# Rational Parametric Algebraic Curves [DK11]

### Project Lead

### Project Duration

01/10/2011 - 30/09/2014### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 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.@

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}

}

**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.@

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}

}

**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}

}

### 2016

[Fuerst]

### Axiomatic Description of Gröbner Reduction

#### Christoph Fuerst

RISC, JKU Linz. PhD Thesis. December 2016. [pdf]@

author = {Christoph Fuerst},

title = {{Axiomatic Description of Gröbner Reduction}},

language = {english},

year = {2016},

month = {December},

translation = {0},

school = {RISC, JKU Linz},

length = {154}

}

**phdthesis**{RISC5388,author = {Christoph Fuerst},

title = {{Axiomatic Description of Gröbner Reduction}},

language = {english},

year = {2016},

month = {December},

translation = {0},

school = {RISC, JKU Linz},

length = {154}

}

[Grasegger]

### A solution method for autonomous first-order algebraic partial differential equations

#### G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Journal of Computational and Applied Mathematics 300, pp. 119-133. 2016. 0377-0427. [doi]@

author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {300},

pages = {119--133},

isbn_issn = {0377-0427},

year = {2016},

refereed = {yes},

length = {15},

url = {http://dx.doi.org/10.1016/j.cam.2015.12.030}

}

**article**{RISC5202,author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {300},

pages = {119--133},

isbn_issn = {0377-0427},

year = {2016},

refereed = {yes},

length = {15},

url = {http://dx.doi.org/10.1016/j.cam.2015.12.030}

}

[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

[Fuerst]

### Computation of Dimension in Filtered Free Modules by Gröbner Reduction

#### Christoph Fuerst, Guenter Landsmann

In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM (ed.), Proceedings of ISSAC '15, pp. 181-188. 2015. 978-1-4503-3435-8. [doi]@

author = {Christoph Fuerst and Guenter Landsmann},

title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},

booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {181--188},

isbn_issn = {978-1-4503-3435-8},

year = {2015},

editor = {ACM},

refereed = {yes},

length = {8},

conferencename = {ISSAC '15},

url = {http://doi.acm.org/10.1145/2755996.2756680}

}

**inproceedings**{RISC5154,author = {Christoph Fuerst and Guenter Landsmann},

title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},

booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {181--188},

isbn_issn = {978-1-4503-3435-8},

year = {2015},

editor = {ACM},

refereed = {yes},

length = {8},

conferencename = {ISSAC '15},

url = {http://doi.acm.org/10.1145/2755996.2756680}

}

[Fuerst]

### Three Examples of Gröbner Reduction over Noncommutative Rings

#### Christoph Fuerst, Guenter Landsmann

Technical report no. 15-16 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2015. [pdf]@

author = {Christoph Fuerst and Guenter Landsmann},

title = {{Three Examples of Gröbner Reduction over Noncommutative Rings}},

language = {english},

abstract = {In this report three classes of noncommutative rings are investigated withemphasis on their properties with respect to reduction relations. TheGröbner basis concepts in these rings, being developed in the literature byseveral authors, are considered and it is shown that the reduction relationscorresponding to these Gröbner bases obey the axioms of a general theoryof Gröbner reduction.},

number = {15-16},

year = {2015},

month = {October},

sponsor = {partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK11},

length = {31},

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**{RISC5178,author = {Christoph Fuerst and Guenter Landsmann},

title = {{Three Examples of Gröbner Reduction over Noncommutative Rings}},

language = {english},

abstract = {In this report three classes of noncommutative rings are investigated withemphasis on their properties with respect to reduction relations. TheGröbner basis concepts in these rings, being developed in the literature byseveral authors, are considered and it is shown that the reduction relationscorresponding to these Gröbner bases obey the axioms of a general theoryof Gröbner reduction.},

number = {15-16},

year = {2015},

month = {October},

sponsor = {partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK11},

length = {31},

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)}

}

[Grasegger]

