# Symbolic and Algebraic Methods for LPDOs [DIFFOP]

### Project Lead

### Project Duration

01/05/2008 - 30/04/2011### 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}

}

### Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems

#### J. Middeke, C. Schneider

To appear in ACM Communications in Computer Algebra, pp. -. 2018. ISSN 1932-2240. Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18). [pdf] [pdf]@

author = {J. Middeke and C.~Schneider},

title = {{Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems}},

language = {english},

journal = {To appear in ACM Communications in Computer Algebra},

pages = {--},

isbn_issn = {ISSN 1932-2240},

year = {2018},

note = {Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18)},

refereed = {yes},

length = {0}

}

**article**{RISC5736,author = {J. Middeke and C.~Schneider},

title = {{Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems}},

language = {english},

journal = {To appear in ACM Communications in Computer Algebra},

pages = {--},

isbn_issn = {ISSN 1932-2240},

year = {2018},

note = {Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18)},

refereed = {yes},

length = {0}

}

### 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

### 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

### Characterization of relative Gröbner bases

#### Christian Dönch

Journal of Symbolic Computation 55, pp. 19-29. 2013. 0747-7171.@

author = {Christian Dönch},

title = {{Characterization of relative Gröbner bases}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {55},

pages = {19--29},

isbn_issn = {0747-7171},

year = {2013},

refereed = {yes},

length = {11}

}

**article**{RISC4738,author = {Christian Dönch},

title = {{Characterization of relative Gröbner bases}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {55},

pages = {19--29},

isbn_issn = {0747-7171},

year = {2013},

refereed = {yes},

length = {11}

}

### 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}

}

### 2011

### A computational view on normal forms of matrices of Ore polynomials

#### Johannes Middeke

Research Institute for Symbolic Computation (RISC). PhD Thesis. July 2011. RISC Technical Report 11-10. [pdf]@

author = {Johannes Middeke},

title = {{A computational view on normal forms of matrices of Ore polynomials}},

language = {english},

abstract = {This thesis treats normal forms of matrices over rings of Orepolynomials. The whole thesis is divided in three parts: First, Orepolynomials are described and basic facts about them arerecalled. This part also includes integro-differential operators as anextended example. Second, in the main part we present one- andtwo-sided normal forms of matrices. More precisely, we deal with thePopov normal form, Hermite normal form and the Jacobson normal form. In the last part, we explore an applicationof matrix normal forms to a problem in control theory.},

year = {2011},

month = {July},

translation = {0},

school = {Research Institute for Symbolic Computation (RISC)},

length = {102},

type = {RISC Technical Report 11-10}

}

**phdthesis**{RISC4377,author = {Johannes Middeke},

title = {{A computational view on normal forms of matrices of Ore polynomials}},

language = {english},

abstract = {This thesis treats normal forms of matrices over rings of Orepolynomials. The whole thesis is divided in three parts: First, Orepolynomials are described and basic facts about them arerecalled. This part also includes integro-differential operators as anextended example. Second, in the main part we present one- andtwo-sided normal forms of matrices. More precisely, we deal with thePopov normal form, Hermite normal form and the Jacobson normal form. In the last part, we explore an applicationof matrix normal forms to a problem in control theory.},

year = {2011},

month = {July},

translation = {0},

school = {Research Institute for Symbolic Computation (RISC)},

length = {102},

type = {RISC Technical Report 11-10}

}

### A toolbox for the analysis of linear systems with delays

#### Felix Antritter and Johannes Middeke

In: Proceedings of the 50th IEEE Conference on Decision and Control (CDC 2011), , Proceedings of 50th IEEE Conference on Decision and Control (CDC 2011), pp. -. December 2011. Orlando, Florida, USA, IEEE,@

author = {Felix Antritter and Johannes Middeke},

title = {{A toolbox for the analysis of linear systems with delays}},

booktitle = {{Proceedings of the 50th IEEE Conference on Decision and Control (CDC 2011)}},

