Abstract
Throughout, K shall denote a field. We show that a Gröbner basis G ⊆ K[X 1,..., X n ] has greatest common divisor 1 precisely if this holds for the set of leading power products of G. As a corollary a result of Buchberger concerning the reducibility of S-Polynomials to 0 can be refined to a necessary and sufficient criterion.
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References
B Buchberger B. “Gröbner Bases: An Algorithm Approach to Polynomial Ideal Theory”, p. 184-232 in N. K. Bose (ed.): Multidimensional System Theory, Reidel (1985).
H Hungerford T. Algebra, Graduate Texts in Mathematics 73, Springer (1974).
K Kovačec A. “ Buchberger’s Theory of Gröbner bases”, unpublished LATEX notes written at occasion of the Algebra-conference at Lisbon/Portugal, June 1988.
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Kovačec, A. (1990). Relatively Prime Gröbner Bases and Reducibility of S-Polynomials. In: Almeida, J., Bordalo, G., Dwinger, P. (eds) Lattices, Semigroups, and Universal Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2608-1_16
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DOI: https://doi.org/10.1007/978-1-4899-2608-1_16
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