Implicit function theorem and its consequence

Question

Let $f:{\Bbb C}^{n+1}\to {\Bbb C}^n$ be a map defined by homogeneous polynomials. There is a point $p\in f^{-1}(0)$ and a neighbourhood $U$ of $p$ in $f^{-1}(0)$, such that $d(f)$ has rank $n$ at every point in $U-\{p\}$. Then, by the implicit function theorem $f^{-1}(0)$ is a graph in some neighbourhood of every point in $U-\{p\}$.

Can we conclude that, $f^{-1}(0)$ is also a graph in some neighbourhood of $p$ (even though $d(f)$ may have rank $< n$ at $p$)?

Answer 1

No. A counterexample is given by $n=1$, $f(x,y)=x^2-y^2$, and $p=(0,0)$.