language = {english},

pages = {--},

address = {Orlando, Florida, USA},

isbn_issn = {?},

year = {2011},

month = {December},

editor = {?},

refereed = {yes},

organization = {IEEE},

length = {0},

conferencename = {50th IEEE Conference on Decision and Control (CDC 2011)}

}

**inproceedings**{RISC4447,author = {Felix Antritter and Johannes Middeke},

title = {{A toolbox for the analysis of linear systems with delays}},

booktitle = {{Proceedings of the 50th IEEE Conference on Decision and Control (CDC 2011)}},

language = {english},

pages = {--},

address = {Orlando, Florida, USA},

isbn_issn = {?},

year = {2011},

month = {December},

editor = {?},

refereed = {yes},

organization = {IEEE},

length = {0},

conferencename = {50th IEEE Conference on Decision and Control (CDC 2011)}

}

### Rational general solutions of parametrizable AODEs

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

Publicationes Mathematicae Debrecen, pp. 573-587. 2011. ISSN 0033-3883.@

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

title = {{Rational general solutions of parametrizable AODEs}},

language = {english},

journal = {Publicationes Mathematicae Debrecen},

pages = {573--587},

isbn_issn = {ISSN 0033-3883},

year = {2011},

refereed = {yes},

length = {15}

}

**article**{RISC4439,author = {L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of parametrizable AODEs}},

language = {english},

journal = {Publicationes Mathematicae Debrecen},

pages = {573--587},

isbn_issn = {ISSN 0033-3883},

year = {2011},

refereed = {yes},

length = {15}

}

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

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

In: Proceedings Forth Internat. Conf. on Mathematical Aspects of Computer and Information Sciences (MACIS 2011), 111 (ed.), pp. 93-100. 2011. 111.@

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

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

booktitle = {{Proceedings Forth Internat. Conf. on Mathematical Aspects of Computer and Information Sciences (MACIS 2011)}},

language = {english},

pages = {93--100},

isbn_issn = {111},

year = {2011},

editor = {111},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4441,author = {Y. Huang and L.X.C. Ngo and F. Winkler},

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

booktitle = {{Proceedings Forth Internat. Conf. on Mathematical Aspects of Computer and Information Sciences (MACIS 2011)}},

language = {english},

pages = {93--100},

isbn_issn = {111},

year = {2011},

editor = {111},

refereed = {yes},

length = {8}

}

### Linear partial differential equations and linear partial differential operators in computer algebra

#### shemyakova, winkler

In: Numerical and Symbolic Scientific Computing - Progress and Prospects, Springer Verlag (ed.), pp. 333-358. 2011. Springer Verlag, ISSN 0943-853X.@

author = {shemyakova and winkler},

title = {{Linear partial differential equations and linear partial differential operators in computer algebra}},

booktitle = {{Numerical and Symbolic Scientific Computing - Progress and Prospects}},

language = {english},

pages = {333--358},

publisher = {Springer Verlag},

isbn_issn = {ISSN 0943-853X},

year = {2011},

editor = {Springer Verlag },

refereed = {yes},

length = {26}

}

**incollection**{RISC4431,author = {shemyakova and winkler},

title = {{Linear partial differential equations and linear partial differential operators in computer algebra}},

booktitle = {{Numerical and Symbolic Scientific Computing - Progress and Prospects}},

