It avoids some (hopefully many...) of the monomials by taking only those not
divisible by leading monomials of the groebner basis. (These monomials form
a basis of the quotient space, which is the most basic property of groebner
bases.)
From what I can see, the final result may be smaller, but it still has
to run the loop for each monomial, so it doesn't seem to me to be any
more time efficient (just space efficient).
Sure. But I think (possibly contrary to what I said earlier), staircase
really isn't the problem. If the result is huge then the next parts
(calling reduced(), solving the linear system) are going to take ages as
well.
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