# Differential Elimination Theory [DET]

### Project Description

F.Winkler.
Budget: 269.472,–

### Project Duration

01/09/2003 - 30/04/2007

## Publications

[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.
@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}
}

[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), , pp. 101-104. 2016. 978-84-608-9024-9.
@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}
}

[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.
@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.
@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, , Lecture Notes in Computer Science 9301, pp. 479-492. 2015. Springer International Publishing, ISSN 0302-9743. [url]
@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},
}

[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.
@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.
@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}
}

[Judakova]

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

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

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

### 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.
@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), , pp. 33-34. 2012. 978-84-8138-770-4.
@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}
}

[Buchberger]

### Algorithms in Symbolic Computation

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

In: Hagenberg Research, , Chapter 1, pp. 5-62. 2009. Springer, 978-3-642-02126-8. [url] [pdf]
@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},
url = {https://doi.org/10.1007/978-3-642-02127-5_2}
}

[Winkler]

### 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.
@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}
}

[Winkler]

### On Computing Groebner Bases in Rings of Differential Operators with Coefficients in a Ring

#### M. Zhou, F. Winkler

Mathematics in Computer Science 1(2), pp. 211-223. 2007. ISSN 1661-8270. [pdf]
@article{RISC3482,
author = {M. Zhou and F. Winkler},
title = {{On Computing Groebner Bases in Rings of Differential Operators with Coefficients in a Ring}},
language = {english},
journal = {Mathematics in Computer Science},
volume = {1},
number = {2},
pages = {211--223},
isbn_issn = {ISSN 1661-8270},
year = {2007},
refereed = {yes},
length = {12}
}
[Zhou]

### Computing difference-differential Groebner Bases and difference-differential dimension polynomials

#### Meng Zhou, Franz Winkler

Technical report no. 07-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 2007. [pdf] [pdf]
@techreport{RISC3042,
author = {Meng Zhou and Franz Winkler},
title = {{Computing difference-differential Groebner Bases and difference-differential dimension polynomials}},
language = {english},
abstract = {Dfference-differential Groebner bases and the algorithms were introduced byM.Zhou and F.Winkler (2006). In this paper we will make further investigationsfor the key concept of S-polynomials in the algorithm and we will improve tech-nically the algorithm. Then we apply the algorithm to compute the difference-differential dimension polynomial of a difference-differential module and of asystem of linear partial difference-differential equations. Also, in cyclic modulecase, we present an algorithm to compute the difference-differential dimensionpolynomials in two variables with the Groebner basis.},
number = {07-01},
year = {2007},
month = {January},
keywords = {difference-differential Groebner basis, S-polynomial, difference-differential dimension polynomial},
length = {18},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

[Lopez Diaz]

### Symbolic Methods for Factoring Linear Differential Operators

#### Glauco Alfredo Lopez Diaz

RISC. PhD Thesis. February 2006. [pdf] [ps]
@phdthesis{RISC2871,
author = {Glauco Alfredo Lopez Diaz},
title = {{Symbolic Methods for Factoring Linear Differential Operators}},
language = {english},
abstract = {A survey of symbolic methods for factoring linear differential operators is given. Starting from basic notions -- ring of operators, differential Galois theory -- methods for finding rational and exponential solutions that can provide first order right-hand factors are considered. Subsequently several known algorithms for factorization are presented. These include Singer's eigenring factorization algorithm, factorization via Newton polygons, van Hoeij's methods for local factorization, and an adapted version of Pade approximation.In addition a procedure based on pure algebraic methods for factoring second order linear partial differential operators is developed. Splitting an operator of this kind reduces to solving a system of linear algebraic equations. Those solutions which satisfy a certain differential condition, immediately produce linear factors of the operator. The method applies also to operators of third order, thereby resulting in a more complicated system of equations. In contrast to the second order case, differential equations must also be solved, which, in particular cases, are simplified with the aid of characteristic sets.Finally, complete decomposition into linear factors of ordinary differential operators of arbitrary order is discussed. A splitting formula is developed, provided that a linear basis of solutions is available. This theoretical representation is valuable in understanding the nature of the classical Beke algorithm and its variants like the algorithm LODEF by Schwarz and the Beke-Bronstein algorithm.},
year = {2006},
month = {February},
translation = {0},
school = {RISC},
keywords = {Symbolic Differential Computation},
sponsor = {DET Project, Fonds zur F\"orderung der wiss. Forschung, P16357-N04.},
length = {112}
}
[Lopez Diaz]

