# Can we find unions of the sets which solve multivariate polynomial equation systems and what are they called?

In algebraic geometry we have a concept called algebraic variety which is a multivariate polynomial equation system

$$\cases{p_1(x_1,x_2,\cdots,x_k)=0\\p_2(x_1,x_2,\cdots,x_k)=0\\\vdots\\p_n(x_1,x_2,\cdots,x_k)=0}$$

or more specifically the solution set to such an equation system.

From a geometry perspective this is rather powerful and expressive.

Already with one equation we can express many shapes, with varieties we can select cups of these sets.

But even more powerful would be if we could also express unions. does there exist any arithmetic or algebraic operation which allows us to take the union of a number of solution sets to multivariate polynomial equation systems?

To each variety $$X \subseteq k^n$$ (where $$k$$ is an algebraically closed field) we can attach a radical ideal $$I(X)$$ defined by
$$I(X) = \{f \in k[x_1,...,x_n] | \forall x \in X, f(x) = 0\}$$
If $$X$$ and $$Y$$ are algebraic varieties,
$$I(X \cup Y) = I(X) \cap I(Y)$$ since it must be functions that vanish on both sets.