# Proving and Solving over the Reals [SFB F1303-1]

### Project Description

J. Schicho.

Budget. 178.775,– Eur.

### Project Lead

### Project Duration

01/04/1998 - 31/03/2001### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2015

[Schicho]

### Factorization of Rational Motions: A Survey with Examples and Applications

#### Z. Li, , T. Rad, J. Schicho, H.-P. Schroecker

In: Proc. IFToMM 14, S.-H. Chang et al. (ed.), pp. 833-840. 2015. ISBN 978-986-04-6098-8.@

author = {Z. Li and and T. Rad and J. Schicho and H.-P. Schroecker},

title = {{Factorization of Rational Motions: A Survey with Examples and Applications}},

booktitle = {{Proc. IFToMM 14}},

language = {english},

pages = {833--840},

isbn_issn = {ISBN 978-986-04-6098-8},

year = {2015},

editor = {S.-H. Chang et al.},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC5236,author = {Z. Li and and T. Rad and J. Schicho and H.-P. Schroecker},

title = {{Factorization of Rational Motions: A Survey with Examples and Applications}},

booktitle = {{Proc. IFToMM 14}},

language = {english},

pages = {833--840},

isbn_issn = {ISBN 978-986-04-6098-8},

year = {2015},

editor = {S.-H. Chang et al.},

refereed = {yes},

length = {8}

}

### 2014

[Schicho]

### Foreword to the Special Issue on Computational Algebraic Geometry

#### S. Di Rocco, J. Schicho

Math. Comp. Sci. 8, pp. 117-118. 2014. ISSN: 1661-8270.@

author = {S. Di Rocco and J. Schicho},

title = {{Foreword to the Special Issue on Computational Algebraic Geometry}},

language = {english},

journal = {Math. Comp. Sci.},

volume = {8},

pages = {117--118},

isbn_issn = { ISSN: 1661-8270},

year = {2014},

refereed = {yes},

length = {2}

}

**article**{RISC5122,author = {S. Di Rocco and J. Schicho},

title = {{Foreword to the Special Issue on Computational Algebraic Geometry}},

language = {english},

journal = {Math. Comp. Sci.},

volume = {8},

pages = {117--118},

isbn_issn = { ISSN: 1661-8270},

year = {2014},

refereed = {yes},

length = {2}

}

### 2013

[Schicho]

### A Regularization Approach for Estimating the Type of a plane Curve Singularity

#### M. Hodorog, J. Schicho

Theor. Comp. Sci. 479, pp. 99-119. 2013. 0304-3975.@

author = {M. Hodorog and J. Schicho},

title = {{A Regularization Approach for Estimating the Type of a plane Curve Singularity}},

language = {english},

journal = {Theor. Comp. Sci.},

volume = {479},

pages = {99--119},

isbn_issn = {0304-3975},

year = {2013},

refereed = {yes},

length = {21}

}

**article**{RISC4920,author = {M. Hodorog and J. Schicho},

title = {{A Regularization Approach for Estimating the Type of a plane Curve Singularity}},

language = {english},

journal = {Theor. Comp. Sci.},

volume = {479},

pages = {99--119},

isbn_issn = {0304-3975},

year = {2013},

refereed = {yes},

length = {21}

}

[Schicho]

### Effective Methods in Algebraic Geometry

#### A. Dickenstein, S. Di Rocco, E. Hubert, J. Schicho (eds.)

J. Symb. Comp. 151, pp. 1-114. 2013. 0747-7171.@

author = {A. Dickenstein and S. Di Rocco and E. Hubert and J. Schicho (eds.)},

title = {{Effective Methods in Algebraic Geometry}},

language = {english},

journal = {J. Symb. Comp.},

volume = {151},

pages = {1--114},

isbn_issn = {0747-7171},

year = {2013},

refereed = {yes},

length = {114}

}

**article**{RISC4921,author = {A. Dickenstein and S. Di Rocco and E. Hubert and J. Schicho (eds.)},

title = {{Effective Methods in Algebraic Geometry}},

language = {english},

journal = {J. Symb. Comp.},

volume = {151},

pages = {1--114},

isbn_issn = {0747-7171},

year = {2013},

refereed = {yes},

length = {114}

}

[Schicho]

