Let $d \in \mathbb { N } ^ { * }$ and $f$ a polynomial function on $\mathbb { R } ^ { d }$. Suppose that the function $f$ is nonzero. Show that $f ^ { - 1 } ( \mathbb { R } \backslash \{ 0 \} )$ is dense in $\mathbb { R } ^ { d }$.
\textit{Hint: You may use the fact that a nonzero polynomial in one variable has only finitely many roots.}