Can every ideal be decomposed into a triangular Groebner basis?

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Groebner basis calculations are an important technique for solving systems of nonlinear equations. Despite being (in the worst case), computationally intractable, they seem to be effective solutions for a multitude of problems.

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?
 

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  • #2
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AFAIK you can obtain and use reduced Gröbner bases. Triangularity is not a concept in the context that I knew of.
 

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