language = {english},

pages = {333--358},

publisher = {Springer Verlag},

isbn_issn = {ISSN 0943-853X},

year = {2011},

editor = {Springer Verlag },

refereed = {yes},

length = {26}

}

### What can Symbolic Computation contribute to Mathematics?

#### F. Winkler

In: Proceedings SYNASC 2011,, aaa (ed.), pp. 1-14. 2011. 1111.@

author = {F. Winkler},

title = {{What can Symbolic Computation contribute to Mathematics?}},

booktitle = {{Proceedings SYNASC 2011,}},

language = {english},

pages = {1--14},

isbn_issn = {1111},

year = {2011},

editor = {aaa},

refereed = {yes},

length = {15}

}

**inproceedings**{RISC4440,author = {F. Winkler},

title = {{What can Symbolic Computation contribute to Mathematics?}},

booktitle = {{Proceedings SYNASC 2011,}},

language = {english},

pages = {1--14},

isbn_issn = {1111},

year = {2011},

editor = {aaa},

refereed = {yes},

length = {15}

}

### 2010

### Relations between Groebner bases, differential Groebner bases, and differential characteristic sets

#### Christian Aistleitner

Johannes Kepler University, Linz, Austria. Diploma Thesis. December 2010. [pdf]@

author = {Christian Aistleitner},

title = {{Relations between Groebner bases, differential Groebner bases, and differential characteristic sets}},

language = {english},

abstract = {While Gröbner bases classically focus on purely algebraic settings, Gröbner basis literature followed the general trend of the last decades to also incorporate differential settings, which resulted in the notion of differential Gröbner bases. In the differential setting, there is also the much older, but different notion of differential characteristec sets. Although those three methods of elimination theory are closely related, literature does not provide a comparison of those methods.The main contribution of this diploma thesis is such a comparison.Additionally, we give a presentation of Gröbner bases, differential Gröbner bases, and differential characteristic sets using a unified notation system that allows to easily identify and exhibit differences and matches between the different methods.},

year = {2010},

month = {December},

translation = {0},

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

length = {99}

}

**mastersthesis**{RISC4229,author = {Christian Aistleitner},

title = {{Relations between Groebner bases, differential Groebner bases, and differential characteristic sets}},

language = {english},

abstract = {While Gröbner bases classically focus on purely algebraic settings, Gröbner basis literature followed the general trend of the last decades to also incorporate differential settings, which resulted in the notion of differential Gröbner bases. In the differential setting, there is also the much older, but different notion of differential characteristec sets. Although those three methods of elimination theory are closely related, literature does not provide a comparison of those methods.The main contribution of this diploma thesis is such a comparison.Additionally, we give a presentation of Gröbner bases, differential Gröbner bases, and differential characteristic sets using a unified notation system that allows to easily identify and exhibit differences and matches between the different methods.},

year = {2010},

month = {December},

translation = {0},

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

length = {99}

}

### Bivariate Difference-differential Dimension Polynomials and Their Computation in Maple

#### Christian Doench, Franz Winkler

In: Proceedings of the 8th International Conference on Applied Informatics, Attila Egri-Nagy, Emőd Kovács, Gergely Kovásznai, Gábor Kusper, Tibor Tómács (ed.), pp. 211-218. 2010. ISBN 978-963-9894-72-3. [pdf]@

author = {Christian Doench and Franz Winkler},

title = {{Bivariate Difference-differential Dimension Polynomials and Their Computation in Maple}},

booktitle = {{Proceedings of the 8th International Conference on Applied Informatics}},

language = {english},

pages = {211--218},

isbn_issn = {ISBN 978-963-9894-72-3},

year = {2010},

editor = {Attila Egri-Nagy and Emőd Kovács and Gergely Kovásznai and Gábor Kusper and Tibor Tómács},

refereed = {no},

length = {8}

}

**inproceedings**{RISC4382,author = {Christian Doench and Franz Winkler},

title = {{Bivariate Difference-differential Dimension Polynomials and Their Computation in Maple}},

booktitle = {{Proceedings of the 8th International Conference on Applied Informatics}},

language = {english},

pages = {211--218},

isbn_issn = {ISBN 978-963-9894-72-3},

year = {2010},

editor = {Attila Egri-Nagy and Emőd Kovács and Gergely Kovásznai and Gábor Kusper and Tibor Tómács},

refereed = {no},

length = {8}

}