### Symbolic Solutions of First-Order Algebraic ODEs

#### G. Grasegger, F. Winkler

In: Computer algebra and polynomials, J. Gutierrez, J. Schicho, M. Weimann (ed.), Lecture Notes in Computer Science 8942, pp. 94-104. 2015. Springer International Publishing, ISSN 0302-9743. [doi]@

author = {G. Grasegger and F. Winkler},

title = {{Symbolic Solutions of First-Order Algebraic ODEs}},

booktitle = {{Computer algebra and polynomials}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {8942},

pages = {94--104},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {J. Gutierrez and J. Schicho and M. Weimann},

refereed = {yes},

length = {11},

url = {http://dx.doi.org/10.1007/978-3-319-15081-9_5}

}

**inproceedings**{RISC5018,author = {G. Grasegger and F. Winkler},

title = {{Symbolic Solutions of First-Order Algebraic ODEs}},

booktitle = {{Computer algebra and polynomials}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {8942},

pages = {94--104},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {J. Gutierrez and J. Schicho and M. Weimann},

refereed = {yes},

length = {11},

url = {http://dx.doi.org/10.1007/978-3-319-15081-9_5}

}

[Grasegger]

### A solution method for autonomous first-order algebraic partial differential equations in several variables

#### G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Technical report no. 15-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2015. [pdf]@

author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations in several variables}},

language = {english},

number = {15-01},

year = {2015},

length = {22},

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**{RISC5117,author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations in several variables}},

language = {english},

number = {15-01},

year = {2015},

length = {22},

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)}

}

[Grasegger]

### Symbolic solutions of first-order algebraic differential equations

#### Georg Grasegger

Johannes Kepler University Linz. PhD Thesis. 06 2015. [url]@

author = {Georg Grasegger},

title = {{Symbolic solutions of first-order algebraic differential equations}},

language = {english},

year = {2015},

month = {06},

translation = {0},

school = {Johannes Kepler University Linz},

length = {154},

url = {http://epub.jku.at/obvulihs/content/titleinfo/753082}

}

**phdthesis**{RISC5160,author = {Georg Grasegger},

title = {{Symbolic solutions of first-order algebraic differential equations}},

language = {english},

year = {2015},

month = {06},

translation = {0},

school = {Johannes Kepler University Linz},

length = {154},

url = {http://epub.jku.at/obvulihs/content/titleinfo/753082}

}

[Grasegger]

### Rational General Solutions of First-Order Algebraic ODEs

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

Technical report no. 15-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2015.@

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

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

language = {english},

number = {15-18},

year = {2015},

length = {25},

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**{RISC5183,author = {N.T. Vo and G. Grasegger and F. Winkler},

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

language = {english},

number = {15-18},

year = {2015},

length = {25},

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)}

}

### 2014

[Fuerst]

### The Concept of Gröbner Reduction for Dimension in filtered free modules

#### Christoph Fuerst, Guenter Landsmann

Technical report no. 14-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2014. [pdf]@

author = {Christoph Fuerst and Guenter Landsmann},

title = {{The Concept of Gröbner Reduction for Dimension in filtered free modules}},

language = {english},

abstract = {We present the concept of Gröbner reduction that is a Gröbner basistechnique on filtered free modules. It allows to compute the dimensionof a filtered free module viewn as a K-vector space. We apply the de-veloped technique to the computation of a generalization of Hilbert-typedimension polynomials in K[X] as well as in finitely generated difference-differential modules. The latter allows us to determine a multivariatedimension polynomial where we partition the set of derivations and theset of automorphism in a difference-differential ring in an arbitrary way.},

number = {14-12},

year = {2014},

month = {October},

length = {13},

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**{RISC5068,author = {Christoph Fuerst and Guenter Landsmann},

title = {{The Concept of Gröbner Reduction for Dimension in filtered free modules}},

language = {english},

abstract = {We present the concept of Gröbner reduction that is a Gröbner basistechnique on filtered free modules. It allows to compute the dimensionof a filtered free module viewn as a K-vector space. We apply the de-veloped technique to the computation of a generalization of Hilbert-typedimension polynomials in K[X] as well as in finitely generated difference-differential modules. The latter allows us to determine a multivariatedimension polynomial where we partition the set of derivations and theset of automorphism in a difference-differential ring in an arbitrary way.},

number = {14-12},

year = {2014},

month = {October},

length = {13},

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)}

}

[Grasegger]

