# Multivariate Symbolic Asymptotics [DK13]

### Project Lead

### Project Duration

01/10/2011 - 31/12/2015### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2016

[Kauers]

### Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra

#### Manuel Kauers, Peter Paule, Greg Reid

ACM Communications in Computer Algebra 50(Issue 1), pp. 27-34. March 2016. 1932-2240. [pdf]@

author = {Manuel Kauers and Peter Paule and Greg Reid},

title = {{Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {Issue 1},

pages = {27--34},

isbn_issn = {1932-2240},

year = {2016},

month = {March},

refereed = {no},

length = {8}

}

**article**{RISC5284,author = {Manuel Kauers and Peter Paule and Greg Reid},

title = {{Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {Issue 1},

pages = {27--34},

isbn_issn = {1932-2240},

year = {2016},

month = {March},

refereed = {no},

length = {8}

}

### 2015

[Huang]

### An Improved Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms

#### Shaoshi Chen, Hui Huang, Manuel Kauers, Ziming Li

arxiv. Technical report no. 1501.04668, 2015. [pdf]@

author = {Shaoshi Chen and Hui Huang and Manuel Kauers and Ziming Li},

title = {{An Improved Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms}},

language = {english},

abstract = { The Abramov-Petkovsek reduction computes an additive decomposition of ahypergeometric term, which extends the functionality of the Gosper algorithmfor indefinite hypergeometric summation. We improve the Abramov-Petkovsekreduction so as to decompose a hypergeometric term as the sum of a summableterm and a non-summable one. The improved reduction does not solve anyauxiliary linear difference equation explicitly. It is also moreefficient thanthe original reduction according to computational experiments.Based on thisreduction, we design a new algorithm to compute minimal telescopers forbivariate hypergeometric terms. The new algorithm can avoid the costlycomputation of certificates.},

number = {1501.04668},

year = {2015},

institution = {arxiv},

length = {8}

}

**techreport**{RISC5102,author = {Shaoshi Chen and Hui Huang and Manuel Kauers and Ziming Li},

title = {{An Improved Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms}},

language = {english},

abstract = { The Abramov-Petkovsek reduction computes an additive decomposition of ahypergeometric term, which extends the functionality of the Gosper algorithmfor indefinite hypergeometric summation. We improve the Abramov-Petkovsekreduction so as to decompose a hypergeometric term as the sum of a summableterm and a non-summable one. The improved reduction does not solve anyauxiliary linear difference equation explicitly. It is also moreefficient thanthe original reduction according to computational experiments.Based on thisreduction, we design a new algorithm to compute minimal telescopers forbivariate hypergeometric terms. The new algorithm can avoid the costlycomputation of certificates.},

number = {1501.04668},

year = {2015},

institution = {arxiv},

length = {8}

}

[Koutschan]

### Integral D-Finite Functions

#### Manuel Kauers, Christoph Koutschan

arxiv. Technical report no. 1501.03691, 2015. [pdf]@

author = {Manuel Kauers and Christoph Koutschan},

title = {{Integral D-Finite Functions}},

language = {english},

abstract = { We propose a differential analog of the notion of integral closure ofalgebraic function fields. We present an algorithm for computing theintegralclosure of the algebra defined by a linear differential operator. Ouralgorithmis a direct analog of van Hoeij's algorithm for computing integral bases ofalgebraic function fields.},

number = {1501.03691},

year = {2015},

institution = {arxiv},

length = {8}

}

**techreport**{RISC5101,author = {Manuel Kauers and Christoph Koutschan},

title = {{Integral D-Finite Functions}},

language = {english},

abstract = { We propose a differential analog of the notion of integral closure ofalgebraic function fields. We present an algorithm for computing theintegralclosure of the algebra defined by a linear differential operator. Ouralgorithmis a direct analog of van Hoeij's algorithm for computing integral bases ofalgebraic function fields.},

number = {1501.03691},

year = {2015},

institution = {arxiv},

length = {8}

}

### 2014

[Johansson]

