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Rational-functions telescopers: blending creative telescoping with hermite reduction

Rational-functions telescopers: blending creative telescoping with hermite reduction Bostan, Chen, Chyzak, Li Rational-Functions Telescopers: Blending Creative Telescoping with Hermite Reduction⋆ Alin Bostan1 , Shaoshi Chen1,2 , Fr´d´ric Chyzak1 , and Ziming Li2 e e 1 Algorithms project, INRIA Paris-Rocquencourt, France. 2 Key Lab of Mathematics Mechanization, AMSS, Beijing, China. The long-term goal initiated in this work is to obtain fast algorithms and implementations for de nite integration in the framework of (di €erential) creative telescoping introduced in [1]. Our approach bases on complexity analysis, by obtaining tight degree bounds on the various di €erential operators and polynomials involved in the method and its variants. To make the problem more tractable, we restrict in this work to the integration of rational functions. Indeed, by considering a more constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction [3]. The rational class already has many applications, for instance in combinatorics, where many non-trivial problems are encoded as diagonals of rational formal power series, themselves expressible as integrals. Given a rational function f ˆ K(x, y) (in characteristic zero), the core of (di €erential) creative telescoping consists in obtaining a linear di €erential operator L in K(x) Dx and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGSAM Bulletin Association for Computing Machinery

Rational-functions telescopers: blending creative telescoping with hermite reduction

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Publisher
Association for Computing Machinery
Copyright
The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc.
Subject
Differential-algebraic equations
ISSN
0163-5824
DOI
10.1145/1838599.1838603
Publisher site
See Article on Publisher Site

Abstract

Bostan, Chen, Chyzak, Li Rational-Functions Telescopers: Blending Creative Telescoping with Hermite Reduction⋆ Alin Bostan1 , Shaoshi Chen1,2 , Fr´d´ric Chyzak1 , and Ziming Li2 e e 1 Algorithms project, INRIA Paris-Rocquencourt, France. 2 Key Lab of Mathematics Mechanization, AMSS, Beijing, China. The long-term goal initiated in this work is to obtain fast algorithms and implementations for de nite integration in the framework of (di €erential) creative telescoping introduced in [1]. Our approach bases on complexity analysis, by obtaining tight degree bounds on the various di €erential operators and polynomials involved in the method and its variants. To make the problem more tractable, we restrict in this work to the integration of rational functions. Indeed, by considering a more constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction [3]. The rational class already has many applications, for instance in combinatorics, where many non-trivial problems are encoded as diagonals of rational formal power series, themselves expressible as integrals. Given a rational function f ˆ K(x, y) (in characteristic zero), the core of (di €erential) creative telescoping consists in obtaining a linear di €erential operator L in K(x) Dx and

Journal

ACM SIGSAM BulletinAssociation for Computing Machinery

Published: Jul 29, 2010

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