# 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## Partners

### The Austrian Science Fund (FWF)

## 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

[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. [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}

}

[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, Schloss Hagenberg, 4232 Hagenberg, Austria. 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 = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**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 = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

[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}

}

### 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, Schloss Hagenberg, 4232 Hagenberg, Austria. 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 = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**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 = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 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]

### 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}

}

### 2009

[Buchberger]

### 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

[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.@

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}

}

### 2007

[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]@

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},

sponsor = {FWF project P16357-N04},

length = {18},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**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},

sponsor = {FWF project P16357-N04},

length = {18},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2005

[Athale]

### 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}

}

[Gu]

### Preprocessing for Finite Element Discretizations of Geometric Problems

#### Hong Gu, Martin Burger

Technical report no. 05-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. May 2005. accepted to be published in proceeding of SNC'05. [ps]@

author = {Hong Gu and Martin Burger},

title = {{Preprocessing for Finite Element Discretizations of Geometric Problems}},

language = {english},

abstract = {In this paper, we use finite element method to approximate thesolutions of parameter-dependent geometric problems, andinvestigate the possibility of using symbolic methods as a preprocessing step.The main idea of our approach is to construct suitable finite elementdiscretizations of the nonlinear elliptic equations leading tosystems of algebraic equations, which can be subsequently solved bysymbolic computation within the tolerance of computer algebra software. The prolongation of the preprocessed symbolicsolution can serve as a starting value for a numerical iterative method ona finer grid.A motivation for this approach is that usual numerical iterations (e.g. viaNewton-type or fixed-point iterations) may diverge if no appropriate initialvalues are available.Moreover, such a purely numerical approach will not find all solutions of thediscretized problem if there are more than one. A final motivation for the use ofsymbolic methodsis the fact that all discrete solutions can be obtained as functions of unknownparameters. In this paper, we focus on a special class ofpartial differential equations derived from geometric problems. A mainchallenge in this class is the fact that the polynomial structure of thenonlinearity is not explicit in thedivergence form usually used for finite element discretization. As a consequence,the discrete form would always yield some non-polynomialterms. We therefore consider two different discretizations, namely a polynomialreformulation before discretization and a direct discretization of the divergence form withpolynomial approximation of the discrete system.In order to perform a detailed analysis and convergence theory of the discretizationmethods we investigate some model problems related to mean-curvaturetype equations.},

number = {05-02},

year = {2005},

month = {May},

note = {accepted to be published in proceeding of SNC'05},

keywords = {Finite element methods, symbolic computation, preprocessing, Newton iteration, multigrid, polynomial equations},

sponsor = {SFB F013/F1304, F1308},

length = {19},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC2464,author = {Hong Gu and Martin Burger},

title = {{Preprocessing for Finite Element Discretizations of Geometric Problems}},

language = {english},

abstract = {In this paper, we use finite element method to approximate thesolutions of parameter-dependent geometric problems, andinvestigate the possibility of using symbolic methods as a preprocessing step.The main idea of our approach is to construct suitable finite elementdiscretizations of the nonlinear elliptic equations leading tosystems of algebraic equations, which can be subsequently solved bysymbolic computation within the tolerance of computer algebra software. The prolongation of the preprocessed symbolicsolution can serve as a starting value for a numerical iterative method ona finer grid.A motivation for this approach is that usual numerical iterations (e.g. viaNewton-type or fixed-point iterations) may diverge if no appropriate initialvalues are available.Moreover, such a purely numerical approach will not find all solutions of thediscretized problem if there are more than one. A final motivation for the use ofsymbolic methodsis the fact that all discrete solutions can be obtained as functions of unknownparameters. In this paper, we focus on a special class ofpartial differential equations derived from geometric problems. A mainchallenge in this class is the fact that the polynomial structure of thenonlinearity is not explicit in thedivergence form usually used for finite element discretization. As a consequence,the discrete form would always yield some non-polynomialterms. We therefore consider two different discretizations, namely a polynomialreformulation before discretization and a direct discretization of the divergence form withpolynomial approximation of the discrete system.In order to perform a detailed analysis and convergence theory of the discretizationmethods we investigate some model problems related to mean-curvaturetype equations.},

number = {05-02},

year = {2005},

month = {May},

note = {accepted to be published in proceeding of SNC'05},

keywords = {Finite element methods, symbolic computation, preprocessing, Newton iteration, multigrid, polynomial equations},

sponsor = {SFB F013/F1304, F1308},

length = {19},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

[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]@

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}

}

**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]@

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}

}

**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}

}