Injectivity of polynomials over finite discrete dynamical systems

TL;DR

This paper characterizes univariate injective polynomials over finite discrete dynamical systems, providing a coefficient-based criterion and an efficient algorithm for solving related polynomial equations.

Contribution

It offers a new algebraic characterization of injective polynomials and introduces a polynomial-time algorithm for solving their equations.

Findings

Characterization of injective polynomials via coefficient form

Development of a polynomial-time algorithm for solving polynomial equations

Application to algebraic decomposition of dynamical systems

Abstract

The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering. The decomposition of dynamics into simpler subsystems allows us to simplify this analysis (or design). Here we focus on an algebraic approach to decomposition, based on alternative and synchronous execution as the sum and product operations; this gives rise to polynomial equations (with a constant side). In this article we focus on univariate, injective polynomials, giving a characterization in terms of the form of their coefficients and a polynomial-time algorithm for solving the associated equations.