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

摘要

Let R be a local Artin principal ideal ring, Rx the polynomial ring over R with indeterminate x. Let #pi# be an element of R such that is the unique maximal ideal of R. Let I be a zero-dimensional ideal of Rx, and sq root I 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 dim_(R/) (I : sq root I)/I=dim_(R/) Rx/sq root I. 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.