may return afalse result, but the probability of this event can be made arbitrarily small. Thisevaluation-interpolation approach is classical in computer algebra for avoiding theproblem of coefficient swell.Algorithm 2 depends on the derivation δ , which is not a field operation, so theconversion to an evaluation-interpolation algorithm is not completely straightfor-ward.7.2.1. Principle.
Let u be in Q and p be a prime number. Let ν be the partialfunction Q ( t ) → F p , which consists in evaluating t in u and reducing modulo p . Thefunction ν is extended coefficient-wise to Q ( t )[ x ], Ω, matrices, etc.Let f be a polynomial in Z [ t ][ x ], and ν ( f ) be its evaluation in F p [ x ]. Wecan consider the reductions [ ] r associated to f , but also the evaluated reduction,denoted [ ] νr , associated to ν ( f ), over F p . Given α ∈ T n +1 , and for generic values of p and u , the evaluations ν ( α ) and ν ([ α ] r ) are defined and ν ([ α ] r ) = [ ν ( α )] νr . However,the value of ν ( δa ) for some form a cannot be deduced from ν ( a ), so Algorithm 2requires an adaptation to fit into an evaluation-interpolation scheme.As in Section 6, let R = a/f q be a rational function in Q ( t ), homogeneous ofdegree − n − x . Let α be aω . Once the value of r isfixed, Algorithm 2 computes the terms of the sequence ( ρ i ) i ∈ N , defined by ρ = [ α ] r and ρ i +1 = [ δ ( ρ i )] r , until it finds a linear dependency relation between the ρ i . Fora prime p and an evaluation point u , can we compute ν ( ρ i ) using only operationsin F p ? The answer seems to be negative, but there are two ways to circumvent thisissue.The first one is to define ρ i to be [ δ i ( α )] r . With this definition, the principle andthe halting condition deg ρ i (cid:54) nN of Algorithm 2 remain valid. And given ν ( δ i ( α )),which is certainly easy to compute, it is possible in this case to compute ν ( ρ i ) usingonly operations in F p . This approach is feasible but it becomes terrible if i reacheshigh values: indeed, the degree of δ i ( α ) is deg α + iN .Another approach is to compute the matrix of the linear map, say m , such that ρ i +1 = ρ δi + m ( ρ i ) , where ρ δi denotes the component-wise differentiation of ρ i , as opposed to δ ( ρ i ) whichis ρ δi − f δ ρ i . Such a linear map exists and its matrix in a certain basis can becomputed by evaluation-interpolation. Algorithm 4.
Computation of annihilating operators of the periods of a rational function,randomized evaluation-interpolation method
Input — R = a/f q a rational function in Q ( t )( x ), homogeneous of degree − n − x Output — L ∈ K h δ i an annihilating operator of H R , with high probability procedure PicardFuchs ( a/f q ) loop p ← random prime numberCompute M , ρ and Mat M m , as defined in §7.2.2, over F p ( t ) by repeatedevaluation of t and rational interpolation.Compute ρ , ρ , . . . over F p ( t ), with ρ i +1 = ρ δi − m ( ρ i ), until finding arelation ρ n + P n − i =0 a i ρ i = 0 over F p ( t ).Using the Chinese remainder theorem and computations modulo previousvalues of p , try to lift the a i in Q ( t ). if possible thenreturn the lifting.7.2.2. The matrix of δ . Let J r be the image [ T n +1 ] r of the reduction map [ ] r . Byconstruction, the reduction [ ] r is idempotent, that is [ α ] r = α for all α ∈ J r . Theevaluation-interpolation algorithm relies on the following property of the reductionmap [ ] r : Proposition 33.
The space J r is stable under component-wise differentiation.Sketch of the proof. This is a consequence of the fact that J r is generated by mono-mials. More precisely, let E be the, finite of infinite, minimal sequence ( b , . . . )of monomials of T n +1 which generates T n +1 / ker[ ] r ; minimal with respect to thelexicographic order on sequences of monomials, where the monomials are comparedwith ≺ . Then E is a basis of J r containing only monomials. (cid:3) As a consequence [ δ ( ρ )] r = ρ δ − [ f δ ρ ] r , for all ρ ∈ J r .Let M be the least set of monomials of T n +1 such that Vect M contains ρ andis stable under the map m : ρ [ f δ ρ ] r , and let B be the matrix in Q ( t ) M×M ofthe map m | Vect M in the basis M . For generic values of p and u , the basis M ,the matrix ν ( B ) and ν ( ρ ) are all computable using only operations in F p , oncegiven ν ( f ), ν ( f δ ) and ν ( α ). Once M , B and ρ are reconstructed over Q ( t ), the ρ i are easily computed with ρ i +1 = ρ δi − m ( ρ i ), and the minimal operator L = P i a i ( t ) δ i such that P i a i ( t ) ρ i = 0 can be deduced. It seems to be a good idea to reconstruct B and ρ over F p ( t ) and compute L modulo p , and only then to use several moduli toreconstruct L over Q ( t ). The full procedure is summarized by Algorithm 4.7.2.3. Estimation of the probability of success.
