On intersections and stable intersections of tropical hypersurfaces
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 9-15.

We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless that stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize to them and the tropicalization of their intersection.

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Accepted:
Published online:
DOI: 10.5802/alco.327
Classification: 14T10, 14T15
Keywords: tropical geometry, tropical hypersurfaces, stable intersections.

Ren, Yue 1

1 Department of Mathematics Durham University Upper Mountjoy Campus Durham DH1 3LE United Kingdom
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ren, Yue. On intersections and stable intersections of tropical hypersurfaces. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 9-15. doi : 10.5802/alco.327. https://alco.centre-mersenne.org/articles/10.5802/alco.327/

[1] Bogart, T.; Jensen, A. N.; Speyer, D.; Sturmfels, B.; Thomas, R. R. Computing tropical varieties, J. Symbolic Comput., Volume 42 (2007) no. 1-2, pp. 54-73 | DOI | MR | Zbl

[2] Deng, Yiyang; Hampton, Marshall Equilateral chains and cyclic central configurations of the planar five-body problem, J. Nonlinear Sci., Volume 33 (2023) no. 1, Paper no. 4, 18 pages | DOI | MR | Zbl

[3] Hampton, Marshall; Jensen, Anders Finiteness of spatial central configurations in the five-body problem, Celestial Mech. Dynam. Astronom., Volume 109 (2011) no. 4, pp. 321-332 | DOI | MR | Zbl

[4] Hampton, Marshall; Jensen, Anders Nedergaard Finiteness of relative equilibria in the planar generalized N-body problem with fixed subconfigurations, J. Geom. Mech., Volume 7 (2015) no. 1, pp. 35-42 | DOI | MR | Zbl

[5] Hampton, Marshall; Moeckel, Richard Finiteness of relative equilibria of the four-body problem, Invent. Math., Volume 163 (2006) no. 2, pp. 289-312 | DOI | MR | Zbl

[6] Jensen, Anders; Sommars, Jeff; Verschelde, Jan Computing tropical prevarieties in parallel, PASCO 2017—International Workshop on Parallel Symbolic Computation, ACM, New York (2017) | DOI | MR

[7] Jensen, Anders Nedergaard Tropical Homotopy Continuation, 2016 | arXiv

[8] Maclagan, Diane; Sturmfels, Bernd Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015, xii+363 pages | DOI | MR

[9] Maclagan, Diane; Yu, Josephine Higher connectivity of tropicalizations, Math. Ann., Volume 384 (2022) no. 1-2, pp. 775-788 | DOI | MR | Zbl

[10] Mak, Cheuk Yu; Ruddat, Helge Tropically constructed Lagrangians in mirror quintic threefolds, Forum Math. Sigma, Volume 8 (2020), Paper no. e58, 55 pages | DOI | MR | Zbl

[11] Markwig, Thomas; Ren, Yue Computing tropical varieties over fields with valuation, Found. Comput. Math., Volume 20 (2020) no. 4, pp. 783-800 | DOI | MR | Zbl

[12] Yu, Josephine Do most polynomials generate a prime ideal?, J. Algebra, Volume 459 (2016), pp. 468-474 | DOI | MR | Zbl

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