# Is Buchberger's algorithm or Wu's method valuable theoretically when we have the Tarski–Seidenberg theorem?

Is Buchberger's algorithm or Wu's method valuable theoretically when we have the Tarski–Seidenberg theorem？ In other words, could the Tarski–Seidenberg theorem subsume Buchberger's algorithm and Wu's method?

Actually, algorithms of both Buchberger and Wu may be just deduced from variants of Hilbert's Nullstellensatz.

• I'm not sure what point you are trying to make when you say "B and W may be just deduced from variants of HN." Is there some related question you mean to be asking with this sentence? – Joshua Grochow Oct 2 '17 at 15:38

Second, Tarski-Seidenberg is for semi-algebraic sets over the reals (that is, allowing $\leq, <, =, \neq$), whereas Buchberger's algorithm works not only for the reals, but for polynomials over any field, or even over other rings (such as $\mathbb{Z}$). With minor modifications, Buchberger's algorithm even works in various noncommutative analogues of polynomial rings.
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