Existence Problem of Telescopers for Rational Functions in Three Variables

Chen, Shaoshi; Du, Lei; Zhu, Chaochao

doi:10.1145/3326229.3326231

Cited by 8 publications

(4 citation statements)

References 55 publications

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“…Then f has a telescoper of type (D t , S x , D y ) if and only if r does. By Theorem 4.21 in [13] or Theorem 4.43 in [12], we have that r has a telescoper of type (D t , S x , D y ) if and only if for each i with 1 ≤ i ≤ I , either α i is D t -separable in F(t, x ) or β i ∈ F(t ) and α i ∈ F(t, x )(β i ) has a telescoper of type (D t , S x ). The existence problem of telescopers of type (D t , S x ) in F(t, x )(β ) with β ∈ F(t ) has been solved in [16].…”

Section: The Algebraic Casementioning

confidence: 86%

Separability Problems in Creative Telescoping

Chen

Feng

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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For given multivariate functions specified by algebraic, differential or difference equations, the separability problem is to decide whether they satisfy linear differential or difference equations in one variable. In this paper, we will explain how separability problems arise naturally in creative telescoping and present some criteria for testing the separability for several classes of special functions, including rational functions, hyperexponential functions, hypergeometric terms, and algebraic functions. CCS CONCEPTS• Computing methodologies → Algebraic algorithms.

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Section: The Algebraic Casementioning

confidence: 86%

Separability Problems in Creative Telescoping

Chen

Feng

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…Then f has a telescoper of type (D t , S x , D y ) if and only if r does. By Theorem 4.21 in [12] or Theorem 4.43 in [11], we have r has a telescoper of type (D t , S x , D y ) if and only if for each i with 1 ≤ i ≤ I, either α i is D t -separable in F(t, x) or β i ∈ F(t) and α i ∈ F(t, x)(β i ) has a telescoper of type (D t , S x ). The existence problem of telescopers of type (D t , S x ) in F(t, x)(β) with β ∈ F(t) has been solved in [15].…”

Section: The Algebraic Casementioning

confidence: 86%

Some open problems related to creative telescoping

Chen

Kauers

2017

J Syst Sci Complex

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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. In this paper, we present a selection of open problems in this context. We would be curious to hear about any substantial progress on any of these problems.

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“…All of the above work only focused on the problem for bivariate functions of a special class. The criteria on the existence of telescopers beyond the bivariate case were given in [17][18][19]. We will solve in this paper the existence problem of telescopers for general rational functions in several discrete variables.…”

Section: Related Work On Symbolic Summationmentioning

confidence: 99%

Shift Equivalence Testing of Polynomials and Symbolic Summation of Multivariate Rational Functions

Chen¹,

Du²,

Fang³

2022

Preprint

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The Shift Equivalence Testing (SET) of polynomials is deciding whether two polynomials p(x 1 , . . . , x m ) and q(x 1 , . . . , x m ) satisfy the relation p(x 1 + a 1 , . . . , x m + a m ) = q(x 1 , . . . , x m ) for some a 1 , . . . , a m in the coefficient field. The SET problem is one of basic computational problems in computer algebra and algebraic complexity theory, which was reduced by Dvir, Oliverira and Shpilka in 2014 to the Polynomial Identity Testing (PIT) problem. This paper presents a general scheme for designing algorithms to solve the SET problem which includes Dvir-Oliverira-Shpilka's algorithm as a special case. With the algorithms for the SET problem over integers, we give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of isotropy groups of polynomials introduced by Sato in 1960s. Our results can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

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Existence Problem of Telescopers for Rational Functions in Three Variables

Cited by 8 publications

References 55 publications

Separability Problems in Creative Telescoping

Separability Problems in Creative Telescoping

Some open problems related to creative telescoping

Shift Equivalence Testing of Polynomials and Symbolic Summation of Multivariate Rational Functions

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