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The algorithm terminates because it is consistently increasing the size of the monomial ideal generated by the leading terms of our set ''F'', and [[Dickson's lemma]] (or the [[Hilbert basis theorem]]) guarantees that any such ascending chain must eventually become constant. Unfortunately, it may take a very long time to terminate, corresponding to the fact that [[Gröbner bases]] can be ''extremely'' large. Thus, it has large storage requirements ([[space complexity]]). Also, the [[time complexity]] of the algorithm is doubly exponential in the input data, which implies that its worst-case behavior can be very slow.
Further methods for computing Gröbner bases include the [[Faugère F4 algorithm]], based on the same mathematics as the Buchberger algorithm, and involutive approaches, based on ideas from [[
==See also==
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