Polynomial representation of intersection of polynomials

Question

How to minimally represent intersection of two degree $d$ polynomials intersecting at $d^2$ points as a single polynomial?

Answer 1

Since:$$a^2+b^2=0$$implies that both $a=0$ and $b=0$, the equality$$(y-p(x))^2+(y-q(x))^2=0$$implies both that $y=p(x)$ and $y=q(x)$—that is, it represents the intersection of the curves.

It has degree $2d$ (and there is a $y^2$ in there), and I'm not sure if it has minimal degree.

(If you want the union, I think this would work:)$$(y-p(x))(y-q(x))=0$$