Dr. Jose Capco

Research Area

Algebraic Geometry in Kinematics

Other than my own working area, I am also interested in number theory (and its extensions: algorithmic/algebraic/analytic number theories), cryptography, real algebraic geometry (my PhD thesis), commutative algebra in general (worked a bit with Galois theory on commutative rings), graph theory, computational geometry, fractal sets (looked a bit on Julia sets from esoteric fields), elliptic curves, Lie algebra and mathematical chess problems (see Noam Elkies on enumerative chess problems).

Publications

2020

[Capco]

Two remarks on sums of squares with rational coefficients

Jose Capco, Claus Scheiderer

Banach Center Publications 121, pp. 25-36. 2020. 0137-6934. [doi]

[bib]

@article{RISC5700,
author = {Jose Capco and Claus Scheiderer},
title = {{ Two remarks on sums of squares with rational coefficients}},
language = {english},
journal = {Banach Center Publications},
volume = {121},
pages = {25--36},
isbn_issn = {0137-6934},
year = {2020},
refereed = {yes},
length = {11},
url = {https://doi.org/10.4064/bc121-2}
}

[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]

[bib]

@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}
}

2019

[Capco]

Sum of Squares over Rationals

J. Capco, C. Scheiderer

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

[bib]

@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}
}

2018

[Capco]

Generalizing some Results in Field Theory for Rings

Jose Capco

Communications in Algebra, pp. 0-12. 2018. 0092-7872. Preprint. [pdf]

[bib]

@article{RISC5375,
author = {Jose Capco},
title = {{Generalizing some Results in Field Theory for Rings}},
language = {english},
journal = {Communications in Algebra},
pages = {0--12},
isbn_issn = {0092-7872},
year = {2018},
note = {Preprint},
refereed = {no},
length = {13}
}

2017

[Capco]

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/}
}

[Capco]

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/}
}

2015

[Capco]

Odd Collatz Sequence and Binary Representations

Jose Capco

Submitted to the RISC Report Series. RISC Report , 2015. [pdf]

[bib]

@techreport{RISC5321,
author = {Jose Capco},
title = {{Odd Collatz Sequence and Binary Representations}},
language = {english},
abstract = {Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro \cite{And2} and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$ the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$ then the Collatz conjecture is true.},
year = {2015},
howpublished = {RISC Report },
length = {11},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}