2 research outputs found
Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear cellular automata
39 pagesLet be a finite commutative ring, and let be a commutative -algebra. Let and be two -matrices over that have the same characteristic polynomial. The main result of this paper states that the set is finite if and only if the set is finite. We apply this result to the theory of discrete time dynamical systems. Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear cellular automata over the alphabet for , i.e. cellular automata in which the local rule is defined by -matrices with elements in . To prove our main result, we derive an integrality criterion for matrices that is likely of independent interest. Namely, let be any commutative ring (not necessarily finite), and let be a commutative -algebra. Consider any -matrix over . Then, is integral over (that is, there exists a monic polynomial satisfying ) if and only if all coefficients of the characteristic polynomial of are integral over . The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist)