As in my comment denote by $P_n$ the subspace of $\newcommand{\bR}{\mathbb{R}}$ $\bR[x_1,x_2,x_3]$ consisting of polynomials of degree $\leq n$. Consider the map

$$ F: \bR^3\times P_n\times P_n\to\bR^2,\;\;(x,p,q)\mapsto (p(x),q(x).$$

If $n$ is sufficiently large then for any $x\in\bR^3$ the map

$$ P_n\times P_n\ni (p,q)\mapsto (p(x),q(x))\in\bR^2 $$

is a submersion. We deduce that the set

$$\Lambda=\{(x,p,q)\in\bR^3\times P_n\times P_n;\;\;p(x)=q(x)=0\} $$

is a submanifold of codimension $2$. We have a natural map

$$\pi:\Lambda\times P_n\times P_n\to P_n\times P_n,\;\;(x,p,q)\mapsto(p,q). $$

Sard's theorem shows that most $(p,q)\in P_n\times P_n$ is a regular value of $\pi$. The fibers of $\pi$ will generically have dimension $1$. Note that the fiber of $\pi$ over $(p,q)$ is the set $\{p=q=0\}$.

As in my comment denote by $P_n$ the subspace of $\newcommand{\bR}{\mathbb{R}}$ $\bR[x_1,x_2,x_3]$ consisting of polynomials of degree $\leq n$. Consider the map

$$ F: \bR^3\times P_n\times P_n\to\bR^2,\;\;(x,p,q)\mapsto (p(x),q(x)).$$

If $n$ is sufficiently large, then for any $x\in\bR^3$ the map

$$ P_n\times P_n\ni (p,q)\mapsto (p(x),q(x))\in\bR^2 $$

is a submersion. We deduce that the set

$$\Lambda=\{(x,p,q)\in\bR^3\times P_n\times P_n;\;\;p(x)=q(x)=0\} $$

is a submanifold of codimension $2$. We have a natural map

$$\pi:\Lambda\times P_n\times P_n\to P_n\times P_n,\;\;(x,p,q)\mapsto(p,q). $$

Sard's theorem shows that most $(p,q)\in P_n\times P_n$ is a regular value of $\pi$. The fibers of $\pi$ will generically have dimension $1$. Note that the fiber of $\pi$ over $(p,q)$ is the set $\{p=q=0\}$.