Suppose we have two polynomials $p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional curve. But how can I make this statement rigourous? Is there any theorem which gives you the Hausdorff dimension of such a zero set? Indeed, we will have to assume that $p$ and $q$ have no common factors.