# Rational Parametric Algebraic Curves

### Project Lead

### Project Duration

01/10/2008 - 30/09/2011### Project URL

Go to Website## Publications

### 2018

[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.@

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}

}

**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}

}

### 2016

[Grasegger]

### A decision algorithm for rational general solutions of first-order algebraic ODEs

#### G. Grasegger, N.T. Vo, F. Winkler

In: Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016), Universidad de la Rioja, J. Heras and A. Romero (eds.) (ed.), pp. 101-104. 2016. 978-84-608-9024-9.@

author = {G. Grasegger and N.T. Vo and F. Winkler},

title = {{A decision algorithm for rational general solutions of first-order algebraic ODEs}},

booktitle = {{Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016)}},

language = {english},

pages = {101--104},

isbn_issn = {978-84-608-9024-9},

year = {2016},

editor = {Universidad de la Rioja and J. Heras and A. Romero (eds.)},

refereed = {yes},

length = {4}

}

**inproceedings**{RISC5400,author = {G. Grasegger and N.T. Vo and F. Winkler},

title = {{A decision algorithm for rational general solutions of first-order algebraic ODEs}},

booktitle = {{Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016)}},

language = {english},

pages = {101--104},

isbn_issn = {978-84-608-9024-9},

year = {2016},

editor = {Universidad de la Rioja and J. Heras and A. Romero (eds.)},

refereed = {yes},

length = {4}

}

### 2015

[Sendra]

### Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one

#### A. Lastra, J.R. Sendra, L.X.C. Ngô, F. Winkler

Publ.Math.Debrecen(86/1-2), pp. 49-69. 2015. 0033-3883.@

author = {A. Lastra and J.R. Sendra and L.X.C. Ngô and F. Winkler},

title = {{Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one}},

language = {english},

journal = {Publ.Math.Debrecen},

number = {86/1-2},

pages = {49--69},

isbn_issn = {0033-3883},

year = {2015},

refereed = {yes},

length = {21}

}

**article**{RISC5204,author = {A. Lastra and J.R. Sendra and L.X.C. Ngô and F. Winkler},

title = {{Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one}},

language = {english},

journal = {Publ.Math.Debrecen},

number = {86/1-2},

pages = {49--69},

isbn_issn = {0033-3883},

year = {2015},

refereed = {yes},

length = {21}

}

[Sendra]

### Birational transformations preserving rational solutions of algebraic ordinary differential equations

#### L.X.C. Ngô, J.R. Sendra, F. Winkler

J. Computational and Applied Mathematics(286), pp. 114-127. 2015. 0377-0427.@

author = {L.X.C. Ngô and J.R. Sendra and F. Winkler},

title = {{Birational transformations preserving rational solutions of algebraic ordinary differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {286},

pages = {114--127},

isbn_issn = {0377-0427},

year = {2015},

refereed = {yes},

length = {14}

}

**article**{RISC5205,author = {L.X.C. Ngô and J.R. Sendra and F. Winkler},

title = {{Birational transformations preserving rational solutions of algebraic ordinary differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {286},

pages = {114--127},

isbn_issn = {0377-0427},

year = {2015},

refereed = {yes},

length = {14}

}

[Vo]

### Algebraic General Solutions of First Order Algebraic ODEs

#### N. T. Vo, F. Winkler

In: Computer Algebra in Scientific Computing, Vladimir P. Gerdt et. al. (ed.), Lecture Notes in Computer Science 9301, pp. 479-492. 2015. Springer International Publishing, ISSN 0302-9743. [url]@

author = {N. T. Vo and F. Winkler},

title = {{Algebraic General Solutions of First Order Algebraic ODEs}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.},

series = {Lecture Notes in Computer Science},

volume = {9301},

pages = {479--492},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {Vladimir P. Gerdt et. al.},

refereed = {yes},

length = {14},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}

}

**inproceedings**{RISC5194,author = {N. T. Vo and F. Winkler},

title = {{Algebraic General Solutions of First Order Algebraic ODEs}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.},

series = {Lecture Notes in Computer Science},

volume = {9301},

pages = {479--492},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {Vladimir P. Gerdt et. al.},

refereed = {yes},

length = {14},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}

}

### 2013

[Ngo]

### Rational general solutions of higher order algebraic ODEs

#### Y. Huang, L.X.C. Ngo, F. Winkler

J. Systems Science and Complexity (JSSC) 26/2, pp. 261-280. 2013. 1009-6124.@

author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of higher order algebraic ODEs}},

language = {english},

journal = {J. Systems Science and Complexity (JSSC)},

volume = {26/2},

pages = {261--280},

isbn_issn = {1009-6124},

year = {2013},

refereed = {yes},

length = {20}

}

**article**{RISC4640,author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of higher order algebraic ODEs}},

language = {english},

journal = {J. Systems Science and Complexity (JSSC)},

volume = {26/2},

pages = {261--280},

isbn_issn = {1009-6124},

year = {2013},

refereed = {yes},

length = {20}

}

[Ngo]

### Rational general solutions of trivariate rational systems of autonomous ODEs

#### Y. Huang, L.X.C. Ngo, F. Winkler

Mathematics in Computer Science 6/4, pp. 361-374. 2013. 1661-8270.@

author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of trivariate rational systems of autonomous ODEs}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {6/4},

pages = {361--374},

isbn_issn = {1661-8270},

year = {2013},

refereed = {yes},

length = {14}

}

**article**{RISC4641,author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of trivariate rational systems of autonomous ODEs}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {6/4},

pages = {361--374},

isbn_issn = {1661-8270},

year = {2013},

refereed = {yes},

length = {14}

}

### 2012

[Judakova]

