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Creative telescoping for rational functions using the griffiths

Alin Bostan, Pierre Lairez, Bruno Salvy
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13  
We describe a precise and elementary algorithmic version of the Griffiths-Dwork method for the creative telescoping of rational functions.  ...  Creative telescoping algorithms compute linear differential equations satisfied by multiple integrals with parameters.  ...  Christol for many rewarding discussions, and we thank G. Villard and W. Zhou for communicating their complexity results in linear algebra.  ... 
doi:10.1145/2465506.2465935 dblp:conf/issac/BostanLS13 fatcat:i3zxw62hdbbppnuvjpbx4nsc4i

Some Open Problems related to Creative Telescoping [article]

Shaoshi Chen, Manuel Kauers
2016 arXiv   pre-print
Creative telescoping is the method of choice for obtaining information about definite sums or integrals.  ...  It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still subject of ongoing research.  ...  More recently, Griffiths and Dwork [44, 45, 50, 51] gave a method that works for any number of variables but requires some kind of regularity of the denominator.  ... 
arXiv:1609.03768v1 fatcat:4d256sszknhazct3bazjg5l5ea

Creative Telescoping for Holonomic Functions [chapter]

Christoph Koutschan
2013 Texts & Monographs in Symbolic Computation  
the impact of creative telescoping in numerous contexts.  ...  The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application  ...  Further innovations include an algorithm for hyperexponential functions based on Hermite reduction [19] and new algorithm for rational functions [22] using the Griffiths-Dwork method.  ... 
doi:10.1007/978-3-7091-1616-6_7 fatcat:7kmlabkcejfsvpwcjfi6vwrth4

Reduction-Based Creative Telescoping for Algebraic Functions

Shaoshi Chen, Manuel Kauers, Christoph Koutschan
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions.  ...  This algorithm is based on Trager's Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper.  ...  Singer for helpful discussions, and the anonymous referees for their constructive and helpful comments.  ... 
doi:10.1145/2930889.2930901 dblp:conf/issac/ChenKK16 fatcat:mvfcg5dghnbgrfu2pnggevqj6a

Linear Differential Equations as a Data-Structure [article]

Bruno Salvy
2018 arXiv   pre-print
A lot of information concerning solutions of linear differential equations can be computed directly from the equation.  ...  For instance, in order to evaluate at π a function given by its differential equation and initial conditions using this method, one would use as intermediate points the first rational numbers in the sequence  ...  C Creative telescoping is a method introduced by Zeilberger in the 1990s [5, 132, 133] that computes definite integrals or sums with a free parameter, in the sense that it produces linear differential  ... 
arXiv:1811.08616v1 fatcat:uvtvvzspxzgl5bgjsrgqfi5ygm

Computing the volume of compact semi-algebraic sets [article]

Pierre Lairez, Mohab Safey El Din
2019 arXiv   pre-print
This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods.  ...  Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in n and polynomial in the maximum degree  ...  using the GriffithsDwork method. In ISSAC 2013. ACM, 93–100. [40] J. van der Hoeven. 2001.  ... 
arXiv:1904.11705v1 fatcat:j4vvixbozvajve5zkfd4crrrwu

Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems [article]

Stephen Melczer, Bruno Salvy
2020 arXiv   pre-print
We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques.  ...  To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.  ...  Acknowledgments The authors would like to thank Mohab Safey El Din and Éric Schost for several discussions during the preparation of this work, and the anonymous referees for their close readings and helpful  ... 
arXiv:1905.04187v2 fatcat:urrhplwdzng2tj4abeeufccvby

Sequences, modular forms and cellular integrals [article]

Dermot McCarthy, Robert Osburn, Armin Straub
2018 arXiv   pre-print
It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular  ...  In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.  ...  The second author would like to thank both the Institut des HautesÉtudes Scientifiques and the Max-Planck-Institut für Mathematik for their support and Francis Brown for his encouragement during the initial  ... 
arXiv:1705.05586v2 fatcat:aimvzxsxu5dsvctxu2b2ardjty