Computing zero-dimensional schemes

This paper generalizes the Buchberger-M\"oller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the problems which can be solved by the algorithm: an approach which looks to be readily applicable in several areas. Implementation issues are also addressed, especially for computations over~$\bbb Q$ where a trace-lifting paradigm is employed. We give a complexity analysis of the new algorithm for fat points in affine space over~$\bbb Q$. Tables of timings show the new algorithm to be efficient in practice.