There is an algorithm that computes the normal form of p modulo the polynomial ideal generated by the given collection of polynomials with respect to the term order specified by the given collection of integers. The amount of work space used by this algorithm, regarded as a function of the amount of space necessary for writing down the input, is bounded by an exponentially growing function. An exponential and, hence, asymptotically corresponding lower bound on the space needed by every algorithm computing normal forms of polynomials is well established. This implies that the presented algorithm is asymptotically optimal with respect to its space requirement. As in the case of the normal form algorithm, this space requirement asymptotically matches a corresponding lower bound and is, therefore, optimal.
Further applications are in ideal theoretic problems arising in algebraic geometry.