Easy to Check Algebraic Characterizations of Dynamical Properties for Linear CA and Additive CA over a Finite Abelian Group

We focus on how the dynamical properties of any Linear CA over \((\mathbb {Z}/m\mathbb {Z})^n\) are hidden inside the characteristic polynomial of its defining matrix, namely, a polynomial of degree n in the indeterminate t and with Laurent polynomials over \(\mathbb {Z}/m\mathbb {Z}\) as coefficients. In particular, as far as Linear CA over \((\mathbb {Z}/m\mathbb {Z})^n\) are concerned, we review the mostly recent algebraic decidable characterizations of the following properties: injectivity, surjectivity, sensitivity to the initial conditions, equicontinuity, topological transitivity, and positive expansivity. These characterizations are easy to check, i.e., related decision algorithms can be designed is such a way that exponential terms in their computational complexity are avoided as much as possible. In particular, gcd operations are involved, while the prime factor decomposition of m is bypassed. Finally, we recall how such characterizations regarding Linear CA over \((\mathbb {Z}/m\mathbb {Z})^n\) can be exploited to decide the above mentioned dynamical properties for the whole class of Additive CA over a finite abelian group.