Let M , ρ and A = Mat M m asin §7.2.2, computed over Q ( t ). For some u in Q and some prime p , let M , ρ and A be the analogues computed over F p . It is not hard to check that ν (ker[ ] r )equals ker[ ] νr , where ν (ker[ ] r ) is the set of all α in ker[ ] r such that ν ( α ) is defined.Let α be an element of T n +1 , whose coefficients are polynomials in t with integercoefficients. Do we have ν ([ α ] r ) = [ ν ( α )] νr ? The fact that J r is generated by mono-mials implies that [ α ] r equals rem( α, ker[ ] r ), and that [ ν ( α )] νr equals rem( α, ker[ ] νr ).The equality is equivalent to ν (rem( α, ker[ ] r )) = rem( α, ν (ker[ ] r )). A sufficient OMPUTING PERIODS OF RATIONAL INTEGRALS 27 condition is that the set L of leading monomials of elements of ker[ ] r equals theset L of leading monomials of ν (ker[ ] r ). Since M (resp. M ) is the complementof L (resp. L ) in the set of all monomials of T n +1 , we obtain Lemma 34. If M = M then A = ν ( A ) and ρ = ν ( ρ ) . Let P be the probability that M = M . Assume for simplicity that deg α (cid:54) nN and that J r is included in F n T n +1 . Let V be the subspace ker[ ] r ∩ F n +1 T n +1 andlet B be an echelonized basis of V , formed by elements of T n +1 whose coefficientsare in Z [ t ]. For the above equalities to hold, it is enough that for all b in B , theevaluation ν (lc b ) of the leading coefficient of b is not zero.Under the assumption, somewhat excessive, that for random p and u the ν (lc b ),with b ∈ B are independent and uniformly distributed in F p , the probability P equals (1 − p ) B . Of course B (cid:54) dim F n +1 T n +1 anddim F n +1 T n +1 = n +1 X q =0 (cid:18) qN − n (cid:19) (cid:54) ( n + 3 / n +1 N n ( n + 1)! . So that(9) P (cid:62) (cid:18) − p (cid:19) e n N n (cid:62) exp (cid:18) − e n N n p (cid:19) . So we will choose p significantly bigger than e n N n to have P (cid:28)
1. The set M is notcomputed, so it is not possible to compare it with M . However, we can comparethe different M obtained for different values of p and u . Typically, most of themwill be mutually equal—and hopefully equal to M —and a few will differ. We simplydrop the pairs ( p, u ) giving degenerated specialisation M .7.3. Computing partial certificates.
Recall that if
L ∈ K h δ i is an annihilatingoperator of H a/f , a certifate for L is a sequence C , . . . , C n of rational functionsin K [ x , f ] such that L ( a/f ) = n X i =0 ∂ i C i . As already mentioned, a certificate is desirable because is allows to check a posteriori in a simple way that L annihilates H a/f , idependently of the algorithm used toobtain L . However, a certificate is typically huge [9, Rem. 11] and computing a oneis necessarily very costly. A compromise is possible: we may compute a certificatefor each reduction ρ k , as a β k ∈ T n such that(10) ρ k = ( α + D f β if k = 0 δ ( ρ k − ) + D f β k if k (cid:62) . Thus, to check that the output L = P nk =0 a k δ k of Algorithm 4 annihilates H a/f ,it is enough to check Equation 10 for k (cid:54) r and to check that P k a k ρ k = 0.The first checks imply that ρ k ≡ δ k α modulo D f ( T n ), and the last one impliesthat L ( α ) ∈ D f ( T n ), and thus that L annihilates H a/f . Since the ρ k ’s are in F n T n +1 ,the β k are in F n + r T n which ensures that their size is kept reasonnable.It is possible to modify Algorithm 4 to compute these certificates β k . Withthe notations of §7.2.2, it amounts to compute β ∈ T n such that α = ρ + D f β ,and to compute some γ µ ∈ T n , for µ ∈ M , such that m ( µ ) = f δ µ + D f ( γ µ ). Since ρ k − = δ ( ρ k ) + m ( ρ k ), it is possible to compute the β k ’s as linear combinationsof the γ µ ’s.In the evaluation-interpolation scheme, it is possible to compute β and the γ µ ’sover F p , to reconstruct them over F p ( t ), then to compute the β k ’s over F p ( t ) andto reconstruct them over Q ( t ). Of course, it comes at an additional cost but apreliminary implementation seems to show that this cost is reasonnable.8. Application to periods arising from mirror symmetry
