I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have seen used the term "system of equations and inequalities" but to me inequality means "$\leq$" or "$\geq$" and not "$\neq$" so I would prefer not to use it. Is there some commonly accepted term?

I made this CW since I am not sure there is a correct answer.