Univ.-Prof. DI.Dr. Franz Winkler

Research Area

Computer Algebra, Constructive Algebraic Geometry

Office

Schloss Hagenberg
A-4232 Hagenberg im Mühlkreis

Room: 2.5-1

Postal Address

Research Institute for Symbolic Computation
Johannes Kepler University
Altenberger Straße 69
A-4040 Linz, Austria

Ongoing Projects

Symbolic Solutions of Algebraic Differential Equations [ADE-solve]

Project Lead: Franz Winkler

Project Duration: 01/05/2018 - 30/04/2021

Course Id	Title	Registration	Type	Hours	Teachers	Rhythm
326.012 (2019S)	Bachelor Seminar with Bachelor Thesis	Register	SE	2,0	Peter Paule Bruno Buchberger Ralf Hemmecke Tudor Jebelean Teimuraz Kutsia Günter Landsmann Veronika Elisabeth Pillwein Josef Schicho Carsten Schneider Wolfgang Schreiner Wolfgang Windsteiger Franz Winkler	Weekly
326.0KA (2019S)	Commutative algebra and algebraic geometry Further information	Register	VO	4,0	Franz Winkler	Weekly
326.0ZZ (2019S)	Seminar for graduate and doctoral students Winter Semester 2018/19 Further information	Register	SE	2,0	Franz Winkler	Weekly
326.0CA (2019S)	Seminar symbolic computation Computer algebra II Further information	Register	SE	2,0	Franz Winkler	Weekly

Content provided by KUSSS, Johannes Kepler University Linz | E-Mail

Software

CASA

Computer Algebra System for Algebraic Geometry

CASA is a special-purpose system for computational algebra and constructive algebraic geometry. The system has been developed since 1990, and is the ongoing product of the Computer Algebra Group under the direction of Prof. Winkler. It is built on the ...

Authors: Franz Winkler

More Software Website

Publications

2018

Resultants: Algebraic and Differential

S. McCallum, F. Winkler

Technical report no. 18-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. August 2018. [pdf]

[bib]

@techreport{RISC5735,
author = {S. McCallum and F. Winkler},
title = {{Resultants: Algebraic and Differential}},
language = {english},
abstract = {This report summarises ongoing discussions of the authors on the topic of differential resultantswhich have three goals in mind. First, we aim to try to understand existing literature on thetopic. Second, we wish to formulate some interesting questions and research goals based on ourunderstanding of the literature. Third, we would like to advance the subject in one or moredirections, by pursuing some of these questions and research goals. Both authors have somewhatmore background in nondifferential, as distinct from differential, computational algebra. For thisreason, our approach to learning about differential resultants has started with a careful review ofthe corresponding theory of resultants in the purely algebraic (polynomial) case. We try, as faras possible, to adapt and extend our knowledge of purely algebraic resultants to the differentialcase. Overall, we have the hope of helping to clarify, unify and further develop the computationaltheory of differential resultants.},
number = {18-08},
year = {2018},
month = {August},
length = {21},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

Rational general solutions of systems of first-order algebraic partial differential equations

G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

J. Computational and Applied Mathematics(331), pp. 88-103. 2018. ISSN 0377-0427. [pdf]

[bib]

@article{RISC5837,
author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},
title = {{Rational general solutions of systems of first-order algebraic partial differential equations}},
language = {english},
journal = {J. Computational and Applied Mathematics},
number = {331},
pages = {88--103},
isbn_issn = {ISSN 0377-0427},
year = {2018},
refereed = {yes},
length = {16}
}

Deciding the existence of rational general solutions for first-order algebraic ODEs

N.T. Vo, G. Grasegger, F. Winkler

Journal of Symbolic Computation(87), pp. 127-139. 2018. ISSN 0747-7171. [pdf]

[bib]

@article{RISC5838,
author = {N.T. Vo and G. Grasegger and F. Winkler},
title = {{Deciding the existence of rational general solutions for first-order algebraic ODEs}},
language = {english},
journal = {Journal of Symbolic Computation},
number = {87},
pages = {127--139},
isbn_issn = {ISSN 0747-7171},
year = {2018},
refereed = {yes},
length = {13}
}

