Improved Gröbner basis algorithm

Question

I'm just learning about Gröbner bases and the Buchberger algorithm. I have seen chapters in several pieces of literature that deal with improving the Buchberger algorithm, but they never seem to mention the following (in my eyes obvious) improvement:

Let $F=(f_1,...f_n)$ be a set of polynomials. Before starting the Buchberger algorithm, do (I can't figure out how to format this properly):

for $i=1\dots n$:

$G = F - {f_i}$

$h = \operatorname{normalForm}(f_i, G)$

$f_i = h$

So this reduces the basis with respect to itself. Is there some reason this is not allowed/not desireable?

Answer 1

I'm not an expert on the computational complexity aspects of Grobner bases, but I believe what you are proposing is, in practice, actually performed at the start of each step of Buchberger's Algorithm. It is called autoreduction.