2004, VOLUME 10, NUMBER 3, PAGES 23-71

Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences

E. V. Gorbatov


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Standard bases of ideals of the polynomial ring R[X]=R[x1,…,xk] over a commutative Artinian chain ring R that are concordant with the norm on R have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from R. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family LR(I), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram $\mathcal F$ and a full system of $\mathcal F$-monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Gröbner basis of a monic ideal to a standard basis with the same cardinality.

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Last modified: June 2, 2005