Das Unendliche im mathemtischen Alltag

F. Winkler

In: Beiträge des 41. Internationalen Wittgenstein Symposiums, G.M. Mras, P. Weingartner, B. Ritter (ed.), Proceedings of 41. Internationales Wittgenstein Symposium, pp. 285-287. August 2018. ISSN 1022-3398. [pdf]

[bib]

@inproceedings{RISC5840,
author = {F. Winkler},
title = {{Das Unendliche im mathemtischen Alltag}},
booktitle = {{Beiträge des 41. Internationalen Wittgenstein Symposiums}},
language = {english},
pages = {285--287},
isbn_issn = {ISSN 1022-3398},
year = {2018},
month = {August},
editor = {G.M. Mras and P. Weingartner and B. Ritter },
refereed = {yes},
length = {3},
conferencename = {41. Internationales Wittgenstein Symposium}
}

Differential Resultants, in Recent Advances in Algebra, Numerical Analysis and Statistics

S. McCallum, F. Winkler

In: Proc. Internat. Conf. on Mathematics (ICM 2018), R. Bris et al. (ed.), Proceedings of ICM 2018, pp. 1-11. Dezember 2018. Ton Duc Thang University (TDTU), Ho Chi Minh City, Vietnam, ISBN 978-2-7598-9058-3. [url] [pdf]

[bib]

@inproceedings{RISC5842,
author = {S. McCallum and F. Winkler},
title = {{Differential Resultants, in Recent Advances in Algebra, Numerical Analysis and Statistics}},
booktitle = {{Proc. Internat. Conf. on Mathematics (ICM 2018)}},
language = {english},
pages = {1--11},
isbn_issn = {ISBN 978-2-7598-9058-3},
year = {2018},
month = {Dezember},
editor = {R. Bris et al.},
refereed = {yes},
institution = {Ton Duc Thang University (TDTU), Ho Chi Minh City, Vietnam},
length = {11},
conferencename = {ICM 2018},
url = {http://icm2018.tdtu.edu.vn/}
}

2016

A solution method for autonomous first-order algebraic partial differential equations

G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Journal of Computational and Applied Mathematics 300, pp. 119-133. 2016. 0377-0427. [url]

[bib]

@article{RISC5202,
author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},
title = {{A solution method for autonomous first-order algebraic partial differential equations}},
language = {english},
journal = {Journal of Computational and Applied Mathematics},
volume = {300},
pages = {119--133},
isbn_issn = {0377-0427},
year = {2016},
refereed = {yes},
length = {15},
url = {http://dx.doi.org/10.1016/j.cam.2015.12.030}
}

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.

[bib]

@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

Symbolic Solutions of First-Order Algebraic ODEs

G. Grasegger, F. Winkler

In: Computer algebra and polynomials, J. Gutierrez, J. Schicho, M. Weimann (ed.), Lecture Notes in Computer Science 8942, pp. 94-104. 2015. Springer International Publishing, ISSN 0302-9743. [url]

[bib]

@inproceedings{RISC5018,
author = {G. Grasegger and F. Winkler},
title = {{Symbolic Solutions of First-Order Algebraic ODEs}},
booktitle = {{Computer algebra and polynomials}},
language = {english},
series = {Lecture Notes in Computer Science},
volume = {8942},
pages = {94--104},
publisher = {Springer International Publishing},
isbn_issn = {ISSN 0302-9743},
year = {2015},
editor = {J. Gutierrez and J. Schicho and M. Weimann},
refereed = {yes},
length = {11},
url = {http://dx.doi.org/10.1007/978-3-319-15081-9_5}
}

A solution method for autonomous first-order algebraic partial differential equations in several variables

G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Technical report no. 15-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2015. [pdf]

[bib]

@techreport{RISC5117,
author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},
title = {{A solution method for autonomous first-order algebraic partial differential equations in several variables}},
language = {english},
number = {15-01},
year = {2015},
length = {22},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

Rational General Solutions of First-Order Algebraic ODEs

N.T. Vo, G. Grasegger, F. Winkler

Technical report no. 15-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2015.