### Factorization of Rational Curves in the Study Quadric and Revolute Linkages

#### G. Hegedüs, J. Schicho, H.-P. Schröcker

Mech. Mach. Theory 69(1), pp. 142-152. 2013. 0094-114X.@

author = {G. Hegedüs and J. Schicho and H.-P. Schröcker},

title = {{Factorization of Rational Curves in the Study Quadric and Revolute Linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {69},

number = {1},

pages = {142--152},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {11}

}

**article**{RISC4922,author = {G. Hegedüs and J. Schicho and H.-P. Schröcker},

title = {{Factorization of Rational Curves in the Study Quadric and Revolute Linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {69},

number = {1},

pages = {142--152},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {11}

}

[Schicho]

### The Theory of Bonds: A New Method for the Analysis of Linkages

#### G. Hegedüs, J. Schicho, H.-P. Schröcker

Mech. Mach. Theory 70, pp. 404-424. 2013. 0094-114X.@

author = {G. Hegedüs and J. Schicho and H.-P. Schröcker},

title = {{The Theory of Bonds: A New Method for the Analysis of Linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {70},

pages = {404--424},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {21}

}

**article**{RISC4923,author = {G. Hegedüs and J. Schicho and H.-P. Schröcker},

title = {{The Theory of Bonds: A New Method for the Analysis of Linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {70},

pages = {404--424},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {21}

}

[Schicho]

### Classification of angle-symmetric 6R linkages

#### Z. Li, J. Schicho

Mech. Mach. Theory 70, pp. 372-379. 2013. 0094-114X.@

author = {Z. Li and J. Schicho},

title = {{Classification of angle-symmetric 6R linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {70},

pages = {372--379},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {8}

}

**article**{RISC4924,author = {Z. Li and J. Schicho},

title = {{Classification of angle-symmetric 6R linkages}},

language = {english},

journal = {Mech. Mach. Theory},

volume = {70},

pages = {372--379},

isbn_issn = {0094-114X},

year = {2013},

refereed = {yes},

length = {8}

}

[Schicho]

### Computational aspects of gonal maps

#### J. Schicho, F.-O. Schreyer, M. Weimann

AAECC 24, pp. 313-341. 2013. 0938-1279 .@

author = {J. Schicho and F.-O. Schreyer and M. Weimann},

title = {{Computational aspects of gonal maps}},

language = {english},

journal = {AAECC},

volume = {24},

pages = {313--341},

isbn_issn = {0938-1279 },

year = {2013},

refereed = {yes},

length = {29}

}

**article**{RISC4925,author = {J. Schicho and F.-O. Schreyer and M. Weimann},

title = {{Computational aspects of gonal maps}},

language = {english},

journal = {AAECC},

volume = {24},

pages = {313--341},

isbn_issn = {0938-1279 },

year = {2013},

refereed = {yes},

length = {29}

}

### 2007

[Moore]

### Determination of the complete set of statically balanced planar four-bar mechanisms

#### B. Moore, J. Schicho, C. Gosselin

SFB F013. Technical report no. 2007-14, July 2007. [pdf]@

author = {B. Moore and J. Schicho and C. Gosselin},

title = {{Determination of the complete set of statically balanced planar four-bar mechanisms}},

language = {english},

abstract = {In this paper, we present a new method to determine the complete set of statically balanced planar four-bar mechanisms. We formulate the kinematic constraints and the static balancing constraints as algebraic equations over real and complex variables. This leads to the problem of factorization of Laurent polynomials which can be solved using Newton polytopes and Minkowski sums. The result of this process is a set of necessary and sufficient conditions for statically balanced four-bar mechanisms.},

number = {2007-14},

year = {2007},

month = {July},

institution = {SFB F013},

length = {16}

}

**techreport**{RISC3123,author = {B. Moore and J. Schicho and C. Gosselin},

title = {{Determination of the complete set of statically balanced planar four-bar mechanisms}},

language = {english},

abstract = {In this paper, we present a new method to determine the complete set of statically balanced planar four-bar mechanisms. We formulate the kinematic constraints and the static balancing constraints as algebraic equations over real and complex variables. This leads to the problem of factorization of Laurent polynomials which can be solved using Newton polytopes and Minkowski sums. The result of this process is a set of necessary and sufficient conditions for statically balanced four-bar mechanisms.},

number = {2007-14},

year = {2007},

month = {July},

institution = {SFB F013},

length = {16}

}

### 2005

[Schicho]