### Computer algebra methods for pattern recognition: systems with complex order

#### F. Winkler, M. Hudayberdiev, G. Judakova

In: Proceedings INTELS 2012 (Moscow), - (ed.), Proceedings of INTELS 2012, pp. 148-150. 2012. 978-5-93347-432-6.@

author = {F. Winkler and M. Hudayberdiev and G. Judakova},

title = {{Computer algebra methods for pattern recognition: systems with complex order}},

booktitle = {{Proceedings INTELS 2012 (Moscow)}},

language = {english},

pages = {148--150},

isbn_issn = {978-5-93347-432-6},

year = {2012},

editor = {-},

refereed = {yes},

length = {3},

conferencename = {INTELS 2012}

}

**inproceedings**{RISC4639,author = {F. Winkler and M. Hudayberdiev and G. Judakova},

title = {{Computer algebra methods for pattern recognition: systems with complex order}},

booktitle = {{Proceedings INTELS 2012 (Moscow)}},

language = {english},

pages = {148--150},

isbn_issn = {978-5-93347-432-6},

year = {2012},

editor = {-},

refereed = {yes},

length = {3},

conferencename = {INTELS 2012}

}

[Ngo]

### Birational Transformations on Algebraic Ordinary Differential Equations

#### L. X. C. Ngo, J. R. Sendra, F. Winkler

Technical report no. 12-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). December 2012. [pdf]@

author = {L. X. C. Ngo and J. R. Sendra and F. Winkler},

title = {{Birational Transformations on Algebraic Ordinary Differential Equations}},

language = {english},

abstract = {We describe a group of birational transformations acting on the set of algebraic ordinary differential equations (AODEs) of arbitrary order n. This transformation group, by its action, partitions the set of algebraic ODEs into equivalence classes. All the elements in a given equivalence class exhibit the same behavior in terms of rational solvability. For a big family of algebraic ODEs we show how to decide whether the given equation can be transformed into an equivalent autonomous ODE.},

number = {12-18},

year = {2012},

month = {December},

length = {20},

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**{RISC4608,author = {L. X. C. Ngo and J. R. Sendra and F. Winkler},

title = {{Birational Transformations on Algebraic Ordinary Differential Equations}},

language = {english},

abstract = {We describe a group of birational transformations acting on the set of algebraic ordinary differential equations (AODEs) of arbitrary order n. This transformation group, by its action, partitions the set of algebraic ODEs into equivalence classes. All the elements in a given equivalence class exhibit the same behavior in terms of rational solvability. For a big family of algebraic ODEs we show how to decide whether the given equation can be transformed into an equivalent autonomous ODE.},

number = {12-18},

year = {2012},

month = {December},

length = {20},

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)}

}

[Ngo]

### Classification of algebraic ODEs with respect to rational solvability

#### L.X.C. Ngo, J.R. Sendra, F. Winkler

Computational Algebraic and Analytic Geometry, Contemporary Mathematics(572), pp. 193-210. 2012. AMS, 0271-4132.@

author = {L.X.C. Ngo and J.R. Sendra and F. Winkler},

title = {{Classification of algebraic ODEs with respect to rational solvability}},

language = {english},

journal = {Computational Algebraic and Analytic Geometry, Contemporary Mathematics},

number = {572},

pages = {193--210},

publisher = {AMS},

isbn_issn = {0271-4132},

year = {2012},

refereed = {yes},

length = {18}

}

**article**{RISC4637,author = {L.X.C. Ngo and J.R. Sendra and F. Winkler},

title = {{Classification of algebraic ODEs with respect to rational solvability}},

language = {english},

journal = {Computational Algebraic and Analytic Geometry, Contemporary Mathematics},

number = {572},

pages = {193--210},

publisher = {AMS},

isbn_issn = {0271-4132},

year = {2012},

refereed = {yes},

length = {18}

}

[Winkler]

### The role of Symbolic Computation in Mathematics

#### F. Winkler

In: Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012), J.R. Sendra and C. Villarino (ed.), pp. 33-34. 2012. 978-84-8138-770-4.@

author = {F. Winkler},

title = {{The role of Symbolic Computation in Mathematics}},

booktitle = {{Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012)}},

language = {english},

pages = {33--34},

isbn_issn = {978-84-8138-770-4},

year = {2012},

editor = {J.R. Sendra and C. Villarino},

refereed = {yes},

length = {2}

}

**inproceedings**{RISC4638,author = {F. Winkler},

title = {{The role of Symbolic Computation in Mathematics}},

booktitle = {{Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012)}},

language = {english},

pages = {33--34},

isbn_issn = {978-84-8138-770-4},

year = {2012},

editor = {J.R. Sendra and C. Villarino},

refereed = {yes},

length = {2}

}

### 1984

[Winkler]

### The Church-Rosser Property in Computer Algebra and Special Theorem Proving

#### Franz Winkler

RISC, Johannes Kepler University Linz. PhD Thesis. 1984.@

author = {Franz Winkler},

title = {{The Church-Rosser Property in Computer Algebra and Special Theorem Proving}},

language = {english},

year = {1984},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

**phdthesis**{RISC4142,author = {Franz Winkler},

title = {{The Church-Rosser Property in Computer Algebra and Special Theorem Proving}},

language = {english},

year = {1984},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}