The amoeba dimension of a linear space

Draisma, Jan; Eggleston, Sarah; Pendavingh, Rudi; Rau, Johannes; Yuen, Chi Ho

doi:10.1090/proc/16744

The amoeba dimension of a linear space
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by Jan Draisma, Sarah Eggleston, Rudi Pendavingh, Johannes Rau and Chi Ho Yuen;

Proc. Amer. Math. Soc. 152 (2024), 2385-2401

DOI: https://doi.org/10.1090/proc/16744

Published electronically: April 23, 2024

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Abstract:

Given a complex vector subspace $V$ of $\mathbb {C}^n$, the dimension of the amoeba of $V \cap (\mathbb {C}^*)^n$ depends only on the matroid that $V$ defines on the ground set $\{1,\ldots ,n\}$. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.

References

George M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971), 459–469. MR 280489, DOI 10.1090/S0002-9947-1971-0280489-8
William H. Cunningham, Testing membership in matroid polyhedra, J. Combin. Theory Ser. B 36 (1984), no. 2, 161–188. MR 746547, DOI 10.1016/0095-8956(84)90023-6
Jan Draisma, Johannes Rau, and Chi Ho Yuen, The dimension of an amoeba, Bull. Lond. Math. Soc. 52 (2020), no. 1, 16–23. MR 4072029, DOI 10.1112/blms.12301
Sarah Eggleston, The dimension of amoebas of linear spaces, Master’s Thesis, Faculty of Science, University of Bern, 2022.
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong, A faster cutting plane method and its implications for combinatorial and convex optimization, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science—FOCS 2015, IEEE Computer Soc., Los Alamitos, CA, 2015, pp. 1049–1065. MR 3473356, DOI 10.1109/FOCS.2015.68
Mounir Nisse and Frank Sottile, Describing amoebas, Pacific J. Math. 317 (2022), no. 1, 187–205. MR 4440944, DOI 10.2140/pjm.2022.317.187
Johannes Rau, The dimension of an amoeba, LAGARTOS, 2020, https://sites.google.com/site/cotterillethan/ethan-cotterill-eng/latin-american-real-and-tropical-geometry-seminar.
Alexander Schrijver, Combinatorial optimization. Polyhedra and efficiency. Vol. B, Algorithms and Combinatorics, vol. 24, Springer-Verlag, Berlin, 2003. Matroids, trees, stable sets; Chapters 39–69. MR 1956925
Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. MR 1925796, DOI 10.1090/cbms/097