A criterion for annihilating ideals of linear recurring sequences over Galois rings

Let R be a local Artin principal ideal ring, R[x] the polynomial ring over R with indeterminate x. Let pi be an element of R such that (pi) is the unique maximal ideal of R. Let I be a zero-dimensional ideal of R [x], and root1 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 dimR/(pi) I : rootI/I = dimR/(pi) R[x]/rootI. The two sides of the formula can be feasibly computed by some typical algorithms from the theory of Grobner bases. Our result is a solution of Nechaev's Open Problem suggested in [11].