## 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

### Varieties of apolar subschemes of toric surfaces

#### Gallet Matteo, Ranestad Kristian, Villamizar Nelly

Ark. Mat. 56(1), pp. 73-99. 2018. ISSN 0004-2080. [url]**article**{RISC5796,

author = {Gallet Matteo and Ranestad Kristian and Villamizar Nelly},

title = {{Varieties of apolar subschemes of toric surfaces}},

language = {english},

journal = {Ark. Mat.},

volume = {56},

number = {1},

pages = {73--99},

isbn_issn = { ISSN 0004-2080},

year = {2018},

refereed = {yes},

length = {27},

url = {https://doi.org/10.4310/ARKIV.2018.v56.n1.a6}

}

### Graphs with Flexible Labelings

#### G. Grasegger, J. Legerský, J. Schicho

Discrete & Computational Geometry, pp. 1-20. 2018. 1432-0444. arXiv:1708.05298. [url]**article**{RISC5803,

author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings}},

language = {english},

journal = {Discrete & Computational Geometry},

pages = {1--20},

isbn_issn = {1432-0444},

year = {2018},

note = {arXiv:1708.05298},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1007/s00454-018-0026-9}

}

### 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}

}

### Graphs with Flexible Labelings allowing Injective Realizations

#### G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2018. [url]**techreport**{RISC5806,

author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings allowing Injective Realizations}},

language = {english},

year = {2018},

institution = {arXiv},

length = {21},

url = {https://arxiv.org/abs/1811.06709}

}

### Rational general solutions of systems of first-order algebraic partial differential equations

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

J. Computational and Applied Mathematics(331), pp. 88-103. 2018. ISSN 0377-0427. [pdf]**article**{RISC5837,

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

title = {{Rational general solutions of systems of first-order algebraic partial differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {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. [pdf]**article**{RISC5838,

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},

number = {87},

pages = {127--139},

isbn_issn = {ISSN 0747-7171},

year = {2018},

refereed = {yes},

length = {13}

}

### Computation of All Rational Solutions of First-Order Algebraic ODEs

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

Advances in Applied Mathematics(98), pp. 1-24. 2018. ISSN 0196-8858. [pdf]**article**{RISC5839,

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

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

language = {english},

journal = {Advances in Applied Mathematics},

number = {98},

pages = {1--24},

isbn_issn = {ISSN 0196-8858},

year = {2018},

refereed = {yes},

length = {24}

}

### 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}

}

### On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs

#### E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

In: ISSAC '18 Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, C. Arreche (ed.), Proceedings of International Symposium on Symbolic and Algebraic Computation 2018, pp. 55-62. 2018. 978-1-4503-5550-6. [url]**inproceedings**{RISC5804,

author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs}},

booktitle = {{ISSAC '18 Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {55--62},

isbn_issn = {978-1-4503-5550-6},

year = {2018},

editor = {C. Arreche},

refereed = {yes},

length = {8},

conferencename = {International Symposium on Symbolic and Algebraic Computation 2018},

url = {https://doi.org/10.1145/3208976.3208994}

}

### On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

#### E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

arXiv. Technical report, 2018. [url]**techreport**{RISC5810,

author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},

language = {english},

year = {2018},

institution = {arXiv},

length = {22},

url = {https://arxiv.org/abs/1811.12800}

}

### Resultants: Algebraic and Differential

#### S. McCallum, F. Winkler

Technical report no. 18-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. August 2018. [pdf]**techreport**{RISC5735,

author = {S. McCallum and F. Winkler},

title = {{Resultants: Algebraic and Differential}},

language = {english},

abstract = {This report summarises ongoing discussions of the authors on the topic of differential resultantswhich have three goals in mind. First, we aim to try to understand existing literature on thetopic. Second, we wish to formulate some interesting questions and research goals based on ourunderstanding of the literature. Third, we would like to advance the subject in one or moredirections, by pursuing some of these questions and research goals. Both authors have somewhatmore background in nondifferential, as distinct from differential, computational algebra. For thisreason, our approach to learning about differential resultants has started with a careful review ofthe corresponding theory of resultants in the purely algebraic (polynomial) case. We try, as faras possible, to adapt and extend our knowledge of purely algebraic resultants to the differentialcase. Overall, we have the hope of helping to clarify, unify and further develop the computationaltheory of differential resultants.},

number = {18-08},

year = {2018},

month = {August},

length = {21},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Differential Resultants, in Recent Advances in Algebra, Numerical Analysis and Statistics

