If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether the equation is always true. We can similarly decide if a single polynomial inequation is always true over $\mathbb{C}$, since $p(z_1,\ldots,z_n) \ne 0$ for all $z_i$ iff the polynomial is a nonzero constant.
Does the simplicity of randomized checking extend to any systems of multivariate polynomial equations? In particular, is there a simple algorithm for deciding whether $$p(z_1,\ldots,z_n) = 0 \implies q(z_1,\dots,z_n) = 0$$ where $p,q$ are straight line programs over $\mathbb{C}$ with integer coefficients? I'm fairly sure randomness does not provide simple algorithms if we go to arbitrary systems of equations and inequations, but I'm curious where the boundary is between easy and hard.