### Local Parametrization of Cubic Surfaces

#### I. Szil\'agyi and B. J\"uttler and J. Schicho

Journal of Symbolic Computation, pp. 1-24. 2005. ISSN 0747-7171. to appear.@

author = {I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Local Parametrization of Cubic Surfaces}},

language = {english},

abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.},

journal = {Journal of Symbolic Computation},

pages = {1--24},

isbn_issn = {ISSN 0747-7171},

year = {2005},

note = {to appear},

refereed = {yes},

length = {22}

}

**article**{RISC2049,author = {I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Local Parametrization of Cubic Surfaces}},

language = {english},

abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.},

journal = {Journal of Symbolic Computation},

pages = {1--24},

isbn_issn = {ISSN 0747-7171},

year = {2005},

note = {to appear},

refereed = {yes},

length = {22}

}

[Schicho]

### Implicitization and Distance Bounds

#### M. Aigner, I. Szil\'agyi, B. J\"uttler, J. Schicho

In: Mathematic and Visualization, M. Elkadi (ed.), Mathematics and Visualization , pp. 1-14. 2005. Springer, to appear.@

author = {M. Aigner and I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Implicitization and Distance Bounds}},

booktitle = {{Mathematic and Visualization}},

language = {english},

abstract = {In this paper, we combine results concerning the numerical stabilityof the implicitization process for a given planar rational curve, withresults on the the stability of the resulting implicit representation.More precisely, it is shown that for any approximate parameterizationof the given curve, the curve obtained by an approximateimplicitization with a given precision is contained within a certainperturbation region. The results can be generalized to the case ofsurfaces.},

series = {Mathematics and Visualization},

pages = {1--14},

publisher = {Springer},

isbn_issn = {?},

year = {2005},

note = {to appear},

editor = {M. Elkadi},

refereed = {yes},

length = {14}

}

**inproceedings**{RISC2439,author = {M. Aigner and I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Implicitization and Distance Bounds}},

booktitle = {{Mathematic and Visualization}},

language = {english},

abstract = {In this paper, we combine results concerning the numerical stabilityof the implicitization process for a given planar rational curve, withresults on the the stability of the resulting implicit representation.More precisely, it is shown that for any approximate parameterizationof the given curve, the curve obtained by an approximateimplicitization with a given precision is contained within a certainperturbation region. The results can be generalized to the case ofsurfaces.},

series = {Mathematics and Visualization},

pages = {1--14},

publisher = {Springer},

isbn_issn = {?},

year = {2005},

note = {to appear},

editor = {M. Elkadi},

refereed = {yes},

length = {14}

}

[Schicho]

### Numerical Stability of Surface Implicitization

#### J. Schicho, I. Szil\'agyi

Journal of Symbolic Computation, pp. 1-14. 2005. ISSN 0747-7171. to appear.@

author = {J. Schicho and I. Szil\'agyi},

title = {{Numerical Stability of Surface Implicitization}},

language = {english},

abstract = {For a numerically given parametrization we cannot compute an exactimplicit equation, just an approximate one. We introduce acondition number to measure the worst effect on the solution whenthe input data is perturbed by a small amount.},

journal = {Journal of Symbolic Computation},

pages = {1--14},

isbn_issn = {ISSN 0747-7171},

year = {2005},

note = {to appear},

refereed = {yes},

length = {14}

}

**article**{RISC2440,author = {J. Schicho and I. Szil\'agyi},

title = {{Numerical Stability of Surface Implicitization}},

language = {english},

abstract = {For a numerically given parametrization we cannot compute an exactimplicit equation, just an approximate one. We introduce acondition number to measure the worst effect on the solution whenthe input data is perturbed by a small amount.},

journal = {Journal of Symbolic Computation},

pages = {1--14},

isbn_issn = {ISSN 0747-7171},

year = {2005},

note = {to appear},

refereed = {yes},

length = {14}

}

[Szilágyi]

### Symbolic-Numeric Techniques for Cubic Surfaces

#### Ibolya Szilagyi

RISC-Linz. PhD Thesis. July 2005. PhD Thesis. [pdf] [ps]@

author = {Ibolya Szilagyi},

title = {{Symbolic-Numeric Techniques for Cubic Surfaces}},

language = {english},

abstract = {In geometric modelling and related areas algebraic curves/surfacestypically are described either as the zero set of an algebraicequation (implicit representation), or as the image of amap given by rational functions (parametricrepresentation). The availability of both representations oftenresult in more efficient computations.Computational theories and techniques of algebraic geometry infloating point environment are of high interest in geometricmodelling related communities. Therefore, deriving approximatealgorithms that can be applied to numeric data have become a veryactive research area. In this thesis we focus on the twoconversion problems, called implicitization and parametrization,from the numeric point of view.A very important issue in the implicitization problem is theperturbation behavior of parametric objects. For a numericallygiven parametrization we cannot compute an exact implicitequation, just an approximate one. We introduce a conditionnumber of the implicitization problem to measure the worst effecton the solution, when the input data is perturbed by a smallamount. Using this condition number we study the algebraic andgeometric robustness of the implicitization process.Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. We introduce several techniques for generatingsuch parameterizations for nonsingular cubic surfaces. For thisclass of surfaces, it is shown that the local parametrizationproblem can be solved for all points, and any such surface can becovered completely.},

