Creative Telescoping for Parametrised Integration and Summation

Chyzak, Frédéric

doi:10.5802/ccirm.14

Cited by 4 publications

(6 citation statements)

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“…This generalizes what happens for integrals of algebraic functions depending on a parameter h, which always admit a telescoper with rational coefficients in h [12]. These can be algorithmically computed [8], and as they always exist, the algorithms terminate. In the transcendental case, such equations do not always exist, and if they do, it is delicate to bound a priori their order and degree (see section 4).…”

Section: Introductionmentioning

confidence: 55%

Section: Findtelescopermentioning

confidence: 98%

“…When l mod k = 0, the last sum can be combined with the second part, and when l mod k = 0, we have δ = 1 in relation (8). In both cases, such relation would give a non trivial solution of equation (8). Thus if the algorithm returns "None", there is no telescoper of order and degree less than ord, N.…”

Section: Findtelescopermentioning

confidence: 99%

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Symbolic Integration of Hyperexponential 1-Forms

Combot

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We consider the problem of symbolic integration of G(x, y(x))dx where G is rational and y(x) is a non algebraic solution of a differential equation y ′ (x) = F (x, y(x)) with F rational. As y is transcendental, the Galois action generates a family of parametrized integrals I(x, h) = G(x, y(x, h))dx. We prove that I(x, h) is either differentially transcendental or up to parametrization change satisfies a linear differential equation in h with constant coefficients, called a telescoper. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on their order and degree ord, N with complexity Õ(N ω+1 ord ω−1 + Nord ω+3 ). For the specific foliation y = ln x, a more complete algorithm without an a priori bound is presented. Oppositely, non existence of telescoper is proven for a classical planar Hamiltonian system. As an application, we present an algorithm which always finds, if they exist, the Liouvillian solutions of a planar rational vector field, given a bound large enough for some notion of complexity height.

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Section: Introductionmentioning

confidence: 55%

Section: Findtelescopermentioning

confidence: 98%

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Symbolic Integration of Hyperexponential 1-Forms

Combot

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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“…This milder growth is due to the fact that the vanishing locus of the second Symanzik polynomial F is a degree polynomial in P −1 , the motive in ( 19) is associated with a −2-fold Calabi-Yau [22]. One approach for deriving such a system of differential operators is to make use of the creative telescoping algorithm [23,24]. With this algorithm, one can derive a Gröbner basis of differential operators acting on the integrand of the Feynman integral, despite the presence of non-isolated singularities.…”

Section: Computational Challengesmentioning

confidence: 99%

“…The creative telescoping algorithm [23,24] gives the following operators acting on the integrand of (46):…”

Section: Picard-fuchs Operatorsmentioning

confidence: 99%