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The generalized Rabinowitsch trick. (English) Zbl 1396.13025
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 219-229 (2017).
Summary: The famous Rabinowitsch trick for Hilbert’s Nullstellensatz is generalized and used to analyze various properties of a polynomial with respect to an ideal. These properties include, among others, (i) checking whether the polynomial is a zero divisor in the residue class ring defined by the associated ideal and (ii) checking whether the polynomial is invertible in the residue class ring defined by the associated ideal. Just like using the classical Rabinowitsch’s trick, its generalization can also be used to decide whether the polynomial is in the radical of the ideal. Some of the byproducts of this construction are that it is possible to be more discriminatory in determining whether the polynomial is a zero divisor (invertible, respectively) in the quotient ring defined by the ideal, or the quotient ideal constructed by localization using the polynomial. This method also computes the smallest integer which gives the saturation ideal of the ideal with respect to a polynomial. The construction uses only a single Gröbner basis computation to achieve all these results.
For the entire collection see [Zbl 1379.13001].

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13A15 Ideals and multiplicative ideal theory in commutative rings
Full Text: DOI
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