Injectivity of polynomials over finite discrete dynamical systems
For researchers in algebraic approaches to dynamical systems, this provides a theoretical characterization and efficient algorithm for injectivity, though the problem is niche.
The paper characterizes injective univariate polynomials over finite discrete dynamical systems and provides a polynomial-time algorithm for solving associated equations.
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.