Randomized and deterministic algorithms for the dimension of algebraic varieties

We prove old and new results on the complexity of computing the dimension of algebraic varieties. In particular, we show that this problem is NP-complete in the Blum-Shub-Smale model of computation over C, that it admits a s(O(1))D-O(n) deterministic algorithm, and that for systems with integer coefficients it is in the Arthur-Merlin class under the Generalized Riemann Hypothesis. The first two results are based on a general derandomization argument.