## Members

## Bruno Buchberger: Founding Chairman 1987-1998

## Sebastian Falkensteiner

## Günter Landsmann

## Johann Mitteramskogler

## Markus Rosenkranz

## Franz Winkler: Chairman 1998-2008

## Ongoing Projects

### Algebraic Solutions of Fuchsian Differential Equations

## Software

### AGADE

#### A Maple package for computing rational general solutions of first-order algebraic ODEs

The Maple package AGADE implements several methods for computing rational general solutions of first-order algebraic ordinary differential equations and planar rational systems. An advantage of these methods, compared to the standard dsolve-routine in Maple, is that the implemented algorithms provide ...

### FirstOrderSolve

#### A Maple package for computing local and algebraic solutions of first order autonomous AODEs

This Maple package computes power series, Puiseux series and algebraic solutions of first order autonomous ordinary differential equations where the unknown function and its derivative fulfill a polynomial equation. The standard commands of Maple might not find these solutions and ...

## 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}

}

### The algebro-geometric method: Solving algebraic differential equations by parametrizations

#### S. Falkensteiner, J.J. Mitteramskogler, R. Sendra, F. Winkler

Bulletin of the American Mathematical Society, pp. 1-41. 2022. ISSN 0273-0979.**article**{RISC6507,

author = {S. Falkensteiner and J.J. Mitteramskogler and R. Sendra and F. Winkler},

title = {{The algebro-geometric method: Solving algebraic differential equations by parametrizations}},

language = {english},

journal = {Bulletin of the American Mathematical Society},

pages = {1--41},

isbn_issn = {ISSN 0273-0979},

year = {2022},

refereed = {yes},

length = {41}

}

### Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables

#### J. Cano, S. Falkensteiner, D. Robertz, R. Sendra

Journal of Symbolic Computation 114, pp. 1-17. 2022. ISSN 1095-855X. [doi]**article**{RISC6501,

author = {J. Cano and S. Falkensteiner and D. Robertz and R. Sendra},

title = {{Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {114},

pages = {1--17},

isbn_issn = {ISSN 1095-855X},

year = {2022},

refereed = {yes},

length = {17},

url = {http://doi.org/10.1016/j.jsc.2022.04.012}

}

### 2021

### On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry

#### S. Falkensteiner, C. Garay-Lopez, M. Haiech, M. P. Noordman, F. Boulier, Z. Toghani

Journal of Symbolic Computation, pp. 1-22. 2021. ISSN: 0747-7171. [url]**article**{RISC6335,

author = {S. Falkensteiner and C. Garay-Lopez and M. Haiech and M. P. Noordman and F. Boulier and Z. Toghani},

title = {{On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {1--22},

isbn_issn = {ISSN: 0747-7171},

year = {2021},

refereed = {yes},

keywords = {Differential Algebra, Tropical Differential Algebraic Geometry, Power Series Solutions, Newton Polyhedra, Arc Spaces, Tropical Differential Equations, Initial forms of Differential Polynomials},

length = {22},

url = {http://hal.archives-ouvertes.fr/hal-03122437v1/document}

}

### On The Relationship Between Differential Algebra and Tropical Differential Algebraic Geometry

#### F. Boulier, S. Falkensteiner, M.P. Noordman, O.L. Sanchez

In: International Workshop on Computer Algebra in Scientific Computing, F. Boulier, M. England, T. Sadykov, E. Vorozhtsov (ed.), Proceedings of Computer Algebra in Scientific Computing, pp. 62-77. 2021. Springer, ISSN 0302-9743. [doi]**inproceedings**{RISC6337,

author = {F. Boulier and S. Falkensteiner and M.P. Noordman and O.L. Sanchez},

title = {{On The Relationship Between Differential Algebra and Tropical Differential Algebraic Geometry}},

booktitle = {{International Workshop on Computer Algebra in Scientific Computing}},

language = {english},

pages = {62--77},

publisher = {Springer},

isbn_issn = {ISSN 0302-9743},

year = {2021},

editor = {F. Boulier and M. England and T. Sadykov and E. Vorozhtsov},

refereed = {yes},

length = {16},

conferencename = {Computer Algebra in Scientific Computing},

url = {https://doi.org/10.1007/978-3-030-85165-1_5}

}

### 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}

}

### General solutions of first-order algebraic ODEs in simple constant extensions

#### Johann J. Mitteramskogler, Franz Winkler

Technical report no. 21-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2021. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6364,

author = {Johann J. Mitteramskogler and Franz Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

abstract = {If a first-order algebraic ODE is defined over a certain differential field, then the most elementary solution class, in which one can hope to find a general solution, is given by the adjunction of a single arbitrary constant to this field. Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial. As for the opposite direction, we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field. These automorphisms are determined by linear rational functions, i.e. Möbius transformations. Intrinsic properties of rational parametrizations, in combination with the particular shape of such automorphisms, lead to a number of necessary conditions on the existence of general solutions in this solution class. Furthermore, the desired linear rational function can be determined by solving a simple differential system over the ODE’s field of definition. All results are derived in a purely algebraic fashion and apply to any differential field of characteristic zero with arbitrary derivative operator.},