[bib]

@techreport{RISC5183,
author = {N.T. Vo and G. Grasegger and F. Winkler},
title = {{Rational General Solutions of First-Order Algebraic ODEs}},
language = {english},
number = {15-18},
year = {2015},
length = {25},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

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]

[bib]

@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}
}

Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions

G. Grasegger, N.T. Vo, F. Winkler

Technical report no. 15-19 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2015. [pdf]

[bib]

@techreport{RISC5197,
author = {G. Grasegger and N.T. Vo and F. Winkler},
title = {{Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions}},
language = {english},
number = {15-19},
year = {2015},
length = {27},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

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.

[bib]

@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.

[bib]

@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}
}

2014

A solution method for autonomous first-order algebraic partial differential equations

G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

Technical report no. 14-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2014. [pdf]

[bib]

@techreport{RISC4977,
author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},
title = {{A solution method for autonomous first-order algebraic partial differential equations}},
language = {english},
number = {14-03},
year = {2014},
length = {17},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

On Symbolic Solutions of Algebraic Partial Differential Equations

G. Grasegger, A. Lastra, J.R. Sendra, F. Winkler

In: Computer Algebra in Scientific Computing, V.P. Gerdt et al. (ed.), Lecture Notes in Computer Science 8660, pp. 111-120. 2014. Springer International Publishing, ISSN 0302-9743. [url]

[bib]

@inproceedings{RISC5017,
author = {G. Grasegger and A. Lastra and J.R. Sendra and F. Winkler},
title = {{On Symbolic Solutions of Algebraic Partial Differential Equations}},
booktitle = {{Computer Algebra in Scientific Computing}},
language = {english},
series = {Lecture Notes in Computer Science},
volume = {8660},
pages = {111--120},
publisher = {Springer International Publishing},
isbn_issn = {ISSN 0302-9743},
year = {2014},
editor = {V.P. Gerdt et al.},
refereed = {yes},
length = {11},
url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-10515-4_9.pdf}
}

Algebraic differential equations - Rational solutions and beyond

Winkler Franz

Southeast Asian Bulletin of Mathematics(38), pp. 153-162. 2014. 0129-2021.

[bib]

@article{RISC5103,
author = {Winkler Franz},
title = {{Algebraic differential equations - Rational solutions and beyond}},
language = {english},
journal = {Southeast Asian Bulletin of Mathematics},
number = {38},
pages = {153--162},
isbn_issn = {0129-2021},
year = {2014},
refereed = {yes},
length = {9}
}

2013

Rational general solutions of higher order algebraic ODEs

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

J. Systems Science and Complexity (JSSC) 26/2, pp. 261-280. 2013. 1009-6124.

[bib]

@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.

[bib]

@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}
}

Rational General Solution of 1-Dimensional Systems of Autonomous Ordinary Differential Equations

A. Lastra, J. R. Sendra, L. X. C. Ngo, F. Winkler

Technical report no. 13-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. December 2013. [pdf] [pdf]

[bib]

@techreport{RISC4884,
author = {A. Lastra and J. R. Sendra and L. X. C. Ngo and F. Winkler},
title = {{Rational General Solution of 1-Dimensional Systems of Autonomous Ordinary Differential Equations}},
language = {english},
abstract = {An algebro-geometric method for determining the rational solvability of autonomous algebraic ordinary differential equations is extended from single equations of order 1 to systems of equations of arbitrary order but dimension 1. We provide necessary conditions, for the existence of rational solutions, on the degree and on the structure at infinity of the associated algebraic curve. Furthermore, from a rational parametrization of a planar projection of the corresponding space curve one deduces, either by derivation or by lifting the planar parametrization, the existence and actual computation of all rational solutions if they exist. Moreover, if the diferential polynomials are defined over the rational numbers, we can express the rational solutions over the same field of coeficients.},
number = {13-09},
year = {2013},
month = {December},
keywords = {algebraic ordinary differential equations, rational solutions, parametrization of curves},
sponsor = {Austrian Science Fund (FWF): W1214-N15, project DK11},
length = {19},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}