# Symbolic Solutions of Algebraic Differential Equations [ADE-solve]

### Project Lead

### Project Duration

01/05/2018 - 31/05/2021## Partners

### The Austrian Science Fund (FWF)

## 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 ...

Authors: Johann Mitteramskogler

MoreSoftware Website## Publications

### 2023

[Mitteramskogler]

### 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. 2023. ISSN 0273-0979.@

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 = {2023},

refereed = {yes},

length = {41}

}

**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 = {2023},

refereed = {yes},

length = {41}

}

### 2022

[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 115, pp. 53-73. 2022. ISSN: 0747-7171. [doi]@

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},

volume = {115},

pages = {53--73},

isbn_issn = {ISSN: 0747-7171},

year = {2022},

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 = {21},

url = {https://doi.org/10.1016/j.jsc.2022.08.005}

}

**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},

volume = {115},

pages = {53--73},

isbn_issn = {ISSN: 0747-7171},

year = {2022},

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 = {21},

url = {https://doi.org/10.1016/j.jsc.2022.08.005}

}

### 2021

[Falkensteiner]

### 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]@

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}

}

**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}

}

[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 20201414, pp. 89-103. 2021. Springer, Cham, ISSN 1865-0929. [url]@

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}

}

**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}

}

### 2020

[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]@

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}

}

**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}

}

[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]@

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)}

}

**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)}

}

[Sendra]

### Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

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

Mathematics in Computer Science 15, pp. 189-198. 2020. ISSN 16618289. [doi]@

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

title = {{Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {15},

pages = {189--198},

isbn_issn = {ISSN 16618289},

year = {2020},

refereed = {yes},

keywords = {Algebraic autonomous ordinary differential equation, Formal Puiseux series solution, Algebraic solutions, Rational solutions, Convergent solution, Algebraic space curve},

length = {10},

url = {https://doi.org/10.1007/s11786-020-00478-w}

}

**article**{RISC6261,author = {J. Cano and S. Falkensteiner and R. Sendra},

title = {{Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {15},

pages = {189--198},

isbn_issn = {ISSN 16618289},

year = {2020},

refereed = {yes},

keywords = {Algebraic autonomous ordinary differential equation, Formal Puiseux series solution, Algebraic solutions, Rational solutions, Convergent solution, Algebraic space curve},

length = {10},

url = {https://doi.org/10.1007/s11786-020-00478-w}

}

### 2019

[Sendra]

### Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

#### S. Falkensteiner, R. Sendra

Mathematics in Computer Science 14, pp. 327-337. 12 2019. ISSN 1661-8289. [doi]@

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

title = {{Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {14},

pages = {327--337},

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}

}

**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},

volume = {14},

pages = {327--337},

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}

}

### 2018

[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. [doi]@

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}

}

**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}

}