It is a well-known result by A. Reeves and B. Sturmfels that the reduction modulo a marked set of polynomials is Noetherian if and only if the marking is induced from an admissible term ordering. For finite sets of polynomials with a nonadmissible ordering, there is a constructive proof of the existence of an infinite reduction sequence, although a finite one might still be possible. On the base of our specialized software for combinatorics of monomial orderings, we have found some examples for which there is no finite reduction sequence. This is what we call “strong” non-Noetherity. Bibliography: 3 titles.
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A. Reeves and B. Sturmfels, “A note on polynomial reduction,” J. Symbolic Comput., 16, 273–277 (1993).
D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms, 3rd edition, Springer, New York (2007).
J. C. Faugère, P. Gianni, D. Lazard, and T. Mora, “Efficient computation of zero-dimensional Gröbner bases by change of ordering,” J. Symbolic Comput., 16, No. 4, 329–344 (1993).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 373, 2009, pp. 73–76.
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Vassiliev, N., Pavlov, D. Strong non-Noetherity of polynomial reduction. J Math Sci 168, 349–350 (2010). https://doi.org/10.1007/s10958-010-9985-y
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DOI: https://doi.org/10.1007/s10958-010-9985-y