# Explicit Resolution and Related Methods in Algebraic Geometry and Number Theory

### Project Description

The goal of the project is to develop theories and algorithms for solving problems related to the resolution of singularities in algebraic geometry and number theory. The three main subareas of research are the following.

RES:

Computing resolutions for various classes of varieties: The main punching line will be dimension two and three, both in characteristic zero and p. The reason is two-fold. First, we think that effective resolutions of 4-folds is practically too costly in terms of computing resources (although theoretically possible). Second, many interesting applications and phenomena happen to appear already in dimension two and three.

ALG:

Solving problems in algebraic geometry related to resolution: This concerns the canonical class of an algebraic variety, in particular the parametrization problem for rational surfaces, the problem of finding rational and elliptic fibrations, and the problem of constructing canonical embeddings.

NUM:

Solving problems in number theory related to resolution: The problems are related to p-adic completions and global fields, such as the computation of normal bases in number fields, or the existence problem of rational points on Del Pezzo surfaces.

### Project Lead

### Project Duration

01/10/2002 - 01/10/2005### Project URL

Go to Website## Publications

### 2015

### 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.**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

### A simplified game for resolution of singularities

#### J. Schicho

In: The Resolution of Singular Algebraic Varieties, D. Ellwood and H. Hauser and S. Mori and J. Schicho (ed.), pp. 209-224. 2014. Amer. Math. Soc., ISBN 978-0-8218-8982-4.**incollection**{RISC5119,

author = {J. Schicho},

title = {{A simplified game for resolution of singularities}},

booktitle = {{The Resolution of Singular Algebraic Varieties}},

language = {english},

pages = {209--224},

publisher = {Amer. Math. Soc.},

isbn_issn = {ISBN 978-0-8218-8982-4},

year = {2014},

editor = {D. Ellwood and H. Hauser and S. Mori and J. Schicho},

refereed = {yes},

length = {16}

}

### Algebraic approaches to FlipIt

#### J. Schicho, J. Top

In: The Resolution of Singular Algebraic Varieties, D. Ellwood and H. Hauser and S. Mori and J. Schicho (ed.), pp. 319-326. 2014. Amer. Math. Soc., ISBN 978-0-8218-8982-4.**incollection**{RISC5120,

author = {J. Schicho and J. Top},

title = {{Algebraic approaches to FlipIt}},

booktitle = {{The Resolution of Singular Algebraic Varieties}},

language = {english},

pages = {319--326},

publisher = {Amer. Math. Soc.},

isbn_issn = {ISBN 978-0-8218-8982-4},

year = {2014},

editor = {D. Ellwood and H. Hauser and S. Mori and J. Schicho},

refereed = {yes},

length = {8}

}

### 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.**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

### 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.**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}

}

### 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.**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}

}

### 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.**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}

}

### 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.**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}

}

### Classification of angle-symmetric 6R linkages

#### Z. Li, J. Schicho

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

}

### Computational aspects of gonal maps

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

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

### 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]**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

### 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.**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}

}

### Numerical Stability of Surface Implicitization

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

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

}

### 2004

### Algorithmic tests for the normal crossing property

#### G. Bodnar

In: Automated Deduction in Geometry-Proc. of ADG 2002, F. Winkler (ed.), LNAI 2930, pp. 1-20. 2004. Springer, ISBN: 3-540-20927-1.**inproceedings**{RISC415,

author = {G. Bodnar},

title = {{Algorithmic tests for the normal crossing property}},

booktitle = {{Automated Deduction in Geometry--Proc. of ADG~2002}},

language = {english},

series = {LNAI},

volume = {2930},

pages = {1--20},

publisher = {Springer},

isbn_issn = {ISBN: 3-540-20927-1},

year = {2004},

editor = {F. Winkler},

refereed = {yes},

length = {20}

}

### Efficient Desingularization of Reducible Algebraic Sets

#### G. Bodnar

In: Proceedings of ISSAC 2004, J. Gurierrez (ed.), pp. 35-41. 2004. Sheridan Printing, ISBN: 1-58113-827-X.**inproceedings**{RISC416,

author = {G. Bodnar},

title = {{Efficient Desingularization of Reducible Algebraic Sets}},

booktitle = {{Proceedings of ISSAC~2004}},

language = {english},

pages = {35--41},

publisher = {Sheridan Printing},

isbn_issn = {ISBN: 1-58113-827-X},

year = {2004},

editor = {J. Gurierrez},

refereed = {yes},

length = {7}

}

### Numerical Stability of Surface Implicitization

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

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

}

### 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]**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}

}

### 2003

### Computation of blowing up centers

#### G. Bodnar

Journal of Pure and Applied Algebra 179(3), pp. 221-233. 2003. ISSN: 0022-4049.**article**{RISC414,

author = {G. Bodnar},

title = {{Computation of blowing up centers}},

language = {english},

journal = {Journal of Pure and Applied Algebra},

volume = {179},

number = {3},

pages = {221--233},

isbn_issn = {ISSN: 0022-4049},

year = {2003},

refereed = {yes},

length = {13}

}

### 2002

### Unification of blowing up sequences

#### G. Bodnar

In: Actas del EACA-2002, Philippe Giménez (ed.), pp. 111-115. 2002. University of Valladolid, Spain, X.**inproceedings**{RISC417,

author = {G. Bodnar},

title = {{Unification of blowing up sequences}},

booktitle = {{Actas del EACA--2002}},

language = {english},

pages = {111--115},

publisher = {University of Valladolid, Spain},

isbn_issn = {X},

year = {2002},

editor = {Philippe Giménez},

refereed = {no},

length = {5}

}

### 2000

### 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,**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}

}