year = {2005},

month = {July},

note = {PhD Thesis},

translation = {0},

school = {RISC-Linz},

keywords = {implicitization, numerical stability, local parametrization, cubic surface},

sponsor = {RISC PhD scholarship program of the government of Upper Austria, and by the Spezialforschungsbereich (SFB) grant F1303, Austrian Science Foundation (FWF).},

length = {112}

}

**phdthesis**{RISC2472,author = {Ibolya Szilagyi},

title = {{Symbolic-Numeric Techniques for Cubic Surfaces}},

language = {english},

abstract = {In geometric modelling and related areas algebraic curves/surfacestypically are described either as the zero set of an algebraicequation (implicit representation), or as the image of amap given by rational functions (parametricrepresentation). The availability of both representations oftenresult in more efficient computations.Computational theories and techniques of algebraic geometry infloating point environment are of high interest in geometricmodelling related communities. Therefore, deriving approximatealgorithms that can be applied to numeric data have become a veryactive research area. In this thesis we focus on the twoconversion problems, called implicitization and parametrization,from the numeric point of view.A very important issue in the implicitization problem is theperturbation behavior of parametric objects. For a numericallygiven parametrization we cannot compute an exact implicitequation, just an approximate one. We introduce a conditionnumber of the implicitization problem to measure the worst effecton the solution, when the input data is perturbed by a smallamount. Using this condition number we study the algebraic andgeometric robustness of the implicitization process.Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. We introduce several techniques for generatingsuch parameterizations for nonsingular cubic surfaces. For thisclass of surfaces, it is shown that the local parametrizationproblem can be solved for all points, and any such surface can becovered completely.},

year = {2005},

month = {July},

note = {PhD Thesis},

translation = {0},

school = {RISC-Linz},

keywords = {implicitization, numerical stability, local parametrization, cubic surface},

sponsor = {RISC PhD scholarship program of the government of Upper Austria, and by the Spezialforschungsbereich (SFB) grant F1303, Austrian Science Foundation (FWF).},

length = {112}

}

### 2004

[Schicho]

### Local Parametrization of Cubic Surfaces

#### I. Szil\'agyi and B. J\"uttler and J. Schicho

J. Kepler University, Linz. Technical report no. 2004-31, 2004. SFB-Report. [url]@

author = {I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Local Parametrization of Cubic Surfaces}},

language = {english},

abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.},

number = {2004-31},

year = {2004},

institution = {J.~Kepler University, Linz},

keywords = {parametrization, cubics, algorithm, surfaces},

length = {22},

url = {http://www.sfb013.uni-linz.ac.at/},

type = {SFB-Report}

}

**techreport**{RISC2047,author = {I. Szil\'agyi and B. J\"uttler and J. Schicho},

title = {{Local Parametrization of Cubic Surfaces}},

language = {english},

abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.},

number = {2004-31},

year = {2004},

institution = {J.~Kepler University, Linz},

keywords = {parametrization, cubics, algorithm, surfaces},

length = {22},

url = {http://www.sfb013.uni-linz.ac.at/},

type = {SFB-Report}

}

[Schicho]

### Numerical Stability of Surface Implicitization

#### J. schicho, I. Szil\'agyi

J. Kepler University, Linz. Technical report no. 2004-27, 2004. SFB-Report. [url]@

author = {J. schicho and I. Szil\'agyi},

title = {{Numerical Stability of Surface Implicitization}},

language = {english},

abstract = {For a numerically given parametrization we cannot compute an exactimplicit equation, just an approximate one. We introduce acondition number to measure the worst effect on the solution whenthe input data is perturbed by a small amount. Using thiscondition number the perturbation behavior of variousimplicitization methods can be analyzed.},

number = {2004-27},

year = {2004},

institution = {J.~Kepler University, Linz},

length = {14},

url = {http://www.sfb013.uni-linz.ac.at/},

type = {SFB-Report}

}

**techreport**{RISC2046,author = {J. schicho and I. Szil\'agyi},

title = {{Numerical Stability of Surface Implicitization}},

language = {english},

abstract = {For a numerically given parametrization we cannot compute an exactimplicit equation, just an approximate one. We introduce acondition number to measure the worst effect on the solution whenthe input data is perturbed by a small amount. Using thiscondition number the perturbation behavior of variousimplicitization methods can be analyzed.},

number = {2004-27},

year = {2004},

institution = {J.~Kepler University, Linz},

length = {14},

url = {http://www.sfb013.uni-linz.ac.at/},

type = {SFB-Report}

}

### 2000

[Bodnar]

