Recursive sequences and polynomial congruences

Abstract

We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer $m$. We show that the period of such a sequence with characteristic polynomial $f$ can be expressed in terms of the order of $\omega =x+\langle f\rangle$ as a unit in the quotient ring ${\mathbb{Z}}_m\left[\omega \right]={\mathbb{Z}}_m\left[x\right]∕\langle f\rangle$. When $m=p$ is prime, this order can be described in terms of the factorization of $f$ in the polynomial ring ${\mathbb{Z}}_p\left[x\right]$. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree $k\le 5$ in ${\mathbb{Z}}_p\left[x\right]$.