Abstract
Zacharias and Trinks proved that it can be constructively determined whether a finite generating set is a generalized Gröbner basis provided ideals are detachable and syzygies are solvable in the coefficient ring. We develop an abstract rewriting characterization of generalized Gröbner bases and use it to give new proofs of the Spear-Zacharias and Trinks theorems for testing and constructing generalized Gröbner bases. In addition, we use the abstract rewriting characterization to generalize Ayoub's binary approach for testing and constructing Gröbner bases over polynomial rings with Euclidean coefficient rings to arbitrary principal ideal coefficient domains. This also shows that Spear-Zacharias' and Trinks' approach specializes to Ayoub's approach, which was not known before.
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Lankford, D. (1989). Generalized Gröbner bases: Theory and applications. A condensation. In: Dershowitz, N. (eds) Rewriting Techniques and Applications. RTA 1989. Lecture Notes in Computer Science, vol 355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51081-8_110
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DOI: https://doi.org/10.1007/3-540-51081-8_110
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