## Ongoing Projects

### Symbolic-Numeric Techniques for Genus Computation and Parametrization [DK9]

## Software

desing homepage ...

CASA is a special-purpose system for computational algebra and constructive algebraic geometry. The system has been developed since 1990, and is the ongoing product of the Computer Algebra Group under the direction of Prof. Winkler. It is built on the ...

CharSet is an Aldor package written by Christian Aistleitner for differential characteristic set computations. CharSet comes with generic implementations of reduction, Gröbner bases, and differential characteristic set algorithms. Interfaces to the command line, Mathematica and Maple are included. ...

PGB is a software package for computing parametric Gröbner bases and related objects in several domains. It is implemented in the computer algebra system Risa/Asir by Katsusuke Nabeshima. ...

## Publications

### 2018

### Rational General Solutions of Systems of First-Order Partial Differential Equations

#### Georg Grasegger, Alberto Lastra, J. Rafael Sendra, Franz Winkler

Journal of Computational and Applied Mathematics 331, pp. 88-103. 2018. ISSN: 0377-0427.**article**{RISC5509,

author = {Georg Grasegger and Alberto Lastra and J. Rafael Sendra and Franz Winkler},

title = {{Rational General Solutions of Systems of First-Order Partial Differential Equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {331},

pages = {88--103},

isbn_issn = {ISSN: 0377-0427},

year = {2018},

refereed = {yes},

length = {16}

}

### Deciding the Existence of Rational General Solutions for First-Order Algebraic ODEs

#### N.T. Vo, G. Grasegger, F. Winkler

Journal of Symbolic Computation 87, pp. 127-139. 2018. ISSN 0747-7171.**article**{RISC5589,

author = {N.T. Vo and G. Grasegger and F. Winkler},

title = {{Deciding the Existence of Rational General Solutions for First-Order Algebraic ODEs}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {87},

pages = {127--139},

isbn_issn = {ISSN 0747-7171},

year = {2018},

refereed = {yes},

length = {12}

}

### A Computable Extension for Holonomic Functions: DD-Finite Functions

#### Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation, pp. -. 2018. ISSN 0747-7171. accepted.**article**{RISC5731,

author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {ISSN 0747-7171},

year = {2018},

note = {accepted},

refereed = {yes},

length = {0}

}

### The Number of Realizations of a Laman Graph

#### Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]**article**{RISC5700,

author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{The Number of Realizations of a Laman Graph}},

language = {english},

journal = {SIAM Journal on Applied Algebra and Geometry},

volume = {2},

number = {1},

pages = {94--125},

isbn_issn = {2470-6566},

year = {2018},

refereed = {yes},

length = {32},

url = {https://doi.org/10.1137/17M1118312}

}

### 2017

### Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

#### Christoph Fuerst, Alexander Levin

In: Mathematics in Computer Science, W. Koepf (ed.), pp. 1-11. 2017. 1661-8289.**inproceedings**{RISC5432,

author = {Christoph Fuerst and Alexander Levin},

title = {{Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules}},

booktitle = {{Mathematics in Computer Science}},

language = {english},

pages = {1--11},

isbn_issn = {1661-8289},

year = {2017},

editor = {W. Koepf},

refereed = {yes},

length = {11}

}

### An Algebraic-Geometric Method for Computing Zolotarev Polynomials

#### Georg Grasegger, N. Thieu Vo

In: Proceedings of the 2017 international symposium on symbolic and algebraic computation (ISSAC), Burr, M. (ed.), pp. 173-180. 2017. ACM Press, New York, ISBN: 978-1-4503-5064-8.**inproceedings**{RISC5510,

author = {Georg Grasegger and N. Thieu Vo},

title = {{An Algebraic-Geometric Method for Computing Zolotarev Polynomials}},

booktitle = {{Proceedings of the 2017 international symposium on symbolic and algebraic computation (ISSAC)}},

language = {english},

pages = {173--180},

publisher = {ACM Press},

address = {New York},

isbn_issn = {ISBN: 978-1-4503-5064-8},

year = {2017},

editor = {Burr and M.},

refereed = {yes},

length = {8}

}

### Computing the number of realizations of a Laman graph

#### Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]**inproceedings**{RISC5478,

