## Members

## Bruno Buchberger: Founding Chairman 1987-1998

## Eriola Hoxhaj

## Günter Landsmann

## Josef Schicho

## Franz Winkler: Chairman 1998-2008

## 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

### 2022

### On Formal Power Series Solutions of Algebraic Ordinary Differential Equations

#### S. Falkensteiner, Yi Zhang, N. Thieu Vo

Mediterranean Journal of Mathematics 19(74), pp. 1-16. March 2022. ISSN 1660-5446. [doi]**article**{RISC6490,

author = {S. Falkensteiner and Yi Zhang and N. Thieu Vo},

title = {{On Formal Power Series Solutions of Algebraic Ordinary Differential Equations}},

language = {english},

journal = {Mediterranean Journal of Mathematics},

volume = {19},

number = {74},

pages = {1--16},

isbn_issn = {ISSN 1660-5446},

year = {2022},

month = {March},

refereed = {yes},

keywords = {Formal power series, algebraic differential equation.},

length = {16},

url = {https://doi.org/10.1007/s00009-022-01984-w}

}

### Zero-Sum Cycles in Flexible Non-triangular Polyhedra

#### Matteo Gallet, Georg Grasegger, Jan Legerský, Josef Schicho

In: 2nd IMA Conference on Mathematics of Robotics, W. Holderbaum, J.M. Selig (ed.), Springer Proceedings in Advanced Robotics 21, pp. 137-143. 2022. 978-3-030-91351-9. [doi]**inproceedings**{RISC6388,

author = {Matteo Gallet and Georg Grasegger and Jan Legerský and Josef Schicho},

title = {{Zero-Sum Cycles in Flexible Non-triangular Polyhedra}},

booktitle = {{2nd IMA Conference on Mathematics of Robotics}},

language = {english},

series = {Springer Proceedings in Advanced Robotics},

volume = {21},

pages = {137--143},

isbn_issn = {978-3-030-91351-9},

year = {2022},

editor = {W. Holderbaum and J.M. Selig},

refereed = {yes},

length = {7},

url = {https://doi.org/10.1007/978-3-030-91352-6_14}

}

### And Yet it Moves - Paradoxically Moving Linkages in Kinematics

#### J. Schicho

Bulletin AMS 59, pp. 59-95. 2022. ISSN 0273-0979. [doi]**article**{RISC6474,

author = {J. Schicho},

title = {{And Yet it Moves -- Paradoxically Moving Linkages in Kinematics}},

language = {english},

journal = {Bulletin AMS},

volume = {59},

pages = {59--95},

isbn_issn = {ISSN 0273-0979},

year = {2022},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1090/bull/1721}

}

### 2021

### Combinatorics of Bricard's octahedra

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

Comptes Rendus. Mathématique 359(1), pp. 7-38. 2021. Académie des sciences, Paris, ISSN 1631-073X. [doi]**article**{RISC6288,

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

title = {{Combinatorics of Bricard's octahedra}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {359},

number = {1},

pages = {7--38},

publisher = {Académie des sciences, Paris},

isbn_issn = {ISSN 1631-073X},

year = {2021},

refereed = {yes},

length = {32},

url = {https://doi.org/10.5802/crmath.132}

}

### On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere

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

SIAM Journal on Discrete Mathematics 35(1), pp. 325-361. 2021. ISSN 0895-4801. [doi]**article**{RISC6290,

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

title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},

language = {english},

journal = {SIAM Journal on Discrete Mathematics},

volume = {35},

number = {1},

pages = {325--361},

isbn_issn = {ISSN 0895-4801},

year = {2021},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1137/19M1289467}

}

### 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

Journal of Symbolic Computation 102, pp. 189-208. 2021. ISSN 0747-7171. [doi]**article**{RISC5992,

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},

journal = {Journal of Symbolic Computation},

volume = {102},

pages = {189--208},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1016/j.jsc.2019.10.015}

}

### My Life in Computer Algebra

#### F. Winkler

Technical report no. 21-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2021. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6329,

author = {F. Winkler},

title = {{My Life in Computer Algebra}},

language = {english},

abstract = {After having spent more than 40 years in Mathematics, and in particular in Computer Algebra, I recollect stages in my scientific career and I explain the connections between my different areas of interest.},

number = {21-12},

year = {2021},

month = {June},

keywords = {computer algebra, biography},

length = {19},

license = {CC BY 4.0 International},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### 2020