Batyrev and Kreuzer [4] have recently constructed a family of 210 smooth Calabi–Yau varieties of dimension three with Hodge number h , equal to one. Their methodis based on toric varieties of reflexive polytopes. To each variety is associated aone-parameter mirror family of varieties and we look for the Picard-Fuchs equationof a distinguished principal period. This computation is the first step toward thecomputation of other important invariants, like, mirror maps, instanton numbers,etc . The 210 varieties gather together into 68 different classes of diffeomorphicmanifolds [4, table 3]. The principal periods associated to diffeomorphic varietiesneed not coincide but they are typically expected to differ only by a rational changeof variable.In concrete terms, we look for a differential equation satisfied by periods ofrational integrals in the form(11) F ( t ) def = I γ − tg ( x , . . . , x ) d x x d x x d x x d x x , where g is a Laurent polynomial and the integral is taken over the cycle γ definedby | x i | = ε , with ε a small positive real number. Here g is P v x v , where the sumranges over the vertices of a reflexive lattice polytope. For the 210 polytopes underconsideration, Batyrev and Kreuzer claim that F ( t ) satisfies a linear differentialequation of order 4, as a consequence of h , being 1. Moreover, this differentialequation should have maximally unipotent monodromy at t = 0.A power series expansion of the integrand with respect to t shows that(12) F ( t ) = X n ct( g n ) t n , where ct( g n ) stands for the constant term of f n . Batyrev and Kreuzer have computedPicard-Fuchs operators for topologies guessing method presented in the introduction: they computed the power seriesexpansion of F ( t ), using equation (12), until they reached a degree d such that theycould find a non-zero solution to the equation X i =0 d X j =0 a i,j t j θ i · F ( t ) = O ( t d +1)+1 ) . The issue with this technique is not the reconstruction step which can be doneefficiently—with respect to the size of the computed operator—but the computationof the power series expansion: the number of monomials in g k is Θ( k ), so thecomputation of N terms of F ( t ) with this technique take Θ( N ) operations in Z ,and we may add an order of magnitude to reflect the binary complexity. For an introduction to the topic, see [15; 5].
OMPUTING PERIODS OF RATIONAL INTEGRALS 29
Metelitsyn [36] computed four more equations for topologies
Gpu programming. Moreover, Almkvist [1] reports that Straten, Metelitsyn and Schömerhave computed one operator for the topology Minimal equation and crosschecking.
The equations obtained from thealgorithm are not always minimal, for two reasons. Firstly they were obtainedwith r = 2 but a higher value might have caught a lower order equation. Secondly,the algorithm computes an annihilating operator of all the periods of a given rationalfunction; a period associated to a given cycle may satisfy a lower order equation.Nevertheless, once any differential equation L for F ( t ) is obtained, it is easyto compute efficiently thousands of terms of its power series expansion: the rela-tion L ( F ) = 0 translates into a linear recurrence relation on the coefficients of thepower series expansion and the initial conditions are given by Equation (12). Thuswe may try to reconstruct the minimal equation L . By contrast to the guessingmethod, the reconstructed equation L can be proven correct: it is enough to checkthat it is a right divisor of L , and that it annihilates the first few terms of F ( t ).If the power series expansion does not reveal a lower order differential equation,we may conjecture that L is minimal. Proving it may be done using methods byvan Hoeij [30], see §8.2.2 for an example.Since Algorithm 4 is randomized, it is desirable to have criteria to crosscheckthe result. The Picard-Fuchs equations of periods of rational integrals are knownto have strong arithmetic properties: regular singularities with rational exponentsand nilpotent p -curvature for all prime p , with a finite number of exceptions [31].Checking these properties is a good confirmation of the correctness of the output:these properties are so strong that a bad reconstruction would most probably breakthem. In addition, the computation of many terms of the power series expansionof F ( t ) using a annihilating operator L can also be used as a crosschecking: if thecoefficients computed are all integers, as expected in view of Equation (12), this isalso strong indication that the operator is indeed correct.8.2. Description of the results.
In depth treatment is a work in progress withJean-Marie Maillard. This section presents two examples.
