@article{a2bdf83b92cf459f9f7ad359161598f1,
title = "Patchworking oriented matroids",
abstract = "In a previous work, we gave a construction of (not necessarily realisable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a combinatorial analogue of Viro's patchworking to derive a topological representation of the oriented matroid directly from the polyhedral structure of the triangulation. This provides a combinatorial manifestation of patchworking besides tropical algebraic geometry. We achieve this by defining a general homeomorphism-preserving operation on regular cell complexes which acts by merging adjacent cells in the complex together. We then rephrase the patchworking procedure in terms of this process using the theory of tropical oriented matroids.",
keywords = "UT-Hybrid-D",
author = "Marcel Celaya and Georg Loho and Yuen, {Chi Ho}",
note = "Funding Information: The first author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy — The Berlin Mathematics Research Center MATH+(EXC‐2046/1, project ID: 390685689). He is grateful to Josephine Yu for helpful discussions on patchworking, in particular pointing out the reference [ 33 ]. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement n ScaleOpt‐757481). The second author also thanks Xavier Allamigeon, Michael Joswig and Mateusz Skomra for inspiring discussions and Ben Smith for helpful comments. The third author was supported by Trond Mohn Foundation project {\textquoteleft}Algebraic and topological cycles in tropical and complex geometry{\textquoteright}, Croucher Fellowship for Postdoctoral Research, and Netherlands Organisation for Scientific Research Vici grant 639.033.514 during his affiliation to University of Oslo, Brown University and University of Bern, respectively. He also thanks Cheuk Yu Mak for pointing out [ 31 ]. The second and third author both thank the semester program at Institut Mittag‐Leffler for the introduction that started the project. The authors thank Laura Anderson, Jesus De Loera and Kris Shaw for comments on our work, as well as the anonymous referees for their helpful comments and suggestions. Tropical Geometry, Amoebas, and Polyhedra Funding Information: The first author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy — The Berlin Mathematics Research Center MATH+(EXC-2046/1, project ID: 390685689). He is grateful to Josephine Yu for helpful discussions on patchworking, in particular pointing out the reference [33]. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement n∘${}^{\circ}$ ScaleOpt-757481). The second author also thanks Xavier Allamigeon, Michael Joswig and Mateusz Skomra for inspiring discussions and Ben Smith for helpful comments. The third author was supported by Trond Mohn Foundation project {\textquoteleft}Algebraic and topological cycles in tropical and complex geometry{\textquoteright}, Croucher Fellowship for Postdoctoral Research, and Netherlands Organisation for Scientific Research Vici grant 639.033.514 during his affiliation to University of Oslo, Brown University and University of Bern, respectively. He also thanks Cheuk Yu Mak for pointing out [31]. The second and third author both thank the Tropical Geometry, Amoebas, and Polyhedra semester program at Institut Mittag-Leffler for the introduction that started the project. The authors thank Laura Anderson, Jesus De Loera and Kris Shaw for comments on our work, as well as the anonymous referees for their helpful comments and suggestions. Publisher Copyright: {\textcopyright} 2022 The Authors. Journal of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",
year = "2022",
month = dec,
doi = "10.1112/jlms.12667",
language = "English",
volume = "106",
pages = "3545--3576",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Wiley",
number = "4",
}