### Ore Polynomials in Sage

#### Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson

In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho, Martin Weimann (ed.), Lecture Notes in Computer Science , pp. ?-?. 2014. tba. [ps] [pdf]@

author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

**inproceedings**{RISC4944,author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

[Kauers]

### Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph

#### Manuel Kauers, Ryan ODonnell, Li-Yang Tan, Yuan Zhou

In: Proceedings of SODA'14, tba (ed.), pp. ?-?. 2014. tba. [pdf]@

author = {Manuel Kauers and Ryan ODonnell and Li-Yang Tan and Yuan Zhou},

title = {{Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph}},

booktitle = {{Proceedings of SODA'14}},

language = {english},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {tba},

refereed = {yes},

length = {15}

}

**inproceedings**{RISC4829,author = {Manuel Kauers and Ryan ODonnell and Li-Yang Tan and Yuan Zhou},

title = {{Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph}},

booktitle = {{Proceedings of SODA'14}},

language = {english},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {tba},

refereed = {yes},

length = {15}

}

[Kauers]

### On the length of integers in telescopers for proper hypergeometric terms

#### Manuel Kauers, Lily Yen

Journal of Symbolic Computation, pp. ?-?. 2014. ISSN 0747-7171. to appear. [pdf] [ps]@

author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {?--?},

isbn_issn = {ISSN 0747-7171},

year = {2014},

note = {to appear},

refereed = {yes},

length = {15}

}

**article**{RISC4955,author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {?--?},

isbn_issn = {ISSN 0747-7171},

year = {2014},

note = {to appear},

refereed = {yes},

length = {15}

}

[Kauers]

### Computer Algebra

#### Manuel Kauers

In: Handbook of Combinatorics, Miklos Bona (ed.), pp. ?-?. 2014. Taylor and Francis, tba.@

author = {Manuel Kauers},

title = {{Computer Algebra}},

booktitle = {{Handbook of Combinatorics}},

language = {english},

pages = {?--?},

publisher = {Taylor and Francis},

isbn_issn = {tba},

year = {2014},

editor = {Miklos Bona},

refereed = {yes},

length = {59}

}

**incollection**{RISC4956,author = {Manuel Kauers},

title = {{Computer Algebra}},

booktitle = {{Handbook of Combinatorics}},

language = {english},

pages = {?--?},

publisher = {Taylor and Francis},

isbn_issn = {tba},

year = {2014},

editor = {Miklos Bona},

refereed = {yes},

length = {59}

}

[Kauers]

### Bounds for D-Finite Closure Properties

#### Manuel Kauers

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 288-295. 2014. isbn 978-1-4503-2501-1/14/07. [pdf]@

author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {288--295},

isbn_issn = {isbn 978-1-4503-2501-1/14/07},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4989,author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {288--295},

isbn_issn = {isbn 978-1-4503-2501-1/14/07},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

[Kauers]

### Bounds for D-Finite Closure Properties

#### Manuel Kauers

arxiv. Technical report no. 1408.5514, 2014. [pdf]@

author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

language = {english},

number = {1408.5514},

year = {2014},

institution = {arxiv},

length = {8}

}

**techreport**{RISC5041,author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

language = {english},

number = {1408.5514},

year = {2014},

institution = {arxiv},

length = {8}

}

[Kauers]

### Desingularization of Ore Operators

#### Shaoshi Chen, Manuel Kauers, Michael F. Singer

arxiv. Technical report no. 1408.5512, 2014. [pdf]@

author = {Shaoshi Chen and Manuel Kauers and Michael F. Singer},

title = {{Desingularization of Ore Operators}},

language = {english},

number = {1408.5512},

year = {2014},

institution = {arxiv},

length = {11}

}

**techreport**{RISC5042,author = {Shaoshi Chen and Manuel Kauers and Michael F. Singer},

title = {{Desingularization of Ore Operators}},

language = {english},

number = {1408.5512},

year = {2014},

institution = {arxiv},

length = {11}

}

[Kauers]

