## Ongoing Projects

### Security and Safety for Shared AI by Deep Model Design (scch)

Project Lead: Josef Schicho

### European Network GRAPES: learninG, pRocessing And oPtimizing shapES [GRAPES]

Project Lead: Josef Schicho

### Artin Approximation, Arc-Räume, Auflösung von Singularitäten

Project Lead: Josef Schicho

## Publications

### 2021

[Grasegger]

### Combinatorics of Bricard's octahedra

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

Comptes Rendus. Mathématique 359(1), pp. 7-38. 2021. Académie des sciences, Paris, ISSN 1631-073X. [doi]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard's octahedra}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {359},

number = {1},

pages = {7--38},

publisher = {Académie des sciences, Paris},

isbn_issn = {ISSN 1631-073X},

year = {2021},

refereed = {yes},

length = {32},

url = {https://doi.org/10.5802/crmath.132}

}

**article**{RISC6288,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard's octahedra}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {359},

number = {1},

pages = {7--38},

publisher = {Académie des sciences, Paris},

isbn_issn = {ISSN 1631-073X},

year = {2021},

refereed = {yes},

length = {32},

url = {https://doi.org/10.5802/crmath.132}

}

[Grasegger]

### On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

SIAM Journal on Discrete Mathematics 35(1), pp. 325-361. 2021. ISSN 0895-4801. [doi]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},

language = {english},

journal = {SIAM Journal on Discrete Mathematics},

volume = {35},

number = {1},

pages = {325--361},

isbn_issn = {ISSN 0895-4801},

year = {2021},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1137/19M1289467}

}

**article**{RISC6290,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},

language = {english},

journal = {SIAM Journal on Discrete Mathematics},

volume = {35},

number = {1},

pages = {325--361},

isbn_issn = {ISSN 0895-4801},

year = {2021},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1137/19M1289467}

}

[Legersky]

### On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

#### E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

Journal of Symbolic Computation 102, pp. 189-208. 2021. ISSN 0747-7171. [doi]@

author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {102},

pages = {189--208},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1016/j.jsc.2019.10.015}

}

**article**{RISC5992,author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {102},

pages = {189--208},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1016/j.jsc.2019.10.015}

}

### 2020

[Capco]

### Robots, computer algebra and 8 connected components

#### J. Capco, M. Safey el Din, J. Schicho

In: Proceedings of ISSAC 2020, I. Emiris, L. Zhi (ed.), pp. 62-69. 2020. 978-1-4503-7100-1. [url]@

author = {J. Capco and M. Safey el Din and J. Schicho},

title = {{Robots, computer algebra and 8 connected components}},

booktitle = {{Proceedings of ISSAC 2020}},

language = {english},

pages = {62--69},

isbn_issn = {978-1-4503-7100-1},

year = {2020},

editor = {I. Emiris and L. Zhi},

refereed = {yes},

length = {8},

url = {https://arxiv.org/abs/2008.13392}

}

**inproceedings**{RISC6247,author = {J. Capco and M. Safey el Din and J. Schicho},

title = {{Robots, computer algebra and 8 connected components}},

booktitle = {{Proceedings of ISSAC 2020}},

language = {english},

pages = {62--69},

isbn_issn = {978-1-4503-7100-1},

year = {2020},

editor = {I. Emiris and L. Zhi},

refereed = {yes},

length = {8},

url = {https://arxiv.org/abs/2008.13392}

}

[Grasegger]

### Graphs with Flexible Labelings allowing Injective Realizations

#### G. Grasegger, J. Legerský, J. Schicho

Discrete Mathematics 343(6), pp. Art. 111713-. 2020. ISSN 0012-365X. [doi]@

author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings allowing Injective Realizations}},

language = {english},

journal = {Discrete Mathematics},

volume = {343},

number = {6},

pages = {Art. 111713--},

isbn_issn = {ISSN 0012-365X},

year = {2020},

refereed = {yes},

length = {14},

url = {https://doi.org/10.1016/j.disc.2019.111713}

}

**article**{RISC6012,author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings allowing Injective Realizations}},

language = {english},

journal = {Discrete Mathematics},

volume = {343},

number = {6},

pages = {Art. 111713--},

isbn_issn = {ISSN 0012-365X},

year = {2020},

refereed = {yes},

length = {14},

url = {https://doi.org/10.1016/j.disc.2019.111713}

}

[Grasegger]

