Finite phylogenetic complexity of $Z_p$ and invariants for $Z_3$

TL;DR

This paper proves that the phylogenetic complexity of finite abelian groups Z_p, with p prime, is finite, and confirms the conjecture that Z_3 has complexity 3, advancing understanding of this invariant.

Contribution

It establishes the finiteness of phylogenetic complexity for all Z_p with p prime and confirms the specific value for Z_3, resolving key conjectures.

Findings

Phylogenetic complexity of Z_p is finite for prime p.

Confirmed that the complexity of Z_3 equals 3.

Extended known results from Z_2 to all Z_p with p prime.

Abstract

We study phylogenetic complexity of finite abelian groups - an invariant introduced by Sturmfels and Sullivant. The invariant is hard to compute - so far it was only known for $Z_2$, in which case it equals $2$. We prove that phylogenetic complexity of any group $Z_p$, where $p$ is prime, is finite. We also show, as conjectured by Sturmfels and Sullivant, that the phylogenetic complexity of $Z_3$ equals $3$.