### On 3-dimensional lattice walks confined to the positive octant

#### Alin Bostan, Mireille Bousquet-Mélou, Manuel Kauers, Stephen Melczer

Arxiv. Technical report no. 1409.3669, 2014. [pdf]@

author = {Alin Bostan and Mireille Bousquet-Mélou and Manuel Kauers and Stephen Melczer},

title = {{On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

abstract = {Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in {0,±1}2: the generating function is D-finite if and only if a certain group associated with the step set is finite.We explore in this paper the analogous problem for 3-dimensional walks confined to the positive octant. The first difficulty is their number: there are 11074225 non-trivial and non-equivalent step sets in {0,±1}3 (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps.We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 -- the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases.On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure, which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations.This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear. },

number = {1409.3669},

year = {2014},

institution = {Arxiv},

length = {36}

}

**techreport**{RISC5054,author = {Alin Bostan and Mireille Bousquet-Mélou and Manuel Kauers and Stephen Melczer},

title = {{On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

abstract = {Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in {0,±1}2: the generating function is D-finite if and only if a certain group associated with the step set is finite.We explore in this paper the analogous problem for 3-dimensional walks confined to the positive octant. The first difficulty is their number: there are 11074225 non-trivial and non-equivalent step sets in {0,±1}3 (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps.We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 -- the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases.On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure, which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations.This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear. },

number = {1409.3669},

year = {2014},

institution = {Arxiv},

length = {36}

}

[Kauers]

### On 3-dimensional lattice walks confined to the positive octant

#### Alin Bostan, Mireille Bousquet-Melou, Manuel Kauers, Stephen Melczer

Annals of Combinatorics, pp. ??-??. 2014. ISSN 0218-0006. to appear. [pdf]@

author = {Alin Bostan and Mireille Bousquet-Melou and Manuel Kauers and Stephen Melczer},

title = {{ On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

journal = {Annals of Combinatorics},

pages = {??--??},

isbn_issn = {ISSN 0218-0006},

year = {2014},

note = {to appear},

refereed = {yes},

length = {36}

}

**article**{RISC5082,author = {Alin Bostan and Mireille Bousquet-Melou and Manuel Kauers and Stephen Melczer},

title = {{ On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

journal = {Annals of Combinatorics},

pages = {??--??},

isbn_issn = {ISSN 0218-0006},

year = {2014},

note = {to appear},

refereed = {yes},

length = {36}

}

[Koutschan]

### A Generalized Apagodu-Zeilberger Algorithm

#### Shaoshi Chen, Manuel Kauers, Christoph Koutschan

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 107-114. 2014. ISBN 978-1-4503-2501-1. [pdf]@

author = {Shaoshi Chen and Manuel Kauers and Christoph Koutschan},

title = {{A Generalized Apagodu-Zeilberger Algorithm}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {107--114},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC5034,author = {Shaoshi Chen and Manuel Kauers and Christoph Koutschan},

title = {{A Generalized Apagodu-Zeilberger Algorithm}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {107--114},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

### 2013

[Kauers]

### On the length of integers in telescopers for proper hypergeometric terms

#### Manuel Kauers and Lily Yen

ArXiv. Technical report no. 1311.3720, 2013. [pdf]@

author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

abstract = {We show that the number of digits in the integers of a creative telescopingrelation of expected minimal order for a bivariate proper hypergeometric termhas essentially cubic growth with the problem size. For telescopers of higherorder but lower degree we obtain a quintic bound. Experiments suggest thatthese bounds are tight. As applications of our results, we give an improvedbound on the maximal possible integer root of the leading coefficient of atelescoper, and the first discussion of the bit complexity of creativetelescoping.},

number = {1311.3720},

year = {2013},

institution = {ArXiv},

length = {20}

}

**techreport**{RISC4849,author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

abstract = {We show that the number of digits in the integers of a creative telescopingrelation of expected minimal order for a bivariate proper hypergeometric termhas essentially cubic growth with the problem size. For telescopers of higherorder but lower degree we obtain a quintic bound. Experiments suggest thatthese bounds are tight. As applications of our results, we give an improvedbound on the maximal possible integer root of the leading coefficient of atelescoper, and the first discussion of the bit complexity of creativetelescoping.},

number = {1311.3720},

year = {2013},

institution = {ArXiv},

length = {20}

}