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Metadata 
bibkey: gurvichkrncvyalyi2025amoebas
authors: Vladimir Gurvich; Matjaž Krnc; Mikhail Vyalyi
year: 2025
title: "Growing Trees and Amoebas’ Replications"
doi: 10.1007/s00025-025-02421-6
url: https://arxiv.org/abs/2401.07484
claim: "Section 2 and Conjecture 1 state that an amoeba A=(H,m) with extension parameter l is totally confined whenever the maximum degree of the finite tree H exceeds the number of roots k by more than two."
strata_touched:
  - D5/S3/Combinatorics/Graph/Walks/AmoebaConfinement
  - D5/S3/Combinatorics/Graph/Walks/AmoebaHost
license: citation-only
triage: anchor

Growing Trees and Amoebas’ Replications

Vladimir Gurvich, Matjaž Krnc and Mikhail Vyalyi, Growing Trees and Amoebas’ Replications, arXiv:2401.07484v1/v2 and Results in Mathematics (2025). Section 2 defines ordinary amoebas, extensions, total-edge minimum growths and total confinement. Conjecture 1 is the complete finite-tree assertion settled by the corresponding D5 modules.

Verified locator

  • DOI: 10.1007/s00025-025-02421-6
  • DOI URL: https://doi.org/10.1007/s00025-025-02421-6
  • arXiv: https://arxiv.org/abs/2401.07484
  • arXiv v1 HTML: https://arxiv.org/html/2401.07484v1
  • arXiv v2 HTML: https://arxiv.org/html/2401.07484v2
  • Published PDF: https://link.springer.com/content/pdf/10.1007/s00025-025-02421-6.pdf
  • Locator: arXiv v1 PDF page 7, Section 2.1, Conjecture 1; arXiv v2 and published PDF page 11, Conjecture 1. Definitions are on published pages 3–5.

Source statement

The source assertion is: for any finite simple tree H, multiplicity function m, and extension length l, if the maximum degree of H exceeds the number of roots k by more than two, then the amoeba is totally confined. The finite reading retains every initial finite tree containing H, every multiplicity-preserving copy, every simultaneous family of arms and every minimum growth counted by all host edges. Zero lengths, zero multiplicities and k=0 remain included. The published statement and its arXiv versions leave infinite union limits as a separate question; this note records the finite source convention used by the formalization.