Linear recurring sequences over modules
The paper is devoted to the following generalization of the concept of the linear recurring sequence. Let \(M\) be an \(R\)-module over a commutative ring with identity. Let \(M^{\langle k\rangle}\) be the set of all \(k\)-sequences, i.e. functions \(\mu: \mathbb{N}_0^k\to M\). A sequence \(\mu\in M^{\langle k\rangle}\) is called a \(k\)-linear recurring sequence over \(M\) if the annihilator of \(\mu\) in the ring of polynomials of \(k\) variables contains \(k\) monic polynomials \(F_1 (x_1), \dots, F_k (x_k)\) of one variable and with specified multiplication of polynomials and \(k\)-sequences. In the paper some fundamental results of the theory of \(k\)-linear recurring sequences over rings and modules (e.g. quasi-Frobenius modules or Galois rings) are collected, thereby giving an overview of the work done in the direction of the extension of the theory of linear recurrences over fields to this more general setting. (The paper contains a list of references consisting of 112 items).
scientific article; zbMATH DE number 841769
scientific article