### Symbolic Methods for Factoring Linear Differential Operators

#### Glauco Alfredo Lopez Diaz

RISC. Technical report no. 06-02, February 2006. [pdf] [ps]
@techreport{RISC2872,
author = {Glauco Alfredo Lopez Diaz},
title = {{Symbolic Methods for Factoring Linear Differential Operators}},
language = {english},
abstract = {A survey of symbolic methods for factoring linear differential operators is given. Starting from basic notions -- ring of operators, differential Galois theory -- methods for finding rational and exponential solutions that can provide first order right-hand factors are considered. Subsequently several known algorithms for factorization are presented. These include Singer's eigenring factorization algorithm, factorization via Newton polygons, van Hoeij's methods for local factorization, and an adapted version of Pade approximation.In addition a procedure based on pure algebraic methods for factoring second order linear partial differential operators is developed. Splitting an operator of this kind reduces to solving a system of linear algebraic equations. Those solutions which satisfy a certain differential condition, immediately produce linear factors of the operator. The method applies also to operators of third order, thereby resulting in a more complicated system of equations. In contrast to the second order case, differential equations must also be solved, which, in particular cases, are simplified with the aid of characteristic sets.Finally, complete decomposition into linear factors of ordinary differential operators of arbitrary order is discussed. A splitting formula is developed, provided that a linear basis of solutions is available. This theoretical representation is valuable in understanding the nature of the classical Beke algorithm and its variants like the algorithm LODEF by Schwarz and the Beke-Bronstein algorithm.},
number = {06-02},
year = {2006},
month = {February},
institution = {RISC},
keywords = {Symbolic Differential Computation},
sponsor = {DET Project, Fonds zur Foerderung der wiss. Forschung, P16357-N04.},
length = {112}
}

[Winkler]

### On computing Groebner bases in rings of differential operators with coefficients in a ring

#### F.Winkler and M.Zhou

Technical report no. 05-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2005. [pdf]
@techreport{RISC2471,
author = {F.Winkler and M.Zhou},
title = {{On computing Groebner bases in rings of differential operators with coefficients in a ring}},
language = {english},
number = {05-04},
year = {2005},
sponsor = {This work has been supported by the FWF project P16357-N04.},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Winkler]

### Using Groebner bases for determining the equivalence of linear codes and solving the decoding problem

#### M. Borges-Quintana, M. A. Borges-Trenard, E. Martinez-Moro, F. Winkler

Technical report no. 05-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2005. [ps]
@techreport{RISC2478,
author = {M. Borges-Quintana and M. A. Borges-Trenard and E. Martinez-Moro and F. Winkler},
title = {{Using Groebner bases for determining the equivalence of linear codes and solving the decoding problem}},
language = {english},
number = {05-12},
year = {2005},
length = {32},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Winkler]

### Grobner bases in difference-differential modules and their applications

#### Meng Zhou and Franz Winkler

Technical report no. 05-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. October 2005. [ps] [pdf]
@techreport{RISC2734,
author = {Meng Zhou and Franz Winkler},
title = {{Grobner bases in difference-differential modules and their applications}},
language = {english},
abstract = {In this paper we will extend the theory of Gr\"obner bases to difference-differential modules which were introduced by Levin(2000) as a generalization of modules over rings of differential operators. The main goal of this paper is to present and verify algorithms for constructing these Gr\"obner basis counterparts. To this aim we define the concept of ''generalized term order'' on ${\Bbb N}^m \times {\Bbb Z}^n$ and on difference-differential modules. The relation between the Gr\"obner bases and some characteristic sets in the modules is also considered. As applications, we can compute the difference-differential dimension polynomial of a difference-differential module and of a system of linear partial difference-differential equations via the Gr\"obner bases.},
number = {05-14},
year = {2005},
month = {October},
keywords = {Gr\"obner basis, generalized term order, difference-differential module, difference-differential dimension polynomial.},
sponsor = {This work has been supported by the FWF project P16357-N04, while the first author spent a research year at RISC-Linz.},
length = {29},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

[Hillgarter]

### 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.
@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}
}