# Computations on Algebraic Curves and Surfaces [SFB F1304-1]

### Project Description

F. Winkler.

Budget: 242.005,– Eur.

### Project Lead

### Project Duration

10/04/1998 - 31/03/2001### Project URL

Go to Website## Publications

### 2018

### 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

### 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

### 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. [url]@

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}

}

### 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}

}

### 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}

}

### 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

### 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}

}

### 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

### 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}

}

### 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}

}

### 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}

}

### 2009

### Algorithms in Symbolic Computation

#### Peter Paule, Bruno Buchberger, Lena Kartashova, Manuel Kauers, Carsten Schneider, Franz Winkler

In: Hagenberg Research, Bruno Buchberger et al. (ed.), Chapter 1, pp. 5-62. 2009. Springer, 978-3-642-02126-8. [pdf]@

author = {Peter Paule and Bruno Buchberger and Lena Kartashova and Manuel Kauers and Carsten Schneider and Franz Winkler},

title = {{Algorithms in Symbolic Computation}},

booktitle = {{Hagenberg Research}},

language = {english},

chapter = {1},

pages = {5--62},

publisher = {Springer},

isbn_issn = {978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-C},

editor = {Bruno Buchberger et al.},

refereed = {no},

length = {58}

}

**incollection**{RISC3845,author = {Peter Paule and Bruno Buchberger and Lena Kartashova and Manuel Kauers and Carsten Schneider and Franz Winkler},

title = {{Algorithms in Symbolic Computation}},

booktitle = {{Hagenberg Research}},

language = {english},

chapter = {1},

pages = {5--62},

publisher = {Springer},

isbn_issn = {978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-C},

editor = {Bruno Buchberger et al.},

refereed = {no},

length = {58}

}

### 2008

### Rational Algebraic Curves - A Computer Algebra Approach

#### Winkler, Sendra, Perez-Diaz

Algorithms and Computation in Mathematics 22, Rational Algebraic Curves edition, 2008. Springer Verlag Heidelberg, RISC, 978-3-540-73724-7.@

author = {Winkler and Sendra and Perez-Diaz},

title = {{Rational Algebraic Curves - A Computer Algebra Approach}},

language = {english},

series = {Algorithms and Computation in Mathematics},

volume = {22},

publisher = {Springer Verlag Heidelberg},

isbn_issn = {978-3-540-73724-7},

year = {2008},

edition = {Rational Algebraic Curves},

translation = {0},

institution = {RISC},

length = {0}

}

**book**{RISC4014,author = {Winkler and Sendra and Perez-Diaz},

title = {{Rational Algebraic Curves - A Computer Algebra Approach}},

language = {english},

series = {Algorithms and Computation in Mathematics},

volume = {22},

publisher = {Springer Verlag Heidelberg},

isbn_issn = {978-3-540-73724-7},

year = {2008},

edition = {Rational Algebraic Curves},

translation = {0},

institution = {RISC},

length = {0}

}

### 2005

### Symbolic Computation in Number Theory.

#### Rahul Athale

RISC, Johannes Kepler University Linz. PhD Thesis. 2005.@

author = {Rahul Athale},

title = {{Symbolic Computation in Number Theory.}},

language = {english},

year = {2005},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

**phdthesis**{RISC4111,author = {Rahul Athale},

title = {{Symbolic Computation in Number Theory.}},

language = {english},

year = {2005},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

### 2004

### Representation of sari using computer-generated patterns

#### Manasi Athale

RISC-Linz, Johannes Kepler University, Linz, Austria. PhD Thesis. 2004. [pdf]@

author = {Manasi Athale},

title = {{Representation of sari using computer-generated patterns}},

language = {english},

abstract = {Cloth Modelling is contained inmanyapplications: in e-commerce but in films too. A major aspect is put on western clothes, modelling of eastern cloth is not so much considered, especially in the literature. The reason may be that eastern cloth like Indian Sari, live much more from textile structure and draping the textile around a body.In our thesis, we give an overview on geometric modelling methods and texture mapping with respect of cloth modelling and study their applicability with respect to the sari. In fact, none of the know methods is applicable without changes.Sari is an Indian garment, worn by ladies. It is a rectangular, unstitched piece of fabric, woven using cotton, silk or other threads. Sari is usually five meters long in length and around one to one and half meters in width, with designed borders. Now a days, one can buy a sari on the world wide web. But its representation on the computer screen is the main obstacle in getting the idea of its texture and design pattern.The kernel of the thesis is a generalization and adaptation of such methods, where we use a structuring of the textile shape into four major parts. Although a sari is one piece of textile, there are four parts distinguished by their patterns - two Kath, Padar and Ang -, which have to be modelled separately. By draping the textile around a body, i.e., by generating kinds of layers, the sari is worn and so a volumetric approach can model both, the structure of patterns and the draping around the body.The proposed methods are implemented in Mesa, and examplesof patterns for the sari, i.e., the four main parts are modelled and visualized. Pictures of results are shown in the thesis too.},

