Abstract
Standard bases of ideals of the polynomial ring R[X] = R[x 1, …, x k ] 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 L R (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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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 23–71, 2004.
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Gorbatov, E.V. Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences. J Math Sci 139, 6672–6707 (2006). https://doi.org/10.1007/s10958-006-0384-3
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DOI: https://doi.org/10.1007/s10958-006-0384-3