### Flexible placements of graphs with rotational symmetry

#### S.Dewar, G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]@

author = {S.Dewar and G. Grasegger and J. Legerský},

title = {{Flexible placements of graphs with rotational symmetry}},

language = {english},

year = {2020},

institution = {arXiv},

length = {9},

url = {https://arxiv.org/abs/2003.09328}

}

**techreport**{RISC6091,author = {S.Dewar and G. Grasegger and J. Legerský},

title = {{Flexible placements of graphs with rotational symmetry}},

language = {english},

year = {2020},

institution = {arXiv},

length = {9},

url = {https://arxiv.org/abs/2003.09328}

}

[Grasegger]

### Combinatorics of Bricard’s octahedra

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2020. [url]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard’s octahedra}},

language = {english},

year = {2020},

institution = {arXiv},

length = {40},

url = {https://arxiv.org/pdf/2004.01236.pdf}

}

**techreport**{RISC6095,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard’s octahedra}},

language = {english},

year = {2020},

institution = {arXiv},

length = {40},

url = {https://arxiv.org/pdf/2004.01236.pdf}

}

[Grasegger]

### Computing Animations of Linkages with Rotational Symmetry (Media Exposition)

#### Sean Dewar, Georg Grasegger, Jan Legerský

In: 36th International Symposium on Computational Geometry (SoCG 2020), Sergio Cabello and Danny Z. Chen (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 164, pp. 77:1-77:4. 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, ISBN 978-3-95977-143-6. [doi]@

author = {Sean Dewar and Georg Grasegger and Jan Legerský},

title = {{Computing Animations of Linkages with Rotational Symmetry (Media Exposition)}},

booktitle = {{36th International Symposium on Computational Geometry (SoCG 2020)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {164},

pages = {77:1--77:4},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},

address = {Dagstuhl, Germany},

isbn_issn = {ISBN 978-3-95977-143-6},

year = {2020},

editor = {Sergio Cabello and Danny Z. Chen},

refereed = {no},

length = {4},

url = {https://doi.org/10.4230/LIPIcs.SoCG.2020.77}

}

**inproceedings**{RISC6128,author = {Sean Dewar and Georg Grasegger and Jan Legerský},

title = {{Computing Animations of Linkages with Rotational Symmetry (Media Exposition)}},

booktitle = {{36th International Symposium on Computational Geometry (SoCG 2020)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {164},

pages = {77:1--77:4},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},

address = {Dagstuhl, Germany},

isbn_issn = {ISBN 978-3-95977-143-6},

year = {2020},

editor = {Sergio Cabello and Danny Z. Chen},

refereed = {no},

length = {4},

url = {https://doi.org/10.4230/LIPIcs.SoCG.2020.77}

}

[Grasegger]

### FlexRiLoG - A SageMath Package for Motions of Graphs

#### G. Grasegger, J. Legerský

In: Mathematical Software – ICMS 2020, Bigatti A., Carette J., Davenport J., Joswig M., de Wolff T. (ed.), Proceedings of ICMS 2020, Lecture Notes in Computer Science 12097, pp. 442-450. 2020. Springer, Cham, ISBN 978-3-030-52199-8. [doi]@

author = {G. Grasegger and J. Legerský},

title = {{FlexRiLoG - A SageMath Package for Motions of Graphs}},

booktitle = {{ Mathematical Software – ICMS 2020}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {12097},

pages = {442--450},

publisher = {Springer, Cham},

isbn_issn = {ISBN 978-3-030-52199-8},

year = {2020},

editor = {Bigatti A. and Carette J. and Davenport J. and Joswig M. and de Wolff T.},

refereed = {no},

length = {9},

conferencename = {ICMS 2020},

url = {https://doi.org/10.1007/978-3-030-52200-1_44}

}

**inproceedings**{RISC6182,author = {G. Grasegger and J. Legerský},

title = {{FlexRiLoG - A SageMath Package for Motions of Graphs}},

booktitle = {{ Mathematical Software – ICMS 2020}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {12097},

pages = {442--450},

publisher = {Springer, Cham},

isbn_issn = {ISBN 978-3-030-52199-8},

year = {2020},

editor = {Bigatti A. and Carette J. and Davenport J. and Joswig M. and de Wolff T.},

refereed = {no},

length = {9},

conferencename = {ICMS 2020},

url = {https://doi.org/10.1007/978-3-030-52200-1_44}

}

[Grasegger]