number = {21-18},

year = {2021},

month = {September},

keywords = {Algebraic ordinary differential equation, general solution, algebraic curve, rational parametrization},

length = {17},

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)}

}

### Existence and convergence of Puiseux series solutions for first order autonomous differential equations

#### J. Cano, S. Falkensteiner, R. Sendra

Journal of Symbolic Computation 108, pp. 137-151. 2021. ISSN 0747-7171. [doi]**article**{RISC6262,

author = {J. Cano and S. Falkensteiner and R. Sendra},

title = {{Existence and convergence of Puiseux series solutions for first order autonomous differential equations}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {108},

pages = {137--151},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

keywords = {Algebraic differential equation, Algebraic curve, Place, Formal Puiseux series solution, Convergent solution},

length = {15},

url = {http://doi.org/10.1016/j.jsc.2020.06.010}

}

### Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs - A MAPLE Package

#### F. Boulier, J. Cano, S. Falkensteiner, R. Sendra

In: Communications in Computer and Information Science, Rob Corless, Jürgen Gerhard and Ilias Kotsireas (ed.), Proceedings of Maple Conference 20201414, pp. 89-103. 2021. Springer, Cham, ISSN 1865-0929. [url]**inproceedings**{RISC6334,

author = {F. Boulier and J. Cano and S. Falkensteiner and R. Sendra},

title = {{Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs - A MAPLE Package}},

booktitle = {{Communications in Computer and Information Science}},

language = {english},

volume = {1414},

pages = {89--103},

publisher = {Springer, Cham},

isbn_issn = {ISSN 1865-0929},

year = {2021},

editor = {Rob Corless and Jürgen Gerhard and Ilias Kotsireas},

refereed = {yes},

keywords = {Maple, Symbolic computation, Algebraic differential equation, Formal Puiseux series solution, Algebraic solution},

length = {15},

conferencename = {Maple Conference 2020},

url = {doi.org/10.1007/978-3-030-81698-8_7}

}

### 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

### The Sage Package Comb_walks for Walks in the Quarter Plane

#### Antonio Jiménez-Pastor, Alin Bostan, Frédéric Chyzak, Pierre Lairez

ACM Commun. Comput. Algebra 54(2), pp. 30-38. sep 2020. Association for Computing Machinery, New York, NY, USA, 1932-2240. [doi]**article**{RISC6282,

author = {Antonio Jiménez-Pastor and Alin Bostan and Frédéric Chyzak and Pierre Lairez},

title = {{The Sage Package Comb_walks for Walks in the Quarter Plane}},

language = {english},

abstract = {We present in this extended abstract a new software designed to work with generating functions that count walks in the quarter plane. With this software we offer a cohesive package that brings together all the required procedures for manipulating these generating functions, as well as a unified interface to deal with them. We also display results that this package offers on a public webpage.},

journal = {ACM Commun. Comput. Algebra},

volume = {54},

number = {2},

pages = {30--38},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {1932-2240},

year = {2020},

month = {sep},

refereed = {yes},

keywords = {Sage, D-algebraic functions, generating functions, elliptic functions, lattice walks},

length = {9},

url = {https://doi.org/10.1145/3427218.3427220}

}

### Power Series Solutions of AODEs - Existence, Uniqueness, Convergence and Computation

#### S. Falkensteiner

RISC Hagenberg, Johannes Kepler University Linz. PhD Thesis. June 2020. Also available as RISC report no. 20-13. [pdf]**phdthesis**{RISC6120,

author = {S. Falkensteiner},

title = {{Power Series Solutions of AODEs - Existence, Uniqueness, Convergence and Computation}},

language = {english},

year = {2020},

month = {June},

note = {Also available as RISC report no. 20-13},

translation = {0},

school = {RISC Hagenberg, Johannes Kepler University Linz},

length = {146}

}

### The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry

#### S. Falkensteiner, C. Garay-Lopez, M. Haiech, M. P. Noordman, Z. Toghani, F. Boulier

In: ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, Angelos Mantzaflaris (ed.), Proceedings of 45th International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 178-185. 07 2020. Association for Computing Machinery, ISBN 9781450371001.**inproceedings**{RISC6263,

author = {S. Falkensteiner and C. Garay-Lopez and M. Haiech and M. P. Noordman and Z. Toghani and F. Boulier},

title = {{The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry}},

booktitle = {{ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {178--185},

publisher = {Association for Computing Machinery},

isbn_issn = {ISBN 9781450371001},

year = {2020},

month = {07},

annote = {Distinguished Paper Award},

editor = {Angelos Mantzaflaris},

refereed = {yes},

keywords = {Differential Algebra, Power series solutions, Arc spaces, Newton Polytope, Tropical Differential Algebraic Geometry},

length = {8},

conferencename = {45th International Symposium on Symbolic and Algebraic Computation (ISSAC)}

}

### 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}

}

### 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}

}