#### S. McCallum, F. Winkler

In: Proc. Internat. Conf. on Mathematics (ICM 2018), R. Bris et al. (ed.), Proceedings of ICM 2018, pp. 1-11. Dezember 2018. Ton Duc Thang University (TDTU), Ho Chi Minh City, Vietnam, ISBN 978-2-7598-9058-3. [url] [pdf]**inproceedings**{RISC5842,

author = {S. McCallum and F. Winkler},

title = {{Differential Resultants, in Recent Advances in Algebra, Numerical Analysis and Statistics}},

booktitle = {{Proc. Internat. Conf. on Mathematics (ICM 2018)}},

language = {english},

pages = {1--11},

isbn_issn = {ISBN 978-2-7598-9058-3},

year = {2018},

month = {Dezember},

editor = {R. Bris et al.},

refereed = {yes},

institution = {Ton Duc Thang University (TDTU), Ho Chi Minh City, Vietnam},

length = {11},

conferencename = {ICM 2018},

url = {http://icm2018.tdtu.edu.vn/}

}

### Das Unendliche im mathemtischen Alltag

#### F. Winkler

In: Beiträge des 41. Internationalen Wittgenstein Symposiums, G.M. Mras, P. Weingartner, B. Ritter (ed.), Proceedings of 41. Internationales Wittgenstein Symposium, pp. 285-287. August 2018. ISSN 1022-3398. [pdf]**inproceedings**{RISC5840,

author = {F. Winkler},

title = {{Das Unendliche im mathemtischen Alltag}},

booktitle = {{Beiträge des 41. Internationalen Wittgenstein Symposiums}},

language = {english},

pages = {285--287},

isbn_issn = {ISSN 1022-3398},

year = {2018},

month = {August},

editor = {G.M. Mras and P. Weingartner and B. Ritter },

refereed = {yes},

length = {3},

conferencename = {41. Internationales Wittgenstein Symposium}

}

### 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}

}

### The number of realizations of a Laman graph

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

Research Institute for Symbolic Computation (RISC/JKU). Technical report, 2017. [url] [pdf]**techreport**{RISC5418,

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

title = {{The number of realizations of a Laman graph}},

language = {english},

abstract = {Laman graphs model planar frameworks that 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. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems. },

year = {2017},

institution = {Research Institute for Symbolic Computation (RISC/JKU)},

length = {42},

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

}

### 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/}

}

### Validating the Formalization of Theories and Algorithms of Discrete Mathematics by the Computer-Supported Checking of Finite Models

#### Alexander Brunhuemer

Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria. Technical report, September 2017. Bachelor Thesis. [pdf]**techreport**{RISC5485,

author = {Alexander Brunhuemer},

title = {{Validating the Formalization of Theories and Algorithms of Discrete Mathematics by the Computer-Supported Checking of Finite Models}},

language = {english},

abstract = {The goal of this Bachelor’s thesis is the formal specification and implementation of centraltheories and algorithms in the field of discrete mathematics by using the RISC AlgorithmLanguage (RISCAL), developed at the Research Institute for Symbolic Computation (RISC).This specification language and associated software system allow the verification of specifications,by using the concept of finite model checking. Validation on finite models is intendedto serve as a foundation layer for further research on the corresponding generalized theorieson infinite models.This thesis results in a collection of specifications of exemplarily chosen formalized algorithmsof set theory, relation and function theory and graph theory. The algorithms arespecified in different ways (implicit, recursive and procedural), to emphasize the correspondingconnections between them.The evaluation and validation of implemented theories is demonstrated on Dijkstra’s algorithmfor finding a shortest path between vertices in a graph.},

year = {2017},

month = {September},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria},

length = {88},

type = {Bachelor Thesis}

}

### 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}

}