### Bracing frameworks consisting of parallelograms

#### G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]@

author = {G. Grasegger and J. Legerský},

title = {{Bracing frameworks consisting of parallelograms}},

language = {english},

year = {2020},

institution = {arXiv},

length = {20},

url = {https://arxiv.org/abs/2008.11521}

}

**techreport**{RISC6203,author = {G. Grasegger and J. Legerský},

title = {{Bracing frameworks consisting of parallelograms}},

language = {english},

year = {2020},

institution = {arXiv},

length = {20},

url = {https://arxiv.org/abs/2008.11521}

}

[Grasegger]

### Zero-sum cycles in flexible polyhedra

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2020. [url]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Zero-sum cycles in flexible polyhedra}},

language = {english},

year = {2020},

institution = {arXiv},

length = {16},

url = {https://arxiv.org/abs/2009.14041}

}

**techreport**{RISC6211,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Zero-sum cycles in flexible polyhedra}},

language = {english},

year = {2020},

institution = {arXiv},

length = {16},

url = {https://arxiv.org/abs/2009.14041}

}

[Grasegger]

### On the Classification of Motions of Paradoxically Movable Graphs

#### G. Grasegger, J. Legerský, J. Schicho

Journal of Computational Geometry 11(1), pp. 548-575. 2020. ISSN: 1920-180X. [doi]@

author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Classification of Motions of Paradoxically Movable Graphs}},

language = {english},

journal = {Journal of Computational Geometry},

volume = {11},

number = {1},

pages = {548--575},

isbn_issn = {ISSN: 1920-180X},

year = {2020},

refereed = {yes},

length = {27},

url = {https://doi.org/10.20382/jocg.v11i1a22}

}

**article**{RISC6239,author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Classification of Motions of Paradoxically Movable Graphs}},

language = {english},

journal = {Journal of Computational Geometry},

volume = {11},

number = {1},

pages = {548--575},

isbn_issn = {ISSN: 1920-180X},

year = {2020},

refereed = {yes},

length = {27},

url = {https://doi.org/10.20382/jocg.v11i1a22}

}

[Grasegger]

### Counting realizations of {L}aman graphs on the sphere

#### M. Gallet, G. Grasegger, J. Schicho

Electronic Journal of Combinatorics 27(2), pp. 1-18. 2020. 1077-8926 . [url]@

author = {M. Gallet and G. Grasegger and J. Schicho},

title = {{Counting realizations of {L}aman graphs on the sphere}},

language = {english},

journal = {Electronic Journal of Combinatorics},

volume = {27},

number = {2},

pages = {1--18},

isbn_issn = {1077-8926 },

year = {2020},

refereed = {yes},

length = {18},

url = {https://arxiv.org/abs/1903.01145}

}

**article**{RISC6243,author = {M. Gallet and G. Grasegger and J. Schicho},

title = {{Counting realizations of {L}aman graphs on the sphere}},

language = {english},

journal = {Electronic Journal of Combinatorics},

volume = {27},

number = {2},

pages = {1--18},

isbn_issn = {1077-8926 },

year = {2020},

refereed = {yes},

length = {18},

url = {https://arxiv.org/abs/1903.01145}

}

[Li]

### Invertible Paradoxic Loop Structures for Transformable Design

#### Li, Z. and Nawratil, G. and Rist, F. and Hensel, M.

Computer Graphics Forum, pp. 261-275. 2020. 1467-8659. [doi]@

author = {Li and Z. and Nawratil and G. and Rist and F. and Hensel and M.},

title = {{Invertible Paradoxic Loop Structures for Transformable Design}},

language = {english},

journal = {Computer Graphics Forum},

pages = {261--275},

isbn_issn = {1467-8659},

year = {2020},

refereed = {yes},

length = {15},

url = {http://doi.org/10.1111/cgf.13928}

}

**article**{RISC6233,author = {Li and Z. and Nawratil and G. and Rist and F. and Hensel and M.},

title = {{Invertible Paradoxic Loop Structures for Transformable Design}},

language = {english},

journal = {Computer Graphics Forum},

pages = {261--275},

isbn_issn = {1467-8659},

year = {2020},

refereed = {yes},

length = {15},

url = {http://doi.org/10.1111/cgf.13928}

}

[Schicho]

