# Computational Methods for Algebraic and Differential Varieties [SFB F1304-2]

### Project Description

F. Winkler.
Budget: 124.273,– Eur.

### Project Duration

01/04/2001 - 31/03/2004

Go to Website

## 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}
}

[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, , Proceedings of ISSAC '15, pp. 181-188. 2015. 978-1-4503-3435-8. [doi]
@inproceedings{RISC5154,
author = {Christoph Fuerst and Guenter Landsmann},
title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},
booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},
language = {english},
pages = {181--188},
isbn_issn = {978-1-4503-3435-8},
year = {2015},
editor = {ACM},
refereed = {yes},
length = {8},
conferencename = {ISSAC '15},
url = {http://doi.acm.org/10.1145/2755996.2756680}
}
[Fuerst]

### Three Examples of Gröbner Reduction over Noncommutative Rings

#### Christoph Fuerst, Guenter Landsmann

Technical report no. 15-16 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2015. [pdf]
@techreport{RISC5178,
author = {Christoph Fuerst and Guenter Landsmann},
title = {{Three Examples of Gröbner Reduction over Noncommutative Rings}},
language = {english},
abstract = {In this report three classes of noncommutative rings are investigated withemphasis on their properties with respect to reduction relations. TheGröbner basis concepts in these rings, being developed in the literature byseveral authors, are considered and it is shown that the reduction relationscorresponding to these Gröbner bases obey the axioms of a general theoryof Gröbner reduction.},
number = {15-16},
year = {2015},
month = {October},
sponsor = {partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK11},
length = {31},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[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},
}

[Fuerst]

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

#### Christoph Fuerst, Guenter Landsmann

Technical report no. 14-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2014. [pdf]
@techreport{RISC5068,
author = {Christoph Fuerst and Guenter Landsmann},
title = {{The Concept of Gröbner Reduction for Dimension in filtered free modules}},
language = {english},
abstract = {We present the concept of Gröbner reduction that is a Gröbner basistechnique on filtered free modules. It allows to compute the dimensionof a filtered free module viewn as a K-vector space. We apply the de-veloped technique to the computation of a generalization of Hilbert-typedimension polynomials in K[X] as well as in finitely generated difference-differential modules. The latter allows us to determine a multivariatedimension polynomial where we partition the set of derivations and theset of automorphism in a difference-differential ring in an arbitrary way.},
number = {14-12},
year = {2014},
month = {October},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

[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}
}

[Kartaschova]

### Discrete wave turbulence of rotational capillary water waves

#### A. Constantin, E. Kartashova, E. Wahlen

Phys. Fluids submitted, pp. 1-13. 2010. AIP, isbn. [pdf]
@article{RISC3932,
author = {A. Constantin and E. Kartashova and E. Wahlen},
title = {{Discrete wave turbulence of rotational capillary water waves}},
language = {english},
abstract = {We study the discrete wave turbulent regime of capillarywater waves with constant non-zero vorticity. The explicitHamiltonian formulation and the corresponding couplingcoefficient are obtained. We also present the constructionand investigation of resonance clustering. Some physicalimplications of the obtained results are discussed.},
journal = {Phys. Fluids},
volume = {submitted},
pages = {1--13},
publisher = {AIP},
isbn_issn = {isbn},
year = {2010},
refereed = {yes},
length = {13}
}
[Kartaschova]

### Capillary freak waves in He-II as a manifestation of discrete wave turbulent regime

#### E. Kartashova

In: Geophysical Research Abstracts, , Proceedings of EGU 2010 (European Geosciences Union, General Assembly 2010),12, pp. 1889-1889. 2010. issn. [pdf]
@inproceedings{RISC3933,
author = {E. Kartashova},
title = {{Capillary freak waves in He-II as a manifestation of discrete wave turbulent regime}},
booktitle = {{Geophysical Research Abstracts}},
language = {english},
abstract = {Two fundamental findings of the modern theory of wave turbulence are 1) existence of Kolmogorov-Zakharov power energy spectra (KZ-spectra) in $k$-space, \cite{zak2}, and 2) existence of gaps" in KZ-spectra corresponding to the resonance clustering, \cite{K06-1}. Accordingly, three wave turbulent regimes can be singled out - \emph{kinetic} (described by wave kinetic equations and KZ-spectra, in random phase approximation, \cite{ZLF92}); \emph{discrete} (described by a few dynamical systems, with coherent phases corresponding to resonance conditions, \cite{K09b}) and \emph{mesoscopic} (where kinetic and discrete evolution of the wave field coexist, \cite{zak4}).We present an explanation of freak waves appearance in capillary waves in He-II, \cite{ABKL09}, as a manifestation of discrete wave turbulent regime. Implications of these results for other wave systems are briefly discussed.},
volume = {12},
pages = {1889--1889},
isbn_issn = {issn},
year = {2010},
editor = { E. Pelinovsky and C. Kharif },
refereed = {yes},
length = {1},
conferencename = {EGU 2010 (European Geosciences Union, General Assembly 2010),}
}
[Kartaschova]

