A canonical system of generators of a unitary polynomial ideal over a commutative Artinian chain ring

A. A. Nechaev ; D. A. Mikhailov

Diskretnaya Matematika, Tome 13 (2001) no. 4, pp. 3-42

Résumé

Let $R$ be a commutative Artinian chain ring. An ideal $I$ of the ring $\mathcal R _ k=R[x_1,\ldots,x_k]$ is called monic if the quotient ring $\mathcal R_k \setminus I$ is a finitely generated $R$-module. For such ideal a standard basis, called the Canonical Generating System (CGS), is constructed. This basis inherits some good properties of CGS of an ideal of $R[x]$ and the Gröbner basis of a polynomial ideal over a field. In particular, using CGS, it is possible to present an algorithm, which is simpler than the exhaustive search algorithm, for constructing cosets of $\mathcal R_k$ modulo $I$. The CGS allows us to check whether the quotient ring $\mathcal R_k\setminus I$ is a free $R$-module. Moreover, if $R$ is a finite ring there is a formula for calculation of $|\mathcal R_k\setminus I|$ that depends only on numerical parameters of CGS. Applying CGS, we create a generating system of a family of $k$-linear recurring sequences with characteristic ideal $I$ and a criterion of existence of a $k$-linear shift register with this characteristic ideal. This research was supported by the Russian Foundation for Basic Research, grants 99–01–00941 and 99–01–00382.