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I have a problem that I am trying to solve. I have perused the literature and am trying to use Macaulay 2 to find a triangular Groebner basis for my system of quadratic equations. I am capable of finding triangular bases for some systems of nonlinear equations. However, I cannot seem to get Macaulay 2 to give me a triangular basis for the problem I am trying to solve.

My question is simple: does Buchberger's Algorithm or the F4/5 Algorithm terminate with a triangular Groebner basis necessarily? If not, are there some ideals that cannot have a triangular basis?