### Probabilities of incidence between lines and a plane curve over finite field

#### M. Gallet, M. Makhul, J. Schicho

Finite Fields and Their Applications 61, pp. 1-22. 2020. 1071-5797.@

author = {M. Gallet and M. Makhul and J. Schicho},

title = {{Probabilities of incidence between lines and a plane curve over finite field}},

language = {english},

journal = {Finite Fields and Their Applications},

volume = {61},

pages = {1--22},

isbn_issn = {1071-5797},

year = {2020},

refereed = {yes},

length = {22}

}

**article**{RISC6073,author = {M. Gallet and M. Makhul and J. Schicho},

title = {{Probabilities of incidence between lines and a plane curve over finite field}},

language = {english},

journal = {Finite Fields and Their Applications},

volume = {61},

pages = {1--22},

isbn_issn = {1071-5797},

year = {2020},

refereed = {yes},

length = {22}

}

[Schicho]

### On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs

#### V. Bartzos, I. Emiris, J. Schicho

Applicable Algebra in Engineering, Communication and Computing 31, pp. 325-357. 2020. 0938-1279. [url]@

author = {V. Bartzos and I. Emiris and J. Schicho},

title = {{On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs}},

language = {english},

journal = {Applicable Algebra in Engineering, Communication and Computing},

volume = {31},

pages = {325--357},

isbn_issn = {0938-1279},

year = {2020},

refereed = {yes},

length = {32},

url = {https://arxiv.org/abs/2005.14485}

}

**article**{RISC6245,author = {V. Bartzos and I. Emiris and J. Schicho},

title = {{On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs}},

language = {english},

journal = {Applicable Algebra in Engineering, Communication and Computing},

volume = {31},

pages = {325--357},

isbn_issn = {0938-1279},

year = {2020},

refereed = {yes},

length = {32},

url = {https://arxiv.org/abs/2005.14485}

}

[Schicho]

### Counting Projections of Rational Curves

#### M. Gallet, J. Schicho

Israel Journal of Mathematics 240, pp. 503-538. 2020. 0021-2172 . [url]@

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

title = {{Counting Projections of Rational Curves}},

language = {english},

journal = {Israel Journal of Mathematics},

volume = {240},

pages = {503--538},

isbn_issn = {0021-2172 },

year = {2020},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1707.05264}

}

**article**{RISC6246,author = {M. Gallet and J. Schicho},

title = {{Counting Projections of Rational Curves}},

language = {english},

journal = {Israel Journal of Mathematics},

volume = {240},

pages = {503--538},

isbn_issn = {0021-2172 },

year = {2020},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1707.05264}

}

### 2019

[Capco]

### Sum of Squares over Rationals

#### J. Capco, C. Scheiderer

RISC. Technical report, 2019. [url] [pdf]@

author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}

**techreport**{RISC5884,author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}

[Grasegger]

### Graphs with Flexible Labelings

#### G. Grasegger, J. Legerský, J. Schicho

Discrete & Computational Geometry 62(2), pp. 461-480. 2019. 1432-0444. arXiv:1708.05298. [doi]@

author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings}},

language = {english},

journal = {Discrete & Computational Geometry},

volume = {62},

number = {2},

pages = {461--480},

isbn_issn = {1432-0444},

year = {2019},

note = {arXiv:1708.05298},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1007/s00454-018-0026-9}

}

**article**{RISC5803,author = {G. Grasegger and J. Legerský and J. Schicho},

title = {{Graphs with Flexible Labelings}},

language = {english},

journal = {Discrete & Computational Geometry},

volume = {62},

number = {2},

pages = {461--480},

isbn_issn = {1432-0444},

year = {2019},

note = {arXiv:1708.05298},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1007/s00454-018-0026-9}

}

[Legersky]

### Flexible and Rigid Labelings of Graphs

#### Jan Legerský

Research Institute for Symbolic Computation, Johannes Kepler University Linz. PhD Thesis. 2019. [url] [pdf]@

author = {Jan Legerský},

title = {{Flexible and Rigid Labelings of Graphs}},

language = {english},

year = {2019},

translation = {0},

school = {Research Institute for Symbolic Computation, Johannes Kepler University Linz},

length = {108},

url = {https://jan.legersky.cz/project/movablegraphs/}

}

**phdthesis**{RISC5941,author = {Jan Legerský},

title = {{Flexible and Rigid Labelings of Graphs}},

language = {english},

year = {2019},

translation = {0},

school = {Research Institute for Symbolic Computation, Johannes Kepler University Linz},

length = {108},

url = {https://jan.legersky.cz/project/movablegraphs/}

}