year = {2004},

translation = {0},

school = {RISC-Linz, Johannes Kepler University, Linz, Austria},

length = {94}

}

**phdthesis**{RISC2354,author = {Manasi Athale},

title = {{Representation of sari using computer-generated patterns}},

language = {english},

abstract = {Cloth Modelling is contained inmanyapplications: in e-commerce but in films too. A major aspect is put on western clothes, modelling of eastern cloth is not so much considered, especially in the literature. The reason may be that eastern cloth like Indian Sari, live much more from textile structure and draping the textile around a body.In our thesis, we give an overview on geometric modelling methods and texture mapping with respect of cloth modelling and study their applicability with respect to the sari. In fact, none of the know methods is applicable without changes.Sari is an Indian garment, worn by ladies. It is a rectangular, unstitched piece of fabric, woven using cotton, silk or other threads. Sari is usually five meters long in length and around one to one and half meters in width, with designed borders. Now a days, one can buy a sari on the world wide web. But its representation on the computer screen is the main obstacle in getting the idea of its texture and design pattern.The kernel of the thesis is a generalization and adaptation of such methods, where we use a structuring of the textile shape into four major parts. Although a sari is one piece of textile, there are four parts distinguished by their patterns - two Kath, Padar and Ang -, which have to be modelled separately. By draping the textile around a body, i.e., by generating kinds of layers, the sari is worn and so a volumetric approach can model both, the structure of patterns and the draping around the body.The proposed methods are implemented in Mesa, and examplesof patterns for the sari, i.e., the four main parts are modelled and visualized. Pictures of results are shown in the thesis too.},

year = {2004},

translation = {0},

school = {RISC-Linz, Johannes Kepler University, Linz, Austria},

length = {94}

}

### 2003

### A Contribution to the Symmetry Classification Problem for 2nd Order PDEs in one Dependent and two Independent Variables

#### Erik Hillgarter

RISC, Johannes Kepler University Linz. PhD Thesis. 2003.@

author = {Erik Hillgarter},

title = {{A Contribution to the Symmetry Classification Problem for 2nd Order PDEs in one Dependent and two Independent Variables}},

language = {english},

year = {2003},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

**phdthesis**{RISC4115,author = {Erik Hillgarter},

title = {{A Contribution to the Symmetry Classification Problem for 2nd Order PDEs in one Dependent and two Independent Variables}},

language = {english},

year = {2003},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

### 2002

### Implicitization of Algebraic Varieties

#### G.Landsmann

In: Proceedings of the 6th World Multiconference on Systemics, Cybernetics and Informatics, SCI2002, Nagib Callaos (ed.), Computer Science 2 11, pp. 290-296. July 2002. Orlando, Florida, USA, 980-07-8150-1.@

author = {G.Landsmann},

title = {{Implicitization of Algebraic Varieties}},

booktitle = {{Proceedings of the 6th World Multiconference on Systemics, Cybernetics and Informatics, SCI2002}},

language = {english},

series = {Computer Science 2},

volume = {11},

pages = {290--296},

address = {Orlando, Florida, USA},

isbn_issn = {980-07-8150-1},

year = {2002},

month = {July},

editor = {Nagib Callaos},

refereed = {yes},

length = {7}

}

**inproceedings**{RISC408,author = {G.Landsmann},

title = {{Implicitization of Algebraic Varieties}},

booktitle = {{Proceedings of the 6th World Multiconference on Systemics, Cybernetics and Informatics, SCI2002}},

language = {english},

series = {Computer Science 2},

volume = {11},

pages = {290--296},

address = {Orlando, Florida, USA},

isbn_issn = {980-07-8150-1},

year = {2002},

month = {July},

editor = {Nagib Callaos},

refereed = {yes},

length = {7}

}

### 1984

### 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}

}