Abstract
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.
The authors were supported by the NSF DMS-1217054 and NSFC 11371356.
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Notes
- 1.
A set G is a minimal Gröbner basis of I if (1) G is a Gröbner basis of I, and (2) for each \(g\in G\), \(\mathrm{lpp}(g)\) is not divisible by any leading power products of \(G \setminus \{g\}\).
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Acknowledgements
The authors are grateful to Professors Teo Mora and Yosuke Sato for their helpful discussions and suggestions.
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Kapur, D., Sun, Y., Wang, D., Zhou, J. (2017). The Generalized Rabinowitsch Trick. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_14
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