### Robots, computer algebra and 8 connected components

#### J. Capco, M. Safey el Din, J. Schicho

In: Proceedings of ISSAC 2020, I. Emiris, L. Zhi (ed.), pp. 62-69. 2020. 978-1-4503-7100-1. [url]**inproceedings**{RISC6247,

author = {J. Capco and M. Safey el Din and J. Schicho},

title = {{Robots, computer algebra and 8 connected components}},

booktitle = {{Proceedings of ISSAC 2020}},

language = {english},

pages = {62--69},

isbn_issn = {978-1-4503-7100-1},

year = {2020},

editor = {I. Emiris and L. Zhi},

refereed = {yes},

length = {8},

url = {https://arxiv.org/abs/2008.13392}

}

### Computing Animations of Linkages with Rotational Symmetry (Media Exposition)

#### Sean Dewar, Georg Grasegger, Jan Legerský

In: 36th International Symposium on Computational Geometry (SoCG 2020), Sergio Cabello and Danny Z. Chen (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 164, pp. 77:1-77:4. 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, ISBN 978-3-95977-143-6. [doi]**inproceedings**{RISC6128,

author = {Sean Dewar and Georg Grasegger and Jan Legerský},

title = {{Computing Animations of Linkages with Rotational Symmetry (Media Exposition)}},

booktitle = {{36th International Symposium on Computational Geometry (SoCG 2020)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {164},

pages = {77:1--77:4},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},

address = {Dagstuhl, Germany},

isbn_issn = {ISBN 978-3-95977-143-6},

year = {2020},

editor = {Sergio Cabello and Danny Z. Chen},

refereed = {no},

length = {4},

url = {https://doi.org/10.4230/LIPIcs.SoCG.2020.77}

}

### Graphs with Flexible Labelings allowing Injective Realizations

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

Discrete Mathematics 343(6), pp. Art. 111713-. 2020. ISSN 0012-365X. [doi]**article**{RISC6012,

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

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

language = {english},

journal = {Discrete Mathematics},

volume = {343},

number = {6},

pages = {Art. 111713--},

isbn_issn = {ISSN 0012-365X},

year = {2020},

refereed = {yes},

length = {14},

url = {https://doi.org/10.1016/j.disc.2019.111713}

}

### FlexRiLoG - A SageMath Package for Motions of Graphs

#### G. Grasegger, J. Legerský

In: Mathematical Software – ICMS 2020, Bigatti A., Carette J., Davenport J., Joswig M., de Wolff T. (ed.), Proceedings of ICMS 2020, Lecture Notes in Computer Science 12097, pp. 442-450. 2020. Springer, Cham, ISBN 978-3-030-52199-8. [doi]**inproceedings**{RISC6182,

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

title = {{FlexRiLoG - A SageMath Package for Motions of Graphs}},

booktitle = {{ Mathematical Software – ICMS 2020}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {12097},

pages = {442--450},

publisher = {Springer, Cham},

isbn_issn = {ISBN 978-3-030-52199-8},

year = {2020},

editor = {Bigatti A. and Carette J. and Davenport J. and Joswig M. and de Wolff T.},

refereed = {no},

length = {9},

conferencename = {ICMS 2020},

url = {https://doi.org/10.1007/978-3-030-52200-1_44}

}

### Zero-sum cycles in flexible polyhedra

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

arXiv. Technical report, 2020. [url]**techreport**{RISC6211,

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

title = {{Zero-sum cycles in flexible polyhedra}},

language = {english},

year = {2020},

institution = {arXiv},

length = {16},

url = {https://arxiv.org/abs/2009.14041}

}

### On the Classification of Motions of Paradoxically Movable Graphs

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

Journal of Computational Geometry 11(1), pp. 548-575. 2020. ISSN: 1920-180X. [doi]**article**{RISC6239,

