Parametrization of linear recurrence relations by row reduction for sequen- ces over a finite ring

In this paper we address the problem of parametrizing all shortest linear recurrence relations for a finite sequence over a finite ring of the type Zpr , where p is a prime integer and r is a positive integer. In the past, the behavioral approach to systems theory has been successfully used to solve this problem in the field case: its solution is obtained by deriving a kernel representation with minimal row degrees, that is, a row reduced kernel representation. Until recently, the concept of row reducedness was not available in the ring case. The recent paper [Kuijper, Pinto,Polderman, Linear Alg. Appl. 2007] develops this concept and we use these results to solve the open problem of parametrizing all shortest linear recurrence relations for a given finite Zpr -sequence. For the field case it is wellknown that a shortest linear recurrence relation is unique if and only if the number of elements in the sequence exceeds twice the complexity of the sequence. We show that this condition is still necessary in the ring case but no longer sufficient. Moreover, we give necessary and sufficient conditions for uniqueness in the ring case.