Computer Algebra for Geometry

Algebraic varieties are defined by polynomial equations. Computer algebra methods for solving systems of polynomial equations and similar problems form the basis for a computational theory of Algebraic Geometry.

From the preface of J. R. Sendra, F. Winkler, S. Pérez-Díaz. Rational Algebraic Curves - A Computer Algebra Approach, Springer-Verlag Berlin Heidelberg, 2008.

Algebraic curves and surfaces are an old topic of geometric and algebraic investigation. They have found applications for instance in ancient and modern architectural designs, in number theoretic problems, in models of biological shapes, in error-correcting codes, and in cryptographic algorithms. Recently they have gained additional practical importance as central objects in computer aided geometric design. Modern airplanes, cars, and household appliances would be unthinkable without the computational manipulation of algebraic curves and surfaces. Algebraic curves and surfaces combine
fascinating mathematical beauty with challenging computational complexity and wide spread practical applicability.

In this book we treat only algebraic curves, although many of the results and methods can be and in fact have been generalized to surfaces. Being the solution loci of algebraic, i.e. polynomial,
equations in two variables, plane algebraic curves are well suited for being investigated with symbolic computer algebra methods. This is exactly the approach we take in our book. We apply algorithms from computer algebra to the analysis, and manipulation of algebraic curves. To a large extent this amounts to being able to represent these algebraic curves in different ways, such as implicitly by defining polynomials, parametrically by rational functions, or locally parametrically by power series expansions around a point. These representations all have their individual advantages; an implicit representation lets us decide easily whether a given point actually lies on a given curve, a parametric representation allows us to generate points of a given curve over the desired coordinate fields, and with the help of a power series expansion we can for instance overcome the numerical problems of tracing a curve through a singularity.

The central problem in this book is the determination of rational parametrizability of a curve, and, in case it exists, the computation of a good rational parametrization. This amounts to determining the genus of a curve, i.e. its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. Various optimality criteria for rational parametrizations of algebraic curves are discussed. We also point to some applications of these techniques in computer aided geometric design. Many of the symbolic algorithmic methods described in our book are implemented in the program system CASA, which is based on the computer algebra system Maple.

Our book is mainly intended for graduate students specializing in constructive algebraic curve geometry. We hope that researchers wanting to get a quick overview of what can be done with algebraic curves in terms of symbolic algebraic computation will also find this book helpful.

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
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CharSet

Differential Characteristic Set Computations

CharSet is an Aldor package written by Christian Aistleitner for differential characteristic set computations. CharSet comes with generic implementations of reduction, Gröbner bases, and differential characteristic set algorithms. Interfaces to the command line, Mathematica and Maple are included. ...

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PGB

Parametric Gröbner Bases

PGB is a software package for computing parametric Gröbner bases and related objects in several domains. It is implemented in the computer algebra system Risa/Asir by Katsusuke Nabeshima. ...

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Publications

2018

Varieties of apolar subschemes of toric surfaces

Gallet Matteo, Ranestad Kristian, Villamizar Nelly

Ark. Mat. 56(1), pp. 73-99. 2018. ISSN 0004-2080. [url]
[bib]
@article{RISC5796,
author = {Gallet Matteo and Ranestad Kristian and Villamizar Nelly},
title = {{Varieties of apolar subschemes of toric surfaces}},
language = {english},
journal = {Ark. Mat.},
volume = {56},
number = {1},
pages = {73--99},
isbn_issn = { ISSN 0004-2080},
year = {2018},
refereed = {yes},
length = {27},
url = {https://doi.org/10.4310/ARKIV.2018.v56.n1.a6}
}

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.
[bib]
@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}
}

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.
[bib]
@article{RISC5589,
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},
volume = {87},
pages = {127--139},
isbn_issn = {ISSN 0747-7171},
year = {2018},
refereed = {yes},
length = {12}
}

A Computable Extension for Holonomic Functions: DD-Finite Functions

Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation, pp. -. 2018. ISSN 0747-7171. accepted.
[bib]
@article{RISC5731,
author = {Jiménez-Pastor Antonio and Pillwein Veronika},
title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},
language = {english},
journal = {Journal of Symbolic Computation},
pages = {--},
isbn_issn = {ISSN 0747-7171},
year = {2018},
note = {accepted},
refereed = {yes},
length = {0}
}

