Linear Dynamical Systems over Finite Rings

TL;DR

This paper introduces an efficient algorithm to analyze the cycle structure of linear dynamical systems over finite commutative rings, extending classical finite field results to more general algebraic structures.

Contribution

It provides the first known polynomial-time algorithm for determining fixed point behavior of systems over finite rings, generalizing finite field methods.

Findings

Algorithm runs in O(n^3 log(n log(q))) time

Determines fixed point systems over finite rings efficiently

Extends finite field dynamical system analysis to rings

Abstract

The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring $R$ such that $|R|=q$, where $q$ is a positive integer, the algorithm determines whether a given linear system over $R^n$ is a fixed point system or not in time $O(n^3\log(n\log(q)))$.