The amoeba dimension of a linear space
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Given a complex vector subspace V of Cn, the dimension of the amoeba of V ∩(C∗)n depends only on the matroid that V defines on the ground set {1, . . ., 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.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Proceedings of the American Mathematical Society |
| Vol/bind | 152 |
| Udgave nummer | 6 |
| Sider (fra-til) | 2385-2401 |
| Antal sider | 17 |
| ISSN | 0002-9939 |
| DOI | |
| Status | Udgivet - 2024 |
Bibliografisk note
Funding Information:
The first author was partially supported by Swiss National Science Foundation (SNSF) project grant 200021 191981 and by Vici grant 639.033.514 from the Netherlands Organisation for Scientific Research (NWO). The fourth author was supported by the FAPA project \u201CMatroids in tropical geometry\u201D from the Facultad de Ciencias, Universidad de los Andes, Colombia. The fifth author was supported by the Trond Mohn Foundation project \u201CAlgebraic and Topological Cycles in Complex and Tropical Geometries\u201D; he was also supported by the Centre for Advanced Study (CAS) in Oslo, Norway, which funded and hosted the Young CAS research project \u201CReal Structures in Discrete, Algebraic, Symplectic, and Tropical Geometries\u201D during the 2021/2022 and 2022/2023 academic years.
Funding Information:
The first author was partially supported by Swiss National Science Foundation (SNSF) project grant 200021_191981 and by Vici grant 639.033.514 from the Netherlands Organisation for Scientific Research (NWO). The fourth author was supported by the FAPA project \u201CMatroids in tropical geometry\u201D from the Facultad de Ciencias, Universidad de los Andes, Colombia. The fifth author was supported by the Trond Mohn Foundation project \u201CAlgebraic and Topological Cycles in Complex and Tropical Geometries\u201D; he was also supported by the Centre for Advanced Study (CAS) in Oslo, Norway, which funded and hosted the Young CAS research project \u201CReal Structures in Discrete, Algebraic, Symplectic, and Tropical Geometries\u201D during the 2021/2022 and 2022/2023 academic years. This paper grew out of several sources: J.R.\u2019s talk on amoebas [Rau20] where the formula of Theorem 1.3.1 was first conjectured, S.E.\u2019s Master\u2019s thesis [Egg22] at the University of Bern under the supervision of J.D., work by C.H.Y. on a combinatorial analysis of the Jacobian of Log at a general point of a linear space, and R.P.\u2019s work on the matroid M' of Theorems 1.3.2 and 1.3.3. C.H.Y. thanks Kris Shaw for suggesting this problem to him. All authors thank Frank Sottile for discussions on an early version of this work.
Publisher Copyright:
© 2024 American Mathematical Society.
ID: 392563062