The Number of Realizations of a Laman Graph

Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]
[bib]
@article{RISC5700,
author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The Number of Realizations of a Laman Graph}},
language = {english},
journal = {SIAM Journal on Applied Algebra and Geometry},
volume = {2},
number = {1},
pages = {94--125},
isbn_issn = {2470-6566},
year = {2018},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1137/17M1118312}
}

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}
}

2017

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

Christoph Fuerst, Alexander Levin

In: Mathematics in Computer Science, W. Koepf (ed.), pp. 1-11. 2017. 1661-8289.
[bib]
@inproceedings{RISC5432,
author = {Christoph Fuerst and Alexander Levin},
title = {{Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules}},
booktitle = {{Mathematics in Computer Science}},
language = {english},
pages = {1--11},
isbn_issn = {1661-8289},
year = {2017},
editor = {W. Koepf},
refereed = {yes},
length = {11}
}

An Algebraic-Geometric Method for Computing Zolotarev Polynomials

Georg Grasegger, N. Thieu Vo

In: Proceedings of the 2017 international symposium on symbolic and algebraic computation (ISSAC), Burr, M. (ed.), pp. 173-180. 2017. ACM Press, New York, ISBN: 978-1-4503-5064-8.
[bib]
@inproceedings{RISC5510,
author = {Georg Grasegger and N. Thieu Vo},
title = {{An Algebraic-Geometric Method for Computing Zolotarev Polynomials}},
booktitle = {{Proceedings of the 2017 international symposium on symbolic and algebraic computation (ISSAC)}},
language = {english},
pages = {173--180},
publisher = {ACM Press},
address = {New York},
isbn_issn = {ISBN: 978-1-4503-5064-8},
year = {2017},
editor = {Burr and M.},
refereed = {yes},
length = {8}
}

The number of realizations of a Laman graph

Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

Research Institute for Symbolic Computation (RISC/JKU). Technical report, 2017. [url] [pdf]
[bib]
@techreport{RISC5418,
author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The number of realizations of a Laman graph}},
language = {english},
abstract = {Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems. },
year = {2017},
institution = {Research Institute for Symbolic Computation (RISC/JKU)},
length = {42},
url = {http://www.koutschan.de/data/laman/}
}

Computing the number of realizations of a Laman graph

Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]
[bib]
@inproceedings{RISC5478,
author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{Computing the number of realizations of a Laman graph}},
booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},
language = {english},
abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},
volume = {61},
pages = {207--213},
isbn_issn = {ISSN 1571-0653},
year = {2017},
editor = {Vadim Lozin},
refereed = {yes},
keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},
length = {7},
conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},
url = {http://www.koutschan.de/data/laman/}
}

Validating the Formalization of Theories and Algorithms of Discrete Mathematics by the Computer-Supported Checking of Finite Models

Alexander Brunhuemer

Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria. Technical report, September 2017. Bachelor Thesis. [pdf]
[bib]
@techreport{RISC5485,
author = {Alexander Brunhuemer},
title = {{Validating the Formalization of Theories and Algorithms of Discrete Mathematics by the Computer-Supported Checking of Finite Models}},
language = {english},
abstract = {The goal of this Bachelor’s thesis is the formal specification and implementation of centraltheories and algorithms in the field of discrete mathematics by using the RISC AlgorithmLanguage (RISCAL), developed at the Research Institute for Symbolic Computation (RISC).This specification language and associated software system allow the verification of specifications,by using the concept of finite model checking. Validation on finite models is intendedto serve as a foundation layer for further research on the corresponding generalized theorieson infinite models.This thesis results in a collection of specifications of exemplarily chosen formalized algorithmsof set theory, relation and function theory and graph theory. The algorithms arespecified in different ways (implicit, recursive and procedural), to emphasize the correspondingconnections between them.The evaluation and validation of implemented theories is demonstrated on Dijkstra’s algorithmfor finding a shortest path between vertices in a graph.},
year = {2017},
month = {September},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria},
length = {88},
type = {Bachelor Thesis}
}

2016

Axiomatic Description of Gröbner Reduction

Christoph Fuerst

RISC, JKU Linz. PhD Thesis. December 2016. [pdf]
[bib]
@phdthesis{RISC5388,
author = {Christoph Fuerst},
title = {{Axiomatic Description of Gröbner Reduction}},
language = {english},
year = {2016},
month = {December},
translation = {0},
school = {RISC, JKU Linz},
length = {154}
}

