Roots in the semiring of finite deterministic dynamical systems

TL;DR

This paper develops polynomial algorithms for division and root extraction in the algebraic structure of finite deterministic dynamical systems, enabling efficient solutions to related polynomial equations.

Contribution

It introduces two algorithms for division and root extraction in FDDS semiring, facilitating solving polynomial equations involving connected systems.

Findings

Algorithms for division and $k$-th roots in FDDS semiring.

Efficient solutions for equations of the form $AX^k=B$ with connected FDDS.

Advances towards solving general polynomial equations on FDDS.

Abstract

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$. These results are some of the important final steps for solving more general polynomial equations on FDDS.