Problem A: Given a set of polynomial equations $ f_1=0,\ldots,f_m=0 $, where the $ f_i $'s are multivariate polynomials with $ n $ variables over a field $\mathbb F$, decide whether there is a (simultaneous) solution to these equations.
Question: What is the computational complexity of Problem A?
Note: Over finite fields $\mathbb F$, Problem A seems to be NP-complete. But what about over the reals, rationals, complex numbers?