### Towards a Theory of Discrete and Mesoscopic Wave Turbulence

#### E.Kartashova, V. Lvov, S. Nazarenko, I. Procaccia

Submitted to the RISC Report Series. Technical report no. 10-04 in RISC Report Series, February 2010. [pdf]
@techreport{RISC3967,
author = {E.Kartashova and V. Lvov and S. Nazarenko and I. Procaccia},
title = {{Towards a Theory of Discrete and Mesoscopic Wave Turbulence}},
language = {english},
abstract = {This is WORK IN PROGRESS carried out in years 2008-2009 and partly supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the project SCIEnce, Contract No. 026133).Abstract:\emph{Discrete wave turbulence} in bounded media refers to the regular and chaotic dynamics of independent (that is, discrete in $k$-space) resonance clusters consisting of finite (often fairly big) number of connected wave triads or quarters, with exact three- or four-wave resonances correspondingly. "Discreteness" means that for small enough amplitudes there is no energy flow among the clusters. Increasing of wave amplitudes and/or of system size opens new channels of wave interactions via quasi-resonant clusters. This changes statistics of energy exchange between waves and results in new, \emph{mesoscopic} regime of \emph{wave turbulence}, where \emph{discrete wave turbulence} and \emph{kinetic wave turbulence} in unbounded media co-exist, the latter well studied in the framework of wave kinetic equations. We overview in systematic manner and from unified viewpoint some preliminary results of studies of regular and stochastic wave behavior in bounded media, aiming to shed light on their relationships and to clarify their role and place in the structure of a future theory of discrete and mesoscopic wave turbulence, elucidated in this paper. We also formulate a set of yet open questions and problems in this new field of nonlinear wave physics, that awaits for comprehensive studies in the framework of the theory. We hope that the resulting theory will offer very interesting issues both from the physical and the methodological viewpoints, with possible important applications in environmental sciences, fluid dynamics, astronomy and plasma physics.},
year = {2010},
month = {February},
howpublished = {Technical report no. 10-04 in RISC Report Series},
length = {42},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Kartaschova]

### Resonance clustering in wave turbulent regimes: Integrable dynamics

#### E. Kartashova, M. Bustamante

Physica A: Stat. Mech. Appl. submitted, pp. 1-31. 2010. Elsevier, ISSN: 0378-4371. [pdf]
@article{RISC3966,
author = {E. Kartashova and M. Bustamante},
title = {{Resonance clustering in wave turbulent regimes: Integrable dynamics}},
language = {english},
abstract = {Two fundamental facts of the modern wave turbulence theory are1) existence of power energy spectra in $k$-space, and 2) existence of gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic},where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Usinga novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters areintegrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. Wealso identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.},
journal = {Physica A: Stat. Mech. Appl.},
volume = {submitted},
pages = {1--31},
publisher = {Elsevier},
isbn_issn = {ISSN: 0378-4371},
year = {2010},
refereed = {yes},
length = {31}
}
[Kartaschova]

### Turbulence of capillary waves revisited

#### E. Kartashova, A. Kartashov

EPL submitted, pp. 1-6. 2010. isbn. [url]
@article{RISC4026,
author = {E. Kartashova and A. Kartashov},
title = {{Turbulence of capillary waves revisited}},
language = {english},
abstract = {Kinetic regime of capillary wave turbulence is classically regarded in terms of three-wave interactions with the exponent of power energy spectrum being $\nu=-7/4$ (two-dimensional case). We show that a number of assumptions necessary for this regime to occur can not be fulfilled. Four-wave interactions of capillary waves should be taken into account instead, which leads to exponents $\nu=-13/6$ and $\nu=-3/2$ for one- and two-dimensional wavevectors correspondingly. It follows that for general dispersion functions of decay type, three-wave kinetic regime need not prevail and higher order resonances may play a major role.},
journal = {EPL},
volume = {submitted},
pages = {1--6},
isbn_issn = {isbn},
year = {2010},
refereed = {yes},
length = {6},
url = {http://arxiv.org/abs/1005.2067}
}
[Kartaschova]

### Nonlinear Resonance Analysis

#### E. Kartashova

Cambridge edition, 2010. Cambridge University Press, ISBN-13: 9780521763608. [url]
@booklet{RISC4037,
author = {E. Kartashova},
title = {{Nonlinear Resonance Analysis}},
language = {english},
publisher = {Cambridge University Press},
isbn_issn = {ISBN-13: 9780521763608},
year = {2010},
edition = {Cambridge},
translation = {0},
length = {250},
url = {http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521763608}
}

[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. [doi] [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}
}