On a generalization of Stickelberger’s Theorem

Abstract

We prove two versions of Stickelberger’s Theorem for positive dimensions and use them to compute the connected and irreducible components of a complex algebraic variety. If the variety is given by polynomials of degree ≤d in n variables, then our algorithms run in parallel (sequential) time (nlogd)O(1) (dO(n4)). In the case of a hypersurface, the complexity drops to O(n2log2d) (dO(n)). In the proof of the last result we use the effective Nullstellensatz for two polynomials, which we also prove by very elementary methods.