The membership problem for unmixed polynomial ideals is solvable in single exponential time

Abstract

Deciding membership for polynomial ideals represents a classical problem of computational commutative algebra which is exponential space hard. This means that the usual algorithms for the membership problem which are based on linear algebra techniques have doubly exponential sequential worst case complexity. We show that the membership problem has single exponential sequential and polynomial parallel complexity for unmixed ideals. More specific complexity results are given for the special cases of zero-dimensional and complete intersection ideals.