The results are available at http://pierre.lairez.fr/supp/periods . Up to the maximal integral root of the indicial polynomial at zero of the right quotient of L by L . There are two numberings. The first one, used in Table 3 of [4], numbers the 68 differenttopologies, ordered by increasing h , number, covering the 210 smooth Calabi-Yau threefoldswith Picard number 1. The second one, used in the database http://hep.itp.tuwien.ac.at/~kreuzer/math/0802 , numbers in the form v x.y the 198849 reflexive 4D polytopes satisfying anextra property. The letter x indicates the number of vertices. Topology
For the period (11) with g = wxyz + wxy + 1 wxy + wxz + 1 wxz + wyz + zwy + wy + 1 wy + 1 wz + w + 1 w + xzy + yxz + 1 xy + xz + 1 xz + x + 1 x + zy + yz + y + 1 y + z + 1 z , where the first few terms of the power series expansion are F ( t ) = 1 + 22 t + 204 t + 3474 t + 57000 t + 1031080 t + 19368720 t + O ( t ) . I have computed the following Picard-Fuchs equation t (7 t + 1) (25 t − (2 t + 1) (101 t + 43) (3 t + 1) ∂ + 2 t (7 t + 1)(25 t − t + 1) (101 t + 43) (3 t + 1) (848400 t + 1012956 t + 413041 t + 62473 t + 1819 t − ∂ + t (7 t + 1)(25 t − t + 1)(101 t + 43)(3 t + 1)(4627173600 t + 10573386192 t + 10004988192 t + 5027593832 t + 1423146511 t + 219009622 t + 15394840 t + 182234 t − ∂ + (7 t + 1)(25 t − t + 1)(101 t + 43)(3 t + 1)(6169564800 t + 13061530080 t + 11311205016 t + 5112706620 t + 1268815538 t + 164341135 t + 9051543 t + 74605 t − ∂ + 8 t (7 t + 1)(25 t − t + 1)(101 t + 43)(3 t + 1)(192798900 t + 375787872 t + 294032949 t + 116697469 t + 24254991 t + 2406495 t + 81356) , or, with θ = t∂ , in a form which highlights the maximally unipotent monodromy,1849 θ − tθ (142 θ + 890 θ + 574 θ + 129) − t (647269 θ + 2441818 θ + 3538503 θ + 2423953 θ + 650848) − t (7200000 θ + 34423908 θ + 65337898 θ + 57379329 θ + 19251960) − t (37610765 θ + 220029964 θ + 499781264 θ + 511393545 θ + 194039928) − t ( θ + 1)(54978121 θ + 324737370 θ + 665066226 θ + 466789876) − t ( θ + 2)( θ + 1)(185181547 θ + 915931425 θ + 1176131796) − t (138979 θ + 413408)( θ + 3)( θ + 2)( θ + 1) − t ( θ + 4)( θ + 3)( θ + 2)( θ + 1) . This equation satisfies the conditions given by Almkvist, Enckevort, Straten, andZudilin [2] and it is not in their database [44]. The computation took 80 secondsand 30 megabytes of memory on a laptop.Note that formula (11), and homogeneization, give a rational function a/f with f of degree 8 with respect to the integration variables. The change of variableswhich maps x to 1 /x and w to w/y lowers this degree down to 5. This improvesdramatically the computation time. This kind of monomial substitution can befound by random trials and errors. Among the substitutions that lead to degree 5,some are better than others in terms of computation time; but this seems hard topredict. EFERENCES 31
Topology
For the period (11) with f = 1 w + w + 1 x + wx + x + xw + 1 y + wy + 1 xy + wxy + y + yw + xyw + 1 z + wz + xz + 1 yz + wyz + wxyz + z + zw + zx + zwx , where the first few terms of the power series expansion are F ( t ) = 1 + 18 t + 138 t + 2070 t + 29040 t + 452610 t + 7308000 t + O ( t ) , I have computed an annihilating operator of order 6 and degree 29, let us denoteit L , which is too large to be reproduced here. The operator is not of order 4and has not maximally unipotent monodromy. Is it the minimal equation of F ( t )?Van Hoeij has proved that if L admits a right factor of order 4 then the degreeof the coefficients of this factor is at most 88. Thus, admitting that L is indeedan annihilating operator of F ( t ), if the minimal annihilating operator of F ( t ) hasorder 4, it would have degree at most 88. Zero being the only solution to the systemof linear equations X i =0 88 X j =0 a i,j t j f ( i ) ( t ) = O ( t ) , where the unknowns are the a i,j , this shows that the minimal annihilating operatorof F ( t ) is not of order 4. The argument holds for orders 1, 2, 3 and 5 with respectivedegree bounds 10, 16, 45 and 125. This is rather surprising since it contradicts theclaims of Batyrev and Kreuzer. The topology References [1] Gert Almkvist. “The art of finding Calabi-Yau differential equations”. In:
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Inria Saclay, équipe Specfun, France
Current address : Pierre Lairez — Fäk. II, Sekr. 3-2 — Technische Universität zu Berlin —Straße des 17. Juni 136 — 10623 Berlin — Deutschland
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