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So I have understood that Gröbner bases are a pretty big deal and have variety of applications. Namely, if you can represent something with a system of polynomial equations you have the Gröbner bases and anything related to them at your disposal. However, what is the relation of Gröbner basis to other available methods? Is it more like so that any system of polynomial equations can always be solved with a Gröbner bases, but computationally might not be the best possible choice; however alternatives to the G-bases might not provide solution at all?

I am trying to understand how Gröbner bases fit into the bigger picture of optimization and solving problems efficiently.

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There are several improvements you can make on Buchberger's Algorithm for computing Groebner bases, to the point where using other methods usually doesn't make sense. Most computer algebra systems use Groebner bases for their computations since they typically outperform any other method. I think one main reason Groebner bases are so revered is that they provide constructive solutions for many important problems. Take Hilbert's Syzygy Theorem for example. You can prove it using abstract nonsense homological algebra stuff, or you can prove it using Groebner bases. The Groebner basis approach actually gives a recipe for an algorithm to compute finite free resolutions, which is vastly more desirable.

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