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

title = {{On the Classification of Motions of Paradoxically Movable Graphs}},

language = {english},

journal = {Journal of Computational Geometry},

volume = {11},

number = {1},

pages = {548--575},

isbn_issn = {ISSN: 1920-180X},

year = {2020},

refereed = {yes},

length = {27},

url = {https://doi.org/10.20382/jocg.v11i1a22}

}

### Counting realizations of {L}aman graphs on the sphere

#### M. Gallet, G. Grasegger, J. Schicho

Electronic Journal of Combinatorics 27(2), pp. 1-18. 2020. 1077-8926 . [url]**article**{RISC6243,

author = {M. Gallet and G. Grasegger and J. Schicho},

title = {{Counting realizations of {L}aman graphs on the sphere}},

language = {english},

journal = {Electronic Journal of Combinatorics},

volume = {27},

number = {2},

pages = {1--18},

isbn_issn = {1077-8926 },

year = {2020},

refereed = {yes},

length = {18},

url = {https://arxiv.org/abs/1903.01145}

}

### Some structural results on D^n finite functions

#### A. Jimenez-Pastor, V. Pillwein, M.F. Singer

Advances in Applied Mathematics 117, pp. 0-0. June 2020. Elsevier, 0196-8858. [doi] [pdf]**article**{RISC6077,

author = {A. Jimenez-Pastor and V. Pillwein and M.F. Singer},

title = {{Some structural results on D^n finite functions}},

language = {english},

abstract = {D-finite (or holonomic) functions satisfy linear differential equations with polynomial coefficients. They form a large class of functions that appear in many applications in Mathematics or Physics. It is well-known that these functions are closed under certain operations and these closure properties can be executed algorithmically. Recently, the notion of D-finite functions has been generalized to differentially definable or Dn-finite functions. Also these functions are closed under operations such as forming (anti)derivative, addition or multiplication and, again, these can be implemented. In this paper we investigate how Dn-finite functions behave under composition and how they are related to algebraic and differentially algebraic functions.},

journal = {Advances in Applied Mathematics},

volume = {117},

pages = {0--0},

publisher = {Elsevier},

isbn_issn = {0196-8858},

year = {2020},

month = {June},

refereed = {yes},

length = {0},

url = {https://doi.org/10.1016/j.aam.2020.102027}

}

### Symbolic solutions of algebraic ODEs - A comparison of methods

#### Franz Winkler, Johann Mitteramskogler

Publications Mathemticae Debrecen, pp. 0-0. 2020. 0.**article**{RISC6253,

author = {Franz Winkler and Johann Mitteramskogler},

title = {{Symbolic solutions of algebraic ODEs -- A comparison of methods}},

language = {english},

journal = {Publications Mathemticae Debrecen},

pages = {0--0},

isbn_issn = {0},

year = {2020},

refereed = {yes},

length = {0}

}

### A reduction theorem of certain relations modulo p involving modular forms

#### Radu, Cristian-Silviu

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, V. Pillwein, C. Schneider (ed.), pp. 317-337. 2020. Springer, 978-3-030-44558-4. [doi] [pdf]**incollection**{RISC6110,

author = {Radu and Cristian-Silviu},

title = {{A reduction theorem of certain relations modulo p involving modular forms}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

pages = {317--337},

publisher = {Springer},

isbn_issn = {978-3-030-44558-4},

year = {2020},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {21},

url = {https://doi.org/10.1007/978-3-030-44559-1_16}

}

### Probabilities of incidence between lines and a plane curve over finite field

#### M. Gallet, M. Makhul, J. Schicho

Finite Fields and Their Applications 61, pp. 1-22. 2020. 1071-5797.**article**{RISC6073,

author = {M. Gallet and M. Makhul and J. Schicho},

title = {{Probabilities of incidence between lines and a plane curve over finite field}},

language = {english},

journal = {Finite Fields and Their Applications},

volume = {61},

pages = {1--22},

isbn_issn = {1071-5797},

year = {2020},

refereed = {yes},

length = {22}

}

### On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs

#### V. Bartzos, I. Emiris, J. Schicho

Applicable Algebra in Engineering, Communication and Computing 31, pp. 325-357. 2020. 0938-1279. [url]**article**{RISC6245,

author = {V. Bartzos and I. Emiris and J. Schicho},

title = {{On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs}},

language = {english},

journal = {Applicable Algebra in Engineering, Communication and Computing},

volume = {31},

pages = {325--357},

isbn_issn = {0938-1279},

year = {2020},

refereed = {yes},

length = {32},

url = {https://arxiv.org/abs/2005.14485}

}

### Counting Projections of Rational Curves

#### M. Gallet, J. Schicho

Israel Journal of Mathematics 240, pp. 503-538. 2020. 0021-2172 . [url]**article**{RISC6246,

author = {M. Gallet and J. Schicho},

title = {{Counting Projections of Rational Curves}},

language = {english},

journal = {Israel Journal of Mathematics},

volume = {240},

pages = {503--538},

isbn_issn = {0021-2172 },

year = {2020},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1707.05264}

}