The Complexity of Local Dimensions for Constructible Sets

We show that deciding whether an algebraic variety has an irreducible component of codimension at least d is an NP C-complete problem for every xed d (and is in the Arthur-Merlin class if we assume a bit model of computation). However, when d is not xed but is instead part of the input, we show that the problem is not likely to be in NP C or in coNP C. These results are generalized to arbitrary constructible sets. We also study the complexity of a few other related problems. 1. Introduction It was shown in 14] that computing the dimension of algebraic varieties is NP C-complete in the Blum-Shub-Smale model of computation, and that in the bit model this problem is in AM (the Arthur-Merlin complexity class) assuming the Generalized Riemann Hypothesis (GRH). The dimension of a variety is the dimension of its largest irreducible component, and the dimensions of smaller components may also be of interest (see for instance 18]). In this paper we investigate the complexity of computing the dimensions of irreducible components, or more generally of computing