Computer Algebra for Differential Equations

Computer Algebra offers more and more symbolic methods for differential problems. In particular we are developing algorithms for deciding the existence and for computing general solutions of algebraic differential equations in various classes of formulas (rational, radical, algebraic, ...).

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Software

Publications

2021

[Falkensteiner]

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]
[bib]
@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}
}
[Grasegger]

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]
[bib]
@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}
}
[Grasegger]

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]
[bib]
@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}
}
[Sendra]

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 2020, pp. 1-15. 2021. ISSN 1865-0929. [url]
[bib]
@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},
pages = {1--15},
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 = {arxiv.org/abs/2103.03646}
}
[Winkler]

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]
[bib]
@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

[AUTHOR]

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]
[bib]
@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}
}
[Falkensteiner]

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]
[bib]
@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}
}
[Falkensteiner]

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.
[bib]
@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)}
}
[Grasegger]

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]
[bib]
@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}
}
[Grasegger]

Flexible placements of graphs with rotational symmetry

S.Dewar, G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6091,
author = {S.Dewar and G. Grasegger and J. Legerský},
title = {{Flexible placements of graphs with rotational symmetry}},
language = {english},
year = {2020},
institution = {arXiv},
length = {9},
url = {https://arxiv.org/abs/2003.09328}
}
[Grasegger]

Combinatorics of Bricard’s octahedra

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

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6095,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{Combinatorics of Bricard’s octahedra}},
language = {english},
year = {2020},
institution = {arXiv},
length = {40},
url = {https://arxiv.org/pdf/2004.01236.pdf}
}
[Grasegger]

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]
[bib]
@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}
}
[Grasegger]

Bracing frameworks consisting of parallelograms

G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6203,
author = {G. Grasegger and J. Legerský},
title = {{Bracing frameworks consisting of parallelograms}},
language = {english},
year = {2020},
institution = {arXiv},
length = {20},
url = {https://arxiv.org/abs/2008.11521}
}
[Grasegger]

Zero-sum cycles in flexible polyhedra

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

arXiv. Technical report, 2020. [url]
[bib]
@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}
}
[Grasegger]

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]
[bib]
@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}
}
[Grasegger]

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]
[bib]
@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}
}
[Jimenez Pastor]

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]
[bib]
@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}
}
[Jimenez Pastor]

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]
[bib]
@article{RISC6283,
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}
}
[Mitteramskogler]

A comparison of methods for computing rational general solutions of algebraic ODEs

Johann J. Mitteramskogler, Franz Winkler

Technical report no. 20-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2020. [pdf]
[bib]
@techreport{RISC6135,
author = {Johann J. Mitteramskogler and Franz Winkler},
title = {{A comparison of methods for computing rational general solutions of algebraic ODEs}},
language = {english},
number = {20-11},
year = {2020},
length = {21},
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)}
}
[Mitteramskogler]

Symbolic solutions of algebraic ODEs - A comparison of methods

Franz Winkler, Johann Mitteramskogler

Publications Mathemticae Debrecen, pp. 0-0. 2020. 0.
[bib]
@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}
}

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