Exponential space computation of Groebner bases

Given a polynomial ideal and a term order, there is a unique reduced Groebner basis and, for each polynomial, a unique normal form, namely the smallest (with respect to the term order) polynomial in the same coset. We consider the problem of finding this normal form for any given polynomial without prior computation of the Groebner basis. This is done by transforming a representation of the normal form into a system of linear equations and solving this system. Using the ability to find normal forms, we show how to obtain the Groebner basis in exponential space.