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In algebraic geometry, it's called a quasi-projective algebraic set, which by definition is a Zariski open subset of a Zariski closed subset of projective space. (I'm assuming you only have finitely many $p_i$'s and finitely many $q_j$'s.) Since you're using non-homogeneous polynomials, you're starting in affine space, but that's simply projective space in the homogeneous variables $X_0,\ldots,X_n$ with the condition $X_0\ne0$, so it fits into your framework.