### Polynomial Equivalence of Finite Rings

#### G. Grasegger, G. Horváth, K.A. Kearnes

Journal of the Australian Mathematical Society 96(2), pp. 244-257. 2014. 1446-8107. [url]@

author = {G. Grasegger and G. Horváth and K.A. Kearnes},

title = {{Polynomial Equivalence of Finite Rings}},

language = {english},

journal = {Journal of the Australian Mathematical Society},

volume = {96},

number = {2},

pages = {244--257},

isbn_issn = {1446-8107},

year = {2014},

refereed = {yes},

length = {14},

url = {http://journals.cambridge.org/article_S1446788713000645}

}

**article**{RISC4889,author = {G. Grasegger and G. Horváth and K.A. Kearnes},

title = {{Polynomial Equivalence of Finite Rings}},

language = {english},

journal = {Journal of the Australian Mathematical Society},

volume = {96},

number = {2},

pages = {244--257},

isbn_issn = {1446-8107},

year = {2014},

refereed = {yes},

length = {14},

url = {http://journals.cambridge.org/article_S1446788713000645}

}

[Grasegger]

### A solution method for autonomous first-order algebraic partial differential equations

#### G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Technical report no. 14-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2014. [pdf]@

author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations}},

language = {english},

number = {14-03},

year = {2014},

length = {17},

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**{RISC4977,author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{A solution method for autonomous first-order algebraic partial differential equations}},

language = {english},

number = {14-03},

year = {2014},

length = {17},

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)}

}

[Grasegger]

### Radical Solutions of First Order Autonomous Algebraic Ordinary Differential Equations

#### G. Grasegger

In: ISSAC '14: Proceedings of the 39th International Symposium on International Symposium on Symbolic and Algebraic Computation, Katsusuke Nabeshima (ed.), pp. 217-223. 2014. ACM, New York, ISBN 978-1-4503-2501-1. [pdf]@

author = {G. Grasegger},

title = {{Radical Solutions of First Order Autonomous Algebraic Ordinary Differential Equations}},

booktitle = {{ISSAC '14: Proceedings of the 39th International Symposium on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {217--223},

publisher = {ACM},

address = {New York},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {7}

}

**inproceedings**{RISC4990,author = {G. Grasegger},

title = {{Radical Solutions of First Order Autonomous Algebraic Ordinary Differential Equations}},

booktitle = {{ISSAC '14: Proceedings of the 39th International Symposium on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {217--223},

publisher = {ACM},

address = {New York},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {7}

}

[Grasegger]

### On Symbolic Solutions of Algebraic Partial Differential Equations

#### G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

In: Computer Algebra in Scientific Computing, V.P. Gerdt et al. (ed.), Lecture Notes in Computer Science 8660, pp. 111-120. 2014. Springer International Publishing, ISSN 0302-9743. [url]@

author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{On Symbolic Solutions of Algebraic Partial Differential Equations}},

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

language = {english},

series = {Lecture Notes in Computer Science},

volume = {8660},

pages = {111--120},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2014},

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

refereed = {yes},

length = {11},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-10515-4_9.pdf}

}

**inproceedings**{RISC5017,author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},

title = {{On Symbolic Solutions of Algebraic Partial Differential Equations}},

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

language = {english},

series = {Lecture Notes in Computer Science},

volume = {8660},

pages = {111--120},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2014},

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

refereed = {yes},

length = {11},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-10515-4_9.pdf}

}

### 2013

[Grasegger]

### A procedure for solving autonomous AODEs

#### Georg Grasegger

Technical report no. 13-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2013. [pdf]@

author = {Georg Grasegger},

title = {{A procedure for solving autonomous AODEs}},

language = {english},

number = {13-03},

year = {2013},

length = {14},

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**{RISC4729,author = {Georg Grasegger},

title = {{A procedure for solving autonomous AODEs}},

language = {english},

number = {13-03},

year = {2013},

length = {14},

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)}

}