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}
}

An Algebraic-Geometric Method for Computing Zolotarev Polynomials

G. Grasegger, N.T. Vo

Technical report no. 16-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2016. [pdf]
[bib]
@techreport{RISC5271,
author = {G. Grasegger and N.T. Vo},
title = {{An Algebraic-Geometric Method for Computing Zolotarev Polynomials}},
language = {english},
number = {16-02},
year = {2016},
length = {17},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

An Algebraic-Geometric Method for Computing Zolotarev Polynomials — Additional Information

G. Grasegger

Technical report no. 16-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2016. [pdf]
[bib]
@techreport{RISC5340,
author = {G. Grasegger},
title = {{An Algebraic-Geometric Method for Computing Zolotarev Polynomials — Additional Information}},
language = {english},
number = {16-07},
year = {2016},
length = {12},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

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}
}

Representation of hypergeometric products in difference rings

E.D. Ocansey, C. Schneider

ACM Communications in Computer Algebra 50(4), pp. 161-163. 2016. ISSN 1932-2240 . Extended abstract of the poster presentation at ISSAC 2016. [pdf]
[bib]
@article{RISC5316,
author = {E.D. Ocansey and C. Schneider},
title = {{Representation of hypergeometric products in difference rings}},
language = {english},
journal = {ACM Communications in Computer Algebra},
volume = {50},
number = {4},
pages = {161--163},
isbn_issn = {ISSN 1932-2240 },
year = {2016},
note = {Extended abstract of the poster presentation at ISSAC 2016},
refereed = {yes},
length = {3}
}

Rational and Algebraic Solutions of First-Order Algebraic ODEs

N. Thieu Vo

Technical report no. 16-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 12 2016. Thesis Dissertation. [pdf]
[bib]
@techreport{RISC5387,
author = {N. Thieu Vo},
title = {{Rational and Algebraic Solutions of First-Order Algebraic ODEs}},
language = {english},
abstract = {The main aim of this thesis is to study new algorithms for determining polynomial, rational and algebraic solutions of first-order algebraic ordinary differential equations (AODEs). The problem of determining closed form solutions of first-order AODEs has a long history, and it still plays a role in many branches of mathematics. There is a bunch of solution methods for specific classes of such ODEs. However still no decision algorithm for general first-order AODEs exists, even for seeking specific kinds of solutions such as polynomial, rational or algebraic functions. Our interests are algebraic general solutions, rational general solutions, particular rational solutions and polynomial solutions. Several algorithms for determining these kinds of solutions for first-order AODEs are presented.We approach first-order AODEs from several aspects. By considering the derivative as a new indeterminate, a first-order AODE can be viewed as a hypersurface over the ground field. Therefore tools from algebraic geometry are applicable. In particular, we use birational transformations of algebraic hypersurfaces to transform the differential equation to another one for which we hope that it is easier to solve. This geometric approach leads us to a procedure for determining an algebraic general solution of a parametrizable first-order AODE. A general solution contains an arbitrary constant. For the problem of determining a rational general solution in which the constant appears rationally, we propose a decision algorithm for the general class of first-order AODEs.The geometric method is not applicable for studying particular rational solutions. Instead, we study this kind of solutions from combinatorial and algebraic aspects. In the combinatorial consideration, poles of the coefficients of the differential equation play an important role in the estimation of candidates for poles of a rational solution and their multiplicities. An algebraic method based on algebraic function field theory is proposed to globally estimate the degree of a rational solution. A combination of these methods leads us to an algorithm for determining all rational solutions for a generic class of first-order AODEs, which covers every first-order AODEs from Kamke's collection. For polynomial solutions, the algorithm works for the general class of first-order AODEs.},
number = {16-11},
year = {2016},
month = {12},
note = {Thesis Dissertation},
length = {93},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

Rational and Algebraic Solutions of First-Order Algebraic ODEs

N. Thieu Vo

Research Institute for Symbolic Computation. PhD Thesis. 2016. [pdf]
[bib]
@phdthesis{RISC5399,
author = {N. Thieu Vo},
title = {{Rational and Algebraic Solutions of First-Order Algebraic ODEs}},
language = {english},
year = {2016},
translation = {0},
school = {Research Institute for Symbolic Computation},
length = {93}
}

2015

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]
[bib]
@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}
}

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