A Criterion for Annihilating Ideals of Linear Recurring Sequences over Galois Rings

Abstract.

Let R be a local Artin principal ideal ring, R[x] the polynomial ring over R with indeterminate x. Let π be an element of R such that <π> is the unique maximal ideal of R. Let I be a zero-dimensional ideal of R[x], and the radical ideal of I. In this paper we show that I is the annihilating ideal of a linear recurring sequence over R if and only if I satisfies the following formula

The two sides of the formula can be feasibly computed by some typical algorithms from the theory of Gröbner bases. Our result is a solution of Nechaevs Open Problem suggested in [11].