### Automated Resolution of Singularities for Hypersurfaces

#### G. Bodnar, J. Schicho

Journal of Symbolic Computation 30(4), pp. 401-428. 2000. London, ISSN: 0747-7171.@

author = {G. Bodnar and J. Schicho},

title = {{Automated Resolution of Singularities for Hypersurfaces}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {30},

number = {4},

pages = {401--428},

address = {London},

isbn_issn = {ISSN: 0747-7171},

year = {2000},

refereed = {yes},

length = {28}

}

**article**{RISC410,author = {G. Bodnar and J. Schicho},

title = {{Automated Resolution of Singularities for Hypersurfaces}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {30},

number = {4},

pages = {401--428},

address = {London},

isbn_issn = {ISSN: 0747-7171},

year = {2000},

refereed = {yes},

length = {28}

}

[Buchberger]

### F 1303: Proving and Solving Over the Reals

#### B. Buchberger, J. Schicho

In: Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000, B. Buchberger and U.Langer (ed.), pp. 126-142. October 2000. Johannes Kepler University Linz, Austria,@

author = {B. Buchberger and J. Schicho},

title = {{F 1303: Proving and Solving Over the Reals}},

booktitle = {{Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000}},

language = {english},

pages = {126--142},

publisher = {Johannes Kepler University Linz, Austria},

isbn_issn = {?},

year = {2000},

month = {October},

annote = {2000-10-00-C},

editor = {B. Buchberger and U.Langer},

refereed = {no},

length = {17}

}

**incollection**{RISC2405,author = {B. Buchberger and J. Schicho},

title = {{F 1303: Proving and Solving Over the Reals}},

booktitle = {{Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000}},

language = {english},

pages = {126--142},

publisher = {Johannes Kepler University Linz, Austria},

isbn_issn = {?},

year = {2000},

month = {October},

annote = {2000-10-00-C},

editor = {B. Buchberger and U.Langer},

refereed = {no},

length = {17}

}

[Buchberger]

### F 1303: Proving and Solving Over the Reals (Progress Report)

#### B. Buchberger, J. Schicho

In: Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000, B. Buchberger and U.Langer (ed.), pp. 126-142. October 2000. Johannes Kepler University Linz, Austria,@

author = { B. Buchberger and J. Schicho},

title = {{F 1303: Proving and Solving Over the Reals (Progress Report)}},

booktitle = {{Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000}},

language = {english},

pages = {126--142},

publisher = {Johannes Kepler University Linz, Austria},

year = {2000},

month = {October},

annote = {2000-10-00-D},

editor = { B. Buchberger and U.Langer},

refereed = {no},

length = {17}

}

**incollection**{RISC2744,author = { B. Buchberger and J. Schicho},

title = {{F 1303: Proving and Solving Over the Reals (Progress Report)}},

booktitle = {{Special Research Program (SFB) F 013, Numerical and Symbolic Scientific Computing, Proposal for Continuation, Part I: Progress Report, April 1998-September 2000}},

language = {english},

pages = {126--142},

publisher = {Johannes Kepler University Linz, Austria},

year = {2000},

month = {October},

annote = {2000-10-00-D},

editor = { B. Buchberger and U.Langer},

refereed = {no},

length = {17}

}

### 1995

[Schicho]

### Rational Parametrization of Algebraic Surfaces. Symbolic Solution of an Equation in three Variables

#### Josef Schicho

RISC, Johannes Kepler University Linz. PhD Thesis. 1995.@

author = {Josef Schicho},

title = {{Rational Parametrization of Algebraic Surfaces. Symbolic Solution of an Equation in three Variables}},

language = {english},

year = {1995},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}

**phdthesis**{RISC4133,author = {Josef Schicho},

title = {{Rational Parametrization of Algebraic Surfaces. Symbolic Solution of an Equation in three Variables}},

language = {english},

year = {1995},

translation = {0},

school = {RISC, Johannes Kepler University Linz},

length = {0}

}