Growth function for an $n$-valued dynamics

TL;DR

This paper investigates the growth function of a specific $n$-valued group, providing formulas for prime $n$, asymptotic estimates for general $n$, and proposing new open problems in the area.

Contribution

It derives a formula for the growth function when $n$ is prime and establishes polynomial asymptotics for general $n$, advancing understanding of $n$-valued group dynamics.

Findings

Formula for growth function when $n$ is prime

Polynomial asymptotic estimate for general $n$

New problems and hypotheses about growth functions

Abstract

This article answers the question of V.M. Buchstaber about the growth function of a particular $n$-valued group. This question is closely related to discrete integrable systems. In this paper, we will find a formula for the growth function in the case when $n$ is prime. In addition, we will prove a polynomial asymptotic estimate for the growth function in the general case. Finally, we will pose new problems and hypotheses about growth functions.