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, ...).

Publications

2020

[Grasegger]

On the Classification of Motions of Paradoxically Movable Graphs

G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6092,
author = {G. Grasegger and J. Legerský and J. Schicho},
title = {{On the Classification of Motions of Paradoxically Movable Graphs}},
language = {english},
year = {2020},
institution = {arXiv},
length = {27},
url = {https://arxiv.org/abs/2003.11416}
}
[Grasegger]

FlexRiLoG - A SageMath Package for Motions of Graphs

G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6090,
author = {G. Grasegger and J. Legerský},
title = {{FlexRiLoG - A SageMath Package for Motions of Graphs}},
language = {english},
year = {2020},
institution = {arXiv},
length = {9},
url = {https://arxiv.org/abs/2003.12029}
}
[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}
}
[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. [url] [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}
}

2019

[Grasegger]

On the existence of paradoxical motions of generically rigid graphs on the sphere

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

arXiv. Technical report, 2019. [url]
[bib]
@techreport{RISC5977,
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},
year = {2019},
institution = {arXiv},
length = {40},
url = {https://arxiv.org/abs/1908.00467}
}
[Sendra]

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

S. Falkensteiner, R. Sendra

Mathematics in Computer Science, pp. 1-11. 12 2019. ISSN 1661-8289. [url]
[bib]
@article{RISC6035,
author = {S. Falkensteiner and R. Sendra},
title = {{Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places}},
language = {english},
journal = {Mathematics in Computer Science},
pages = {1--11},
isbn_issn = {ISSN 1661-8289},
year = {2019},
month = {12},
refereed = {yes},
keywords = {Algebraic autonomous differential equation, Algebraic curve, Local parametrization, Place, Formal power series solution, Analytic solution},
length = {11},
url = {https://doi.org/10.1007/s11786-019-00431-6}
}
[Winkler]

The Algebro-Geometric Method for Solving Algebraic Differential Equations - A Survey

Franz Winkler

Journal of System Science and Complexity 32, pp. 256-270. 2019. 1009-6124.
[bib]
@article{RISC6027,
author = {Franz Winkler},
title = {{The Algebro-Geometric Method for Solving Algebraic Differential Equations -- A Survey}},
language = {english},
journal = {Journal of System Science and Complexity},
volume = {32},
pages = {256--270},
isbn_issn = {1009-6124},
year = {2019},
refereed = {yes},
length = {15}
}
[Winkler]

The algebro-geometric solution method for algebraic differential equations - An introduction by examples

J.R. Sendra, Franz Winkler

In: Complex Differential and Difference Equations, Proceedings of the School and Conference CDDE, held at Bedlewo, Poland, deGruyter (ed.), pp. 129-146. 2019. Polish Academy of Sciences, deGruyter, 978-3-11-061142-7.
[bib]
@inproceedings{RISC6033,
author = {J.R. Sendra and Franz Winkler},
title = {{The algebro-geometric solution method for algebraic differential equations -- An introduction by examples}},
booktitle = {{Complex Differential and Difference Equations, Proceedings of the School and Conference CDDE, held at Bedlewo, Poland}},
language = {english},
pages = {129--146},
publisher = {Polish Academy of Sciences, deGruyter},
isbn_issn = {978-3-11-061142-7},
year = {2019},
editor = {deGruyter},
refereed = {yes},
length = {18}
}

2018

[Grasegger]

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]
[bib]
@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}
}
[Grasegger]

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.
[bib]
@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}
}
[Grasegger]

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]
[bib]
@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}
}
[Koutschan]

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]
[bib]
@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}
}
[McCallum]

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]
[bib]
@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}
}
[McCallum]

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]
[bib]
@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/}
}
[Vo]

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. March 2018. Elsevier, 0196-8858. [url]
[bib]
@article{RISC5797,
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},
volume = {98},
pages = {1--24},
publisher = {Elsevier},
isbn_issn = {0196-8858},
year = {2018},
month = {March},
refereed = {yes},
keywords = {Ordinary differential equation, Rational solution, Algebraic function field, Rational curve},
length = {24},
url = {https://doi.org/10.1016/j.aam.2018.03.002}
}
[Winkler]

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]
[bib]
@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

[Fuerst]

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.
[bib]
@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}
}
[Grasegger]

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.
[bib]
@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}
}
[Koutschan]

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]
[bib]
@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/}
}

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