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{Computing the number of realizations of a Laman graph}},

booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},

language = {english},

abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},

volume = {61},

pages = {207--213},

isbn_issn = {ISSN 1571-0653},

year = {2017},

editor = {Vadim Lozin},

refereed = {yes},

keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},

length = {7},

conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},

url = {http://www.koutschan.de/data/laman/}

}

### 2016

### Axiomatic Description of Gröbner Reduction

#### Christoph Fuerst

RISC, JKU Linz. PhD Thesis. December 2016. [pdf]**phdthesis**{RISC5388,

author = {Christoph Fuerst},

title = {{Axiomatic Description of Gröbner Reduction}},

language = {english},

year = {2016},

month = {December},

translation = {0},

school = {RISC, JKU Linz},

length = {154}

}

### A solution method for autonomous first-order algebraic partial differential equations

#### G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Journal of Computational and Applied Mathematics 300, pp. 119-133. 2016. 0377-0427. [url]**article**{RISC5202,

author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {300},

pages = {119--133},

isbn_issn = {0377-0427},

year = {2016},

refereed = {yes},

length = {15},

url = {http://dx.doi.org/10.1016/j.cam.2015.12.030}

}

### A decision algorithm for rational general solutions of first-order algebraic ODEs

#### G. Grasegger, N.T. Vo, F. Winkler

In: Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016), Universidad de la Rioja, J. Heras and A. Romero (eds.) (ed.), pp. 101-104. 2016. 978-84-608-9024-9.**inproceedings**{RISC5400,

author = {G. Grasegger and N.T. Vo and F. Winkler},

title = {{A decision algorithm for rational general solutions of first-order algebraic ODEs}},

booktitle = {{Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016)}},

language = {english},

pages = {101--104},

isbn_issn = {978-84-608-9024-9},

year = {2016},

editor = {Universidad de la Rioja and J. Heras and A. Romero (eds.)},

refereed = {yes},

length = {4}

}

### Representation of hypergeometric products in difference rings

#### E.D. Ocansey, C. Schneider

ACM Communications in Computer Algebra 50(4), pp. 161-163. 2016. ISSN 1932-2240 . Extended abstract of the poster presentation at ISSAC 2016. [pdf]**article**{RISC5316,

author = {E.D. Ocansey and C. Schneider},

title = {{Representation of hypergeometric products in difference rings}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {4},

pages = {161--163},

isbn_issn = {ISSN 1932-2240 },

year = {2016},

note = {Extended abstract of the poster presentation at ISSAC 2016},

refereed = {yes},

length = {3}

}

### Rational and Algebraic Solutions of First-Order Algebraic ODEs

#### N. Thieu Vo

Research Institute for Symbolic Computation. PhD Thesis. 2016. [pdf]**phdthesis**{RISC5399,

author = {N. Thieu Vo},

title = {{Rational and Algebraic Solutions of First-Order Algebraic ODEs}},

language = {english},

year = {2016},

translation = {0},

school = {Research Institute for Symbolic Computation},

length = {93}

}

### 2015

### Computation of Dimension in Filtered Free Modules by Gröbner Reduction

#### Christoph Fuerst, Guenter Landsmann

In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM (ed.), Proceedings of ISSAC '15, pp. 181-188. 2015. 978-1-4503-3435-8. [url]**inproceedings**{RISC5154,

author = {Christoph Fuerst and Guenter Landsmann},

title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},

booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {181--188},

isbn_issn = {978-1-4503-3435-8},

year = {2015},

editor = {ACM},

refereed = {yes},

length = {8},

conferencename = {ISSAC '15},

url = {http://doi.acm.org/10.1145/2755996.2756680}

}

### Symbolic Solutions of First-Order Algebraic ODEs

#### G. Grasegger, F. Winkler

In: Computer algebra and polynomials, J. Gutierrez, J. Schicho, M. Weimann (ed.), Lecture Notes in Computer Science 8942, pp. 94-104. 2015. Springer International Publishing, ISSN 0302-9743. [url]**inproceedings**{RISC5018,

author = {G. Grasegger and F. Winkler},

title = {{Symbolic Solutions of First-Order Algebraic ODEs}},

booktitle = {{Computer algebra and polynomials}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {8942},

pages = {94--104},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {J. Gutierrez and J. Schicho and M. Weimann},

refereed = {yes},

length = {11},

url = {http://dx.doi.org/10.1007/978-3-319-15081-9_5}

}

### Symbolic solutions of first-order algebraic differential equations

#### Georg Grasegger

Johannes Kepler University Linz. PhD Thesis. 06 2015. [url]**phdthesis**{RISC5160,

author = {Georg Grasegger},

title = {{Symbolic solutions of first-order algebraic differential equations}},

language = {english},

year = {2015},

month = {06},

translation = {0},

school = {Johannes Kepler University Linz},

length = {154},

url = {http://epub.jku.at/obvulihs/content/titleinfo/753082}

}

### Automated Reasoning in Reduction Rings using the Theorema System

#### A. Maletzky

In: Computer Algebra in Scientific Computing, Vladimir P. Gerdt and Wolfram Koepf and Werner M. Seiler and Evgenii V. Vorozhtsov (ed.), Proceedings of CASC 2015 (September 14-18, Aachen, Germany), LNCS 9301, pp. 305-319. 2015. Springer-Verlag Berlin Heidelberg, ISSN 0302-9743. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-24021-3_23. [url] [pdf]**inproceedings**{RISC5151,

author = {A. Maletzky},

title = {{Automated Reasoning in Reduction Rings using the Theorema System}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

series = {LNCS},

volume = {9301},

pages = {305--319},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn_issn = {ISSN 0302-9743},

year = {2015},

note = {The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-24021-3_23},

editor = {Vladimir P. Gerdt and Wolfram Koepf and Werner M. Seiler and Evgenii V. Vorozhtsov},

refereed = {yes},

length = {15},

conferencename = {CASC 2015 (September 14-18, Aachen, Germany)},

url = {http://dx.doi.org/10.1007/978-3-319-24021-3_23}

}

### A Note on a Problem Proposed by Kim and Lisonek

#### Cristian-Silviu Radu

In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho and Martin Weimann (ed.), Lecture notes in Computer Science , pp. 151-156. 2015. Springer, 978-3-319-15080-2.**incollection**{RISC5156,

author = {Cristian-Silviu Radu},

title = {{A Note on a Problem Proposed by Kim and Lisonek}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

series = {Lecture notes in Computer Science},

pages = {151--156},

publisher = {Springer},

isbn_issn = { 978-3-319-15080-2},

year = {2015},

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

refereed = {yes},

length = {6}

}

### Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one

#### A. Lastra, J.R. Sendra, L.X.C. Ngô, F. Winkler

Publ.Math.Debrecen(86/1-2), pp. 49-69. 2015. 0033-3883.**article**{RISC5204,

author = {A. Lastra and J.R. Sendra and L.X.C. Ngô and F. Winkler},

title = {{Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one}},

language = {english},

journal = {Publ.Math.Debrecen},

number = {86/1-2},

pages = {49--69},

isbn_issn = {0033-3883},

year = {2015},

refereed = {yes},

length = {21}

}

### Birational transformations preserving rational solutions of algebraic ordinary differential equations

#### L.X.C. Ngô, J.R. Sendra, F. Winkler

J. Computational and Applied Mathematics(286), pp. 114-127. 2015. 0377-0427.**article**{RISC5205,

author = {L.X.C. Ngô and J.R. Sendra and F. Winkler},

title = {{Birational transformations preserving rational solutions of algebraic ordinary differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {286},

pages = {114--127},

isbn_issn = {0377-0427},

year = {2015},

refereed = {yes},

length = {14}

}

### Algebraic General Solutions of First Order Algebraic ODEs

#### N. T. Vo, F. Winkler

In: Computer Algebra in Scientific Computing, Vladimir P. Gerdt et. al. (ed.), Lecture Notes in Computer Science 9301, pp. 479-492. 2015. Springer International Publishing, ISSN 0302-9743. [url]**inproceedings**{RISC5194,

author = {N. T. Vo and F. Winkler},

title = {{Algebraic General Solutions of First Order Algebraic ODEs}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.},

series = {Lecture Notes in Computer Science},

volume = {9301},

pages = {479--492},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {Vladimir P. Gerdt